Bott-ehern Secondary
Characteristic Objects andArithmetic Riemann-Roch Theorem-I -Lin Weng

Max-Planck-Institut fUr Mathematik
Gattfried-Claren-5traBe 26
53225 Bann
Germany

MPI / 94-5]

Bott-Chern Secondary Characteristic Objects
andArithInetic Riemann-Roch Theorem(Last Preliminary Version),tin Weng

Part I

Contents

1

Contents
Introduction

Part I. Bott-Chern Secondary Characteristic Objects
Chapter 1.1. Classieal Bott-Cbern Secondary Cbaracteristic Forrns
Cbapter 1.2. Relative Bott-Cbern Secondary Cb81acteristic Fonns Cor Smootb Morpbisms I:
Axioms
Chapter 1.3. Existence of Beat Kernela
Chapter 1.4. Mellin Transform
Chapter 1.5. Loeal Family Index Theorem
Chapter 1.6. Relative Bou-Chern Secondary Characteristic Fonns for Smooth Morphisms
11: Existence
Chapter 1.7. Relative Bott-Chern Secondary Characteristic Currents for Closed Immersions
I: Axioms.
Chapter 1.8. Relative Bott-Chern Secondary Charaeteristic Currents for Closed Immersions
11: Existence
Chapter 1.9. Temary Objecta And Deformation To The Normal Cone

Part 11. Arithmetic Riemann-Roch Theorem
Chapter 11.1. Grothendieck Rie~ann-Roch. Theorem
Chapter 11.2. Arithmetic Intersection Theory
Chapter 11.3. Arithmetic Charaeteri.stic Classes

2

Contents

Chapter 11.4. Arithmetic Riemann-Roch Theorem for Smooth Morphisms
Chapter 11.5. Arithmetic Riemann·Roch Theorem for Closed Immersions
Chapter 11.6. Arithmetic Riemann-Roch Theorem for L.C.I. Marphisms
Chapter 11.7. Graaamannian Graph Construction In Arithmetic Geometry
Chapter 11.8. Arithmetic K -Theory I: ADefinition Of Higher Arithmetic K -Graupe
RefereDcesIntroduction

INTRODUCTION
The main purpose of tbis book is to give b8Bic concepts, tecbniques, and results ofarithmetic geometry in the sense of Arakelov. In partieular, we will give tbe aritbmetie
Riem8lln~Roch theorem for Ioeal eomplete interseetion morphisms, but with one tecbnieal
eondition that the morphiam at infinite place is amootb.
In this introduction, we will illuatrate tbe theory witb one aimplest example: tbe situation in the eategory of eomplex projective manifolds.
We start with the following Riemann-Roeh theorem in tbe sense of Gorthendieek:
Let f : X - Y be a smooth morphism of complex manifold. Then for aoy veetor abeaf E 00 X, we may define tbe puah-out morphisffi of E in tbe sense of K -theory:f K (E) := Li(-1).i R' f. (E) . Theo, at tbe eohomology dass level, we have tbe followiog
GrothendieckooRiemann-Roch theorem with reapect to smooth morphisms:fCH( eh(E) td(TJ)) = ch(fK(E)),

where fCH is tbe natural pusb-out morphism of algebraic eydes, td ia tbe Todd eharacteristic
dass, and TJ ia tbe relative tangent aheaf of f.
On tbe other hand, let i: X c........ Z be a dosed immersion of eomplex manifolds. There
exiBts a natural exa.d sequenee:

For any vector sheaf E 00 X, the direct image i. Eisa eoherent sheaf on Z. Hy classicalsbeaf theory, there exists a vector aheaf resolution of i.E on Z:

Then tbe Grothendieck-Riemann-Roch theorem with respect to closed immersions says that we have the following equality at the level of eohomology classes, Le. io
CH(Z)Q,ch( i. E) i. (td(}/) -1 eh( E)).

=

Tbus by the fact that i. Tl= Ei (-1 YEj , we bavech(E.) = i.(td(}/)-l ch(E)).

Introduction

11

Put the above two situations together, we may have the Grothendieck-RiemannRoch theorem with respect to I.c.i. morphisms:
Let I : X -- Y be a l.c.i. morphism of complex manifold. Then the following diagram
is commutative:cb()td(r, )Ko(X)CH(X)Q.

IK 1

Ko(Y)

-

eh

1 ICH
CH(Y)Q'

With above, put the situation in a simplesi way, we may say thai that arithmetic
Riemann-Roch theorem should be a natural generalization of the above commutative diagram 80 that the above diagram is still commutative after we put the arithmetic notation
at the suitable place correspondingly, that is, the arithmetic Riemann-Roch theorem should
become the following commutative diagram
K~r(x)I~ 1
Ktr(y)
With this in mind, the first thing we need to do ia to give the fundamental concepts andresults in the above picture, such as arithmetic intersection theory, arithemtic characteristicclasses, etc .. In three fundamental papers [GS 90lt [GS 9Ib] and [GS 9Iclt Gillet and Soute
give the arithmetic intereection theory and arithmetic characteristic classes. Next let us
expose them in our situation here.
By a standard result, we know that under the natural Chern character, the algebraic
K-group is isomorphic to the Chow group for any regular variety. In particular, the algebraic
intersection can be introduced use the topological property of algebraic K-theory. Among
ethers, let us just mention that the divisors corresponds to tbe line sheave9, and a1l tbe
theory may be deduced from this very special situation, aB we have the splitting prineiple
for vector sheaves. More precisely, let r. be a line sheaf on X, then its associated algebraic
cyde may be defined by div(s) for a non-trivial rational section 6 of l.: This is a natural
correspondence at the level of (de Rham) cebomology dasses. In order to go further, let us
consider the situation at the level of differential forms. One then knows that we may put
hermitian metrics on r.. Choose ODe hermitian metric, say p. Then we have the first Chern
eharacteristie form Cl(.C,P). It ia well-known that Cl(.C,P) ia a closed differential form andits de Rham class is just the corresponding algebraic cyde dass, which is tben of course
independent on the choiee of the metric. On the other hand, the form itsell doe~ depend on
the meine. Furthermore, we have the following Poincare-Lelong equation:

where ~ is the Dirae symbol. With this, a natural idea is to ehoose (div(8), -logI81~) todefine Cl Ar( r., p). Thus, by the splitting principle, we may lead to define the ari thmetic
eyde in general. In practice, following [GS 90], we define an arithmetic cyde a.s a pair(Z, 9 z) such that Z is an algebraic cycle, and dd C 9 z + tlz is a smooth form. Usually, weeall gZ as a Green's current for Z. Also we have the arithmetic Chow group CHAr(X) by

Introduction

III

letting it he a quotient group generated by the above pairs modulo the relations given by(div(/), -00gI/1 2]), where 1 denotes tbe rational function of certaio irreducible subvarieties.
With this definition about arithmetic cycles, we may introduce the arithmetic intersection. The story about the algebraic cycle is rather obvious: We may choose the algebraic
intersection among them. But the situation for the currents is rather complicated: In general, Green's currents may.have very bad singularities. To control this, from the PoincareLelong equation, we may introduce Greeo's currente with logarithmic eingularities around
Z. It may be shown that for each classes modulo the exact currents for 8 and B, there
is a representative of Green's current with logarithmic singularities. With this control of
singularities, we may introduce the arithmetic intersection by a moving lemma at he levelof Kl-groUPS. For more details, see 11.2.
BBBically, we mayaiso use the splitting principle to introduce the the arithmetic chara.cteristic cl888e8. Since we also wnat that the arithmetic Chern character should otter anatural isomorphism between the arithmetic Chow group aod the arithmetic K-group. In
algebraic geometry, i.e. at tbe level of cobomology c1asses, we define tbe algebraic K-group
K(X) BB the quotient of the group generated by vector sheaves on X modulo the relations
E2 - EI - Es 0 if-+ EI -+ E2 -+ Es -+

=

°

°

is exact. Tbus a natural choice for Ktr(X) should be the quotient group generated bytripies (E, PjWJ), where (E, p) is a bermitian vector sheaf on X, aod w is a 80mmth form.
What should be the relations among them?

Ta find the relations, let us go back to the definition about the arithmetic cydes. We
know that cl(l, p) as a differ:ential form is dependent on the choice of the hermitian metricp. So tbe relations for tbe aritbmetic K -group should reßects tbis change, eince we now
consider the problems at tbe level of differential forms. Note that if in the above algebraicexact sequence, if we let EI = 0, and by the fact the ch is an isomorphism between K(X)and CH(X)QI we see tbat the above problem about the relations for tbe arithmetic K-groupbeeomes the following: Bow to measure the differenee eh( E, p) - eh(E, T)?
To answer the latest problem, we come to a famous theorem given by Bott and Cbern.
Around 1968, in the paper [BG 68], Bott and Cbern eould solve the follwoing partita!
differentiaJ equation
dcfchBc(E, p, T) = eh(E, p) - eh(E, T).
Just from this, we introduce the definition for the arithmetic K-group: The relations arewhere as above, we have the exact sequenee

and Pi are hermitian metrie on E; and the form ehBc(t'"p,) is tbe solution of the following
equation

iv

Introduction

thus we ean finish tha proce!s for introducing arithmetie intersection theory and arithmetie
characteristic classes aod hence give the notation in the two row! of the aritbmetic RiemannRoch theorem diagram above.
Next, we introduee the push-out morphism f~r and ftft. For this, we need to give afactorization for our morphisffi f. Say, f = goi with i : X - Z a regular dosed immersion
and 9 : Z - Y a smooth morphism. We first consider the push-out morphism for arithmetie
cycles: Tbe algebraic cydes are settled by a standard algebraie process. For Green's current,
we then rieed to be careful. For smooth morphisms, we just take the integration along thefibres. For dosed immersion, we may use the arithmetic intersection to do so, after choosing
certain arithmetic cydes for X, viewing aB a subvariety in Z. Thus finally definition comesfrom the moving lemma at K I-level.
In the following, we ooly discuss the situation for smooth morphisms to give adefinitionrfor. For this, we recall the following theory about the classical Bott-Chern secondary
characteristic forms above.

f:

Let E be a vector sheaf of rank r on a complex manifold X. Put a hermitian metric p on
E. Then there exists a unique cODonical connectioIi .6. associated with (E, p). Hence we haveita curvature .6. 2 • In this way, we may define the Chern characteristic form cb( E, p) by firstinentifing End(E) witb tbe metrix algabra Mr(C), then defining it locally as exp(-Ih.6. 2 ).
By"a local dis~ussion, we know that this offers us a global differential form on X. And fromthe Biancbi identity, it is closed. Also it is compatible with the puB-back by any morphism.
Furthermore, from tbe de Rham cohomology theory, the cohomology dass of ch(E, p) doee
not depend on the choice of p. So this claBS offer us a satisfactory answer in algebraic
geometry. On tbe other hand, the form ch(E, p) itself does depend 00 p. Such a kiod of
dependence is given by the dassical Bott-Chern secondary characteristic forms:
We first iotroduce axioms for the cla8sical Bott-Chern secondary characteristic
form, q'JBc(E.,p.), with respect to any power series tf,J, a short exact sequence of vector
sheaves
E. : 0 - Et - &2 - E3 - 0
and hermitian metrics Pi 00 Ej for j == 1,2,3: (it is worthy to mention that here it ia notnecessary to 8BSume that Pt aod P3 are induced from p2.)
Axiom 1. (Downstairs Rule) Let

be a short exact sequence of vector sheaves Qver a complex manifold X with hermitian
1,2,3. Then
metrics Pj on Ej for j=

dxcfxq'JBC(E., p.) = rjJ(E2, P2) - r/J(E I EB E3, PI EB P3)
holds in Ä(X) := EBAP,P(X)/ImB + 8.
Axiom 2. (Functorial Rule) For any morphism f : XI we have

X of complex manifolds,

vIntroduction

Axiom 3. (Uniquenes8 Rule) Ir (&., p.) is split, i.e. (l'2, P2) = (l'I EB l'3, PI EB P3),
then

4JBC(l'.,P.) = O.

.

Then we may state the Bott-Chern theorem a.s the following

Existence Theorem for Classical Bott·Chern Secondary CharacteriBtic Forms.1
Let

be a sbort exact sequence of vector sheaves on a complex manifold X with hermitian
metries Pj on l'j for j = 1, 2, 3. Then for any symmetrie power series 4J, there exists a
unique differential form <PBC (l'., p.) E Ä( X) such that q,BC (l'. t p.) satisfies the axioms
1, 2, and 3 above.
There are several metbods to prove this theorem. The basic idea is that we first forma family of differential forrns phi( DE, , D pa) so that, at t
0, it gi ves phi( &2, P2), while att
00, it becomes tjJ(l'l EB l'3, PI e {>3). then the integration of this form will offer a solution.
For more details, see Cbapter 1.1.

==

What should he tbe relation of tbe classical Bott-Cbern seeondary characteristic formsrand? First we discuss the situation when J is smootb. In algebraic geometry, weknow that IK(E)E(-l)tRA: I.l'. But, in general, RA: 1.& are coberent sheaves. So todefine 1ft, we n~ to note tbe fact that K( X) is gen~rated by J-acyclie vector sheaves.
So in the following, we will ooly consider the situation for such vector sbeaf on X. As aconsequence, we know that IK(E) = I.E, whicb is a vector sheaf on Y. Thus we havea natural element (/.E, I.p) in Ktr(y) for I-acyclic hermitian vector sheaf (E, p) on X.
On the other band, fixed a hermitian metric on the relative tangent vector sheaf Tl of I,
we bave, by Riemann-Roch theorem, another differential form 1.(ch(E,p)td(T1 ,P/)' By
Grothendieck·Riemann-Roch theorem, a.s de Rham cobomology claases, cb(/.E,/.p) is just1.(ch(E, p) td(1j ,PI)' But aB differential forma, tbey are not tbe same in general. Therefore,
as in the situation for the classical Bott-Chern secondary cbaracteristic forms, we may thinktbe difference cb(/.E,/.p) - 1.(cb(E, p) td(Tl,Pl) aB the change of E, p) with tbe action of(I, PI)' Tbis lead U8 to introduce tbe relative Bott-Chern secondary characteristic forII18
witb respect to smooth morphisms by the following axioms:

I:=

Axiom 1. (DoWDstaira Rule) Let I : X ~ Y be a smooth morpbism of Kähler
manifolds with a bermitian metric PIon tbe relative tangent aheaf TJ . Suppose (E,p)
is an /-acyclic hermitian vector sheaf on X, tben on Ä(Y), we have=dY4 ch ec(E,p,/,p/)I. (ch(E ,p) td(Tl' PI» - ch (/.E, f.p)·

Axiom 2. (Base Change RuIe) For auy Hat base change 9 : Y' - Y, we have

VI

Introduction
Here gl denotes the induced morphism of 9 with respeet to I, and similarly for Ig'
That iSt we have the following commutative diagram:

X XyY'..!.!..

I, 1

x
11

Y'

...!......

Y.

Here PI. is the natural metric induced by the fiat base change 9 from PI.
Axiom 3. (Uniqueness With Respect Ta Vector Sheaves) For any ahort exact
sequence of I-acyelic vector sheaves

with hermitian metrics Pi on Ej for i = 1,2,3, let

be the direct image of E. with associated hermitian metrics I.Pi on I.Ei for j
Thenchsc(E'l' P'l; /, PI) - chsc(EI , PI; I, PI) - eh sc (E3 , P3; I, PI)= I. (ehsc(E" p,) td(7j, PI)) - chsc(/.E., f.p,)·= 1,2,3.

Axiom 4. (Uniqueness With Respect To Morphisms) Let f : X - Y aod
9 : Y - W be two smooth morphisms of Kähler manifolds. Let (E,p) be an I-acyelie
hermitian vector sheaf on X such that I.E ia g-acyelic. Thenehec(E,p;g 0 I, P,o/) - ehsc(j.E, /.P; 9, pg) - g.(chBc(E,p; /, PI) td(Tg,pg))=(g 0 !).(ch(E, p) tdBC(!,g)).
Here tdBC(/, g) denotes the elassieal Bott-ehern seeondary characteristic form associated with the following abort exact sequence of the relative hermitian tangent sheaves:o-1j - TgOl - j.7g - O.
With this, similarly a.s the situation for the classical Bott-ehern secondary eharacteristic
forms, we mayaIso have the following

Existence Theorem oe Relative Bott Chern Secondary Characteristic Forms
With Respect To Smooth Morphisms.
Let j : X - Y be a sffiooth morphism of Kähler manifolds with a hermitian metric PI
on the relative tangent sheaf TI' Then for aoy I-acyelie hermitian vector sheaf (E, p),there exists a unique element ehBc(E, P, f, PI) in Ä(N) whieh satisfies the axioms above.
6

For the proof of this theorem, we may imitate the one for the elassical one. Hut now weare working in a infinite dimensional situation, Le. on CCCI(Y, f.E). Then we meet certain
problems. The most important one is that about the convergence. We know that the natural
L 2-connection will not offer us a (good) trace elass. Fortunately, by the work of Bismut

Introduction

Vllabout the local index theorem, we may choose the Bismut superconnection as an ~1ternative
object.
Basically, the Bismut superconnection is tbe limit of the Dirac operator along tbe fibreof f with s blowing up metric proces for the base, Le. the change of the metric is given by
PI + s-lpy. Thus, we may bave the associated heat kernel exp(-Bn. Here the parameter
may be introduced by rescaling the total metric with a factor t. Then Biamut's local index
theorem may be roughly stated as that when t - 0+, the supertrace of tbe restriction of
tbe above heat kernel to the diagonal gives us the differential form 1",(ch(E,p)td(T/,PI))'
On the other hand, a result of Berline aod Vergne asserts that when t - 00, the samedate offer UB tbe differential form cb (f.E, f.p). Tberefore, it is possible to give the above
existence theorem for the relative Bott-Chern secondary characteristic forms with respect tosmootb morpbisms. Surely, during tbis prOCe88, we need a kind of local double transgression
formula, for which we need to introduce the number operator form an intrinsie point of view.
Also in order to overcome the difficult about tbe coovergence for tbe trace c1888 in question,
we need to use a cone construction following Faltings {Fa 92].
The above process coming from tbe index theorem ie rather complicated, wbich ie
originally given by Bismut witb certain tecbnique from st~cb88tic integration. Here we use
tbe beat kernel approach following Berline, Getzler sod Vergne {BGS 92]. In tbis book, wedevote it witb several chapters: horn 1.2 to 1.6.
With the above work ahout Bott-Cben secondary characteristic objects, for any power
series R(z) E R((z]] , we may define tbe associated push.aut rnorphism for the arithmetic
K-group by 1:·Ar(E, p) for I-acyclic bermitian vector (E, p) with(/.E, f.p) + CbBC(l', P; I, PI) + 1.(0, ch(E, p) td(TI' PI) w(R(TI ))),
wbere R(E) ia a additive characteristic class defined by the power series R(z), aod w(Z, gz) :=1
d~gz + 6z. Then tbe arithmetic Riemann-Roch theorem may be stated aB follows
Arithmetic Riemann-Roch theorem For Smooth Morphism:({Fa 92]) Thereexists uoique power series R(z) Buch that for any smooth morphism ofregular arithmetic
varieties / : X ~ Y, the following diagram
K~r(x)

Ihr!
K$'r(y)commu tes ,where I KAr := IR,Ar
K.

CHAr(X)Q'

-

eh

! ItH
CHAr(Y)Q

Tbe proof of tbis tbeorem may be divided iota two steps. First, we consider the situation
for amooth morphisms. We follow {Fa 92]. Witb the same notation as above, for any power
series P, for any smooth morphism I : X - Y of regular arithmetic varieties over anarithmetic ring (A,E,Foo ), any f-acyclic bermitian vector ebeaf (E,p) on X, letErr(E,p;/,pJ;P):= ch~r(IK(E,p))- fCH(chAr(E,p)Td~r(/,PI))'
To prove the theorem, it is sufficient to show that there exiata a unique power series R(x)so thatviii

Introduction

for aoy J-acyclic hermitian vector sheaf (f, p). For this we need some intermediary results.
Proposition 1. Let f : X - Y be a amooth morphism of regular arithmetic varieties
with an Foo-invariant bermitian metric PIon the relative tangent vector aheaf of f.
Then for aoy abort exad sequence of /-acyclic hermitian vector sheaves

with 'Foo-invariant hermitian metrics Pi on Ei for i = 1,2,3, we haveErr(E1 , Pl; f, PI; P) + Err(Eat Pai /, PI i P)

= Err(E2, P2i f, PI; P).

In particular, Err(t, P; f, PI; P) does not depend on the metric p. Moreover, Err(E, P; /, PI; P)I
lies in the a-image of harmonie forrns.
Proposition 2. Let f : X -+ Y and 9 : Y - Z be two smootb morphisITIB of regulararithmetie varieties whicb have Foo-invariant hermitian metries PI, Pg and Pgol on therelative tangent vector sheaves of /, 9 and go/ respectively. Let (E, p) be an f -acyclic
hermitian vector sheaf on X such that /.E is g-acyclic. ThenErr(&, p;g 0 J, P,ol; P):= Err(/.E, I.p; g,P,j P) + g.(Err(E,pj /, PI; P)Td~r(g,p,)).

In particular, Err(E, P; /, PI; P) does not depend on the metric PI'

Remark. BecauBe of these two prop08itionB t we denote Err(&, P; f, PI; P) simply aB
Err(E; /; P).
Proposition 3. There ia a natural morphism
Err : K(XF) -H(Xa)/ p(CH(l,O)(Y))Q'

auch that Err(&; P) = Err(E; fj P).
Proposition 4. Let f : X - Y be a amooth morphism of regular arithmetic varieties
with an Foo-invariant hermitian metric PIon the relative tangent vector aheaf of f·
Then for any Bat base change 9 : Z - Y, we haveg. Err(E; /; P)= Err(gjE; I,; P).

Here we use tbe following diagramx
1/
Y.
Proposition 5. There ie a unique power series R(x) such that for any pi-bundle
p : X = Py(.1') -+ Y,

Err(t:; p; R) = O.

Introduction

IX

Finally, we consider Err for closed immersions. In this CaBe, we have to introduce a
new Err term. That is, let i : X <-+ Z be a closed immersion with the smooth structure
morphisms J : X - Y and 9 : Z - Y of regular arithmetic varieties, then we defineErr(E; i; P) := Err(E; J; P) - Err( i.E; g; P).
Hy Proposition 3, this deftnition makes sense, even through i.E is UBually only a coherent
sheaf.

Proposition 6. Let i : X C-....+ Z be a codimension.one regular closed immersion of reg·
. ular arithmetic varieties over an arithmetic variety Y with smooth structure morphisms
J : X - Y and 9 : Z - Y. Let (E, p) be an J-acyclic hermitian vector sheaf on X suchthat i.E is g·acyclic, thenErr(Ej ij P) O.

=

With this, note that by the deformation to the normal cone theory, at the level ofalgabraic K·theory, any dosed immersion may be deduced from codimension one closed
immersions and the zero section of projctive bundle, thus by Proposition 3 aod the fact thatarithmetic Riemann·Roch theorem holde for identity morphiBms, we see that it 8ufficientfor UB to prove the theorem for projective hundles. Hence we may UBe the induetion on therelative dimension to deduce the result.
In particular, in proving Proposition 6, we need to use a result about tbe deformation
theory for the relative Bott·Cbern secondary characteristic forms with respect to emoothmorphisms. Tbis finally leads UB to introduce an axiom for the ~called Bott-Cbern ternary
characteristic objects. Roughly speaking, the ternary object& measure the change of secoodary characteristic objects. We will not give more details for them, which will be foundin Chapter 1.9.
In order to discU88 tbe arithmetic Riemann-Roch theorem for I.c.i. morpbisms of arithmetic varieties, we need to have a similar discussion as above for clD8ed immersions.
First, we cOD!lider the difference given by the Grothendieck·Riemann·Roch theorem atthe level of differential forms. Put metrics on the exact sequence of normal sheaves. Also,
even through i.E is only a coherent sheaf, we may still put the metrics on Ej. Just as forsmooth morphisms, a natural question is how we can measure the change of ('1, g'1)' after
the action of the closed immersion i, at the level of differential forrns. Similarly, it is for thisreason that we introduce the relative Bott-Chern secondary characteristic currents withrespect to c10sed immersioD!I, ChBC(E, p; i, gd, whicb ia originally given by Bismut, Gillet
and Soule [BGS 91]. (H~re we have to UBe the language of current, as at least formally, thei.-image of a form may be written as the product of this form with the Dirac symbol 6x of
X in Z.) So we have the downstairs rule as follows:

But the situation is not 80 simple. We know that the metrics on E. are not unique and,in general, we cannot control them very weil. In order to introduce the relative Bott.Chern
secondary characteristic currents with resped to c10sed immersions, we need a technical
8B8umption on the metrics, which is nothing but the so--called Bismut condition (A), which

xIntroduction

gives certain compatibility condition for the aBSOciated metrics. We may also give tbe
axioms for the relative Bott-Chern secondary cbaraeteristic currents witb respect to c10sed
immersions. Sirnilarly, we have the existence theorem for them. For more details, see
Chapter 1.7 and Chapter 1.8.
On the other band, for any closed immersion i, we may deforrn it to the zero section ofprojective bundles. Therefbre, one may also bope that there is a ternary theory for cloeed
immmersi9ns. At the same time, tbe zero section is rather simple, it suggests us to use theK08zul complex to make tbe caleulation in a quite precise form. All of this will given in
Chapter 1.9, and was first given in [BGS 91].
Onee we have tbe relative Bott-Chern secondary characteristic objects and a special
Bott-Chern ternary characteristic objects, we may finaUy give tbe arithmetie Riemann-Rochtheorem for l.c.i. morphisms. But sinee at finite place, we eannot ooly use tbe deformation
to tbe normal cone, it is quite natural for UB to use the MacPherson's Grassmannian con·struetion to acbieve tbe final result. For more details, see Chapter 11.5, Cbapter 11.6 andCbapter 11.7.
At the end of this book, we propose adefinition for higher arithmetie K -groups by
Quillen's construction. This will finaHy offer UB aglobaI triangle relation between arithmetc
K-theory, algebraic K~theory and certain analytic homology tbeory, and beuce give theregulator morphisma with their more general meaning: The global morphisms wbich relate
tbe properties of tbe finite part and the properties of the infinite part for an aritbmetic
object.

Part I

Bott-Chern Secondary Characteristic Objects

Chapter 1.1.

1Chapter 1.1
Classical Bott-Chern Secondary Characteristic Forms
In this chapter, we recall basic concepts and results 8B8Ociated with the clBSBical BottChern secondary characteristic forms, which were first glven by Bott and Cbern around196§1BC 6!f
The cl8BBical Bott-Chern secondary characteriBtic form measures the change of various
characteristic forms with respect to different metrics. In this sense, we may think of thecl888ical Bott-ehern secondary characteristic form as a refined version of characteristic forms
in the theory ofChern-Weil, Le. we discU88 the ehern-Weil theory at the level of differential
forms.

§I.l.l. Characteristic Forms
I. 1. La. Connections, Curvatures, and the Canonical Connection
We start with the situation over areal manifold.
Let M he an n-dimensional real Coo manifold and E a Coo complex vector bundle ofrank r over M. AB usual, we letAP(M):= the complex vector space of COO complex p-forms over M;
AP(E):= the complex vector space of COO complex p-forms over M with values in E.

By definition, a connection V on E is a homomorphism
V: AO(E) -A1(E)over C such that, for f E AO(M), a E AO(E),

V(fa) = adf + fVa.
The connection V above may be realized locally as follows:

(1)

2

Classieal Bott-Chern=

Let S (81,' .. , S,.) be a loeal frame field of e over an open subset U C M, so that(i) Sj EAO(Elu) forj= 1, ... ,r;(ii) (81(;r), ... ,8,.(Z)) ia a basis of the fiber E l1 of E at z for eaeb z E U. Then for theeonneetion 'V,,.

'Vs j = LSA:wj,

(2)

A:=l

wJ

with
E A l(U). We eaB the matrix of I-forrns w := (w~) the connection form of'V with respect to the loeal frame field s. Obvioualy, if si is another loeal frame field
Qver U, and if w' is the eonnection form of 'V with respeet to s', then there ia a Coo
matrix-valued function a : U - GL(r; C) such thatS,

= S a,

(3)

and

(4)
We may extend tbe eonneetion V to aC-linear morphism

for p ~ 0 by settingfor 0 E AO(E), q, E AP(M).
We define the curvature of 'V to be(5)= rl.'V

1
aod let R. Tben R is AO(M)-linear. Hence, R is a 2-form on M with the value
in End (E). Using the matrix notation, the curvature form n of V with respeet to theframe field S is defined byThus 0= dw + w /\ w, aod there follows easily the Bianchi identity:
dn=11/\w-w/\w.

Furthermore, if we let 0' be the eurvature form of 'V with respect to the loeal frame s', then

(6)
Globally, let {U, V, ... } be an open covering of M with a loeal frame field Su on eaehGL(rj C) a COO map,

U. Ir U n V i= 0, then on U n V, su = 8 v 9vu, with 9vu : U n V -3Chapter 1.1.

called a transition function. Let wu be the connection form on U with respect to su,
then on U n V, we haveWu gvb w gvu + gvb dgvu.
(7)=

Conversely, given a system of g/(r; C)-valued I-forrns Wu on U satisfying (7), we may obtaina connection 'V on E having Wu as its connection form. Also, if Ou is the curvature form
of'V with respect to su, then we havef2u

=gv~ f2 v gVU(8)on U n v.
From now on, we assurne that M is a complex manifold and Eisa Coo complex vector
bundle of rank r over M. We let

AP"(M):= the vector space of C oo complex (p, q)-forrns over Mj
AP"(E):= the vector space of Coo complex (p,q)-forms over M with values in E.

Thus

AJ:(M) =E AP,9(M), AIc(E) = E AP"(E),p+,=.

p+,=irand there are natural operators

So

d= 8 +a.

Usually, we also introduce cJC as folIows:
1-dC := -.(8 - 8).
411'1

dd c

i-I
= -88
= -08
211'
211"i
i3

is a real operator. Let

A(M) := ~IcAir(M), Ä(M):= A(M)j(lm8 + 1mB).
Let V be a connection of E as above. We may write V = VI,a + VO,1 withHence

Claseical Bott~Chern

4

where

In the language of differential forms, we have the corresponding decompositions
w ;wl,o + WO,I,n = n~'o + nl,1 + nO,~.

Tbe complex vector bUDdle which admits a holomorpbic structure is characterised BS
folIows:
Propos~tion 1 (Newlander-Nirenberg) Let E be a (;CKJ complex vector bundle overa complex manifold M. Then E admits a holomorphic structure, i.e. the transition
fuoetions are holomorphic, if and ooly if there exists a eonnection \I = \10,1 + \11,0 suchthat(\lO,IYZ = \10,1 0 V"0,l = O.

Furthermore, we have VO,l = ä.
The proof of this standard result may be found in any textbook on differential geometry.
Let E be a Coo complex vector bundle over a (real or eomplex) manifold M. A hermitian metric p on E is a C co hermitian inner product on the fibers of Ej usually, we write
this as a pair (E, p). Given a loeal frame field Su = (SI, ... ,sr) of E on U, we let

Then Hu ia a positive definite hermitian matrix at each point of U. We aay that Su ia
a unitary field or an orthonormal frame Reid if H u ia the identity matrix. Under achange of local frame fields Su = Sv 9, where 9 = gUV is the transition funetion, we have
Hu = gt Hv g.
A conneetion V" of ( E, p) is called a hermitian connection if 'V preserves p (or makes
p parallel) in the following sense:(9)
for any {, Tl E A(E).
Proposition 2. Let (E, p) be a holomorphic hermitian vector bundle on a complex
manifold M. Then there exists a unique hermitian connection which preserves theholomorphic structure on E with respect to p. We eall this eonnection the canonicalconnection of (E, p). The curvature of this canonical connection ia of type (1,1).
From now on, we will assurne that every vector bundle ia a holomorphic vector bundle
aod that the connection for a hermitian vector bundle ia the canonical connection. \Ve

Chapter 1.1.

5

will also use the terminology of vector eheaves for locally free sheaves, and will make DO
difference when we use vector bundles and veetor sheaves in the sequel.

I.1.I.b. Characteristic Forms
We start with a fact from algebra: Let B eRbe a eubring, and let tP E B[[Ttl ... , Tr ]]he any symmetrie power series. For every k ~ 0; let ~[I:} he the degree k homogeneous
eomponent of tP. Then there exista a unique polynomial map
cIl[k] : Mr(C) - 4 Csuch that(1) eil [I:] is inva.riant under the eonjugation of GLn(C).
(2) cIl[k] (diag( aI, . . . ,ar)) = tPtk)( aI, . . . 1 ar ).
More generally, for aoy B-algebra A, we define

Furthermore, if I is a nilpotent subalgebra of A, tben we may also define

Thus, if (C, p) is a hermi tian vector sbeaf of rank r on a complex manifold M, and <P isas above, we definetP(C, p) := ~(-Rt,p) E A(M)aB folIows:First, identify End(C) with Mr(C) locally and then apply the eonstruction above toNote that by' the results in the previoUB subsection, especially the equality a.8, weknow that the above procedure for tP( E, p) ie well-defined, eince ~ ie invariant under the
conjugation. Moreover, we have the following

Proposition 1. W~tb the same notation 88 above,(1) t/J(C, p) is a dosed form on M, i.e. d tP(C, p) = O.
(2) For aoy morphism J: N - M, f·(t!J(E,p)) t/J(f·E,J·p).
(3) The de Rham cohomology dass of t/J(E,p) doe5 not depeod on the choice of p, hut
the form 4J(E,p) itself doe5 depend on p.

=

The proof of this prop08ition can be found in aoy standard textbook which contains thetheory of characteristic forms. (For e:xample, assertion I corne! from the Bianchi identity.)

6

Classical Bott-Chern

Here, instead of giving a proof, we would like to mention the following fact: Part 3 of this
proposition is the starting point to introduce the whole story in this book: We understandeverything at the level of differential forms rather than at the level of cohomology classes.
Roughly speaking, the refined version of the eharaeteristie form ljJ(f, p) for finite dimensional veetor sheaves and infinite dimensional vector sheaves, Le. the classical Bott-Chern
seeondary characteristie forms aad the relative Bott-Chern secondary eharacteristlc objects
for both smootb morphis~ aad dosed immersions are tbe central parts of our theory.

§I.l.2. Classical Bott-Chern Secondary Characteristic Forms
1.1.2.3. Axioms for Classical Bott·Chern Secondary Characteristic Forme
Flom above, we know that for aoy hermitian vector sheaf (E,p) on a eomplex manifold
M, we can define the associated characteristic form for any symmetrie power series t/J. We
also know that tbe de Rham cohomology dass of this form does not depend on the choiee ofthe metric, hut tbe form itself does depend on the metric. With tbis in mind, tbe first thing
we have to understand is how the characterietic forms change with respect to hermitian
metrics. Around 196W;~ott and Chern first solved this problem by considering the second
order partial differential equation:;,-

They found that in fact one ean solve this differential equation in A(M). Hence, they gave
the classical Bott-Chern secondary characteristic forms [BC611.
r
Now we introduce axioms for the classical Bott~Chern l!Iecondary characteristic

form, t/Jec(E" p.), with respect to any power series ljJ as in section La, a short exaet sequenceof vector sheaves

E. : 0 -- E1 - E'l -- E3 -0=and herrnitian metrics pj on Ej for j
1,2, 3: (it is worthy to mention that here it is notnecessary to assurne tbat Pl aud P3 are induced from P'l')
Axiom 1. (Downstairs Rule) Let

E.: 0 - Ei - E'l -

[3 -

0

be a sbort exact sequence of vector sheaves over a complex manifold M with hermitian
metrics P; on Cj for j = l, 2, 3. Tben, theie exists an element IjJBC (E., p.) E Ä( M), suchthat
holde.
Axiom 2. (Functorial Rule) For any morphism f : N we have

Al of complex manifolds ,7Chapter 1.1.

Axiom 3. (Uniqueness Rule) If (&., p.) is split, i.e. (&2, P2) = (&1 EB &3, PI EB Pa),
then
4JBC(&.,p.) = O.
Among these axioms, axiom 1 is essential. Furthermore, tPBC measures the change ofthe eharacteristic forrns w.ith respeet to metrics; Indeed, with the degenerate short exact
sequenee obtained by letting &3 = 0, axiom 1 gives the Bott-Chern equation stated at the
beginning of this subsection. In that case, we denote eh BC (&., p.) by eh BC (&1, PI , P2).

L1.2.b. Existence of Classical Bott-Chern Secondary Characteristic Forms
Next we prove the following
Existence Theorem for Classical Bott-Chern Secondary Characteristic Forms.•
Let

E.: 0 -

[1 -

[2 -

[3 -

0

be a short exact sequence of vector sheaves on a complex manifold M witb hermitian
metrics Pi on [J for j
1, 2, 3. Then for any symmetrie power series tP, there exists a
unique differential form tPBC ([ ., p.) E Ä( M) such that <PBC (E., p.) satisfies tbe axioms
1, 2, and 3 above.

=

Proof: There are several different ways to prove this theorem. Here we UBe the pl_deformation tecbnique. (pl means a projective line.) Otber methods will be explained in
tbe following ehapters.
Tbe basic idea of tbe pl-deformation technique ia tbat in order to eonstruet eertain
differential forrns, we introduce a new parameter in pi and then try to find differential forms
with parameter in pl. Finally, we show that our forms are nothing but the integration of
the fonns with parameter in pl over pi with respect to tbe current OoglzI2].
For our purpose here, we first cODBtruct an ~act sequenee Df. on M X pl, called apl-deformation of f. as folIows:
Let 8 be a section of the sheaf OPl (1), such that 8 vanishes at 00 and has tbe value 1
at O. Letwith the natural morphism Idel08 : EI - [1(1). Then we have tbe following exact sequence
on M X pl

For aoy point z E pl, let i, : M - M X pI be amorphism, defined by i,(x) = (x,z).
Then we have

8Classical Bott-Chern(1) i; DE2 ~ E2, if z -# 00.
(2) i~ DE2 ~ EI EB E3.
(3) i~DEl ~ EI'
Using a partition of uoity, we may choose a hermitian metric DP2 on DE2 in such a
way that the isomorphisms (1) aod (2) above become isometries. Hence, we introduce a
new parameter in pi.
Now letSioce
we know thatdMdMt/JBC(E., p.) =i~</J(DE2, Dp2) - i~</J(DE2, Dp2)'
Hence by the functorial properties of characteristic forrns in Prop. l.b.(2), we have axiom
1.

In order to check axiom 2, from the construction above, by Prop. l.b.(2) again, it isenougb to prove tbat, in Ä(M), tbe above conetruction does not depend on the choice of
the·metric DP2 on DE2 • Suppose there exists another choice Dp'2' Coneider the productM x pI x pI"with points (y, z, u). We have the following natural maps:-u

M x pI Ä M X pi X pi ~ M X pi ,
and

-13

M X pI ~ M X pi X pI ~ M X pi
witb
i~2(y, z) := (y, z, u),PI2(Y, z, u) := (Y, z),i;3(y, u) := (y, z, u), P13(Y, z, u) := (y, u).
Also let PI : M X pI - M be the projection to the first factor. Then on the bundle Pi2Dc2,
we may find a metric T such that

(1) (ia2t(pi2D&2, T) ~ (DE2, Dp2);(2) (i~)·(pi2D&2, T) ~ (DE2, Dp2);(3) (iÖ3t(pi2D&2' T) ~ pi(E2, P2)j
(4) (i~t(pi2D&2' T) ~ pi(E i ffi &3, PI $ P3)'
Hence,f OoglzI2}4J(Df2 , Dp2) - }plr [loglzj2}4J(Df2 , Dp;)

}pl

=}pl
f 00glz12] (</J(D&2, Dp2) - </J(D&2, Dp;»=

r 00glz1 (4J«ib 2t(pi2 D t'2' T» - <p«i~t(P~2DE2, T»).
2

}pl

]

Chapter 1.1.

9

Thus by Stokes' formula, we havef ~oglzI2) 4>(DE2 , DP2) - }plf Ooglzl2] tjJ(DE21 Dp~)}pl=}p,.IXPl
f[loglzI 2] [logluI2](dud~(tjJ(pi2DE21 r))).
a

Hut if we let = 8M + 8: + 8 u and 1) = 8M + 8: + 8u be the differentials on M x pI X pI,
then by the fact that characteristic forms are d closed, we havef}p1XPI~oglzI2] [loglul:Z](du~(Ijl(pi2DE21 r)))=}p1XPIf[loglzI 2] ~oglul:Z](d:~(Ijl(pi2DE2, r))).
Thus, using Stokes' formula again, we havef ~oglzl:Z] Ijl(DE2, DP2) - }plf ~oglzI2] 4J(DE2, Dp~)

}pl

=}pl
f [logluI (1jl«i~3r(pi2DE2, r)) - tjJ«i~r(pi2DE2, r))2)= f DogluFl](pi (,p(E:z, P2) - ,p(EI $ E3 , PI EB pa)))
}pl

=}pl
f [loglul 2)(pi (4J( E2, p:z) - 4>(EI EB E3 , PI EB pa»)=0.
Here, in the last step, we use the fact thatis a constant form with resprct to pi. So we have axiom 2.

The proof ofaxiom 3 is rather simple, since in the case that (E.,p.) ia split, we may
choose a metric Dp, wbich does not depend on z.
Finally, we have to prove the uniqueness. For this, let us start from the exact sequence
DE.. By axiom 1, we know that

Hence we havef ~oglzI2) dMxpl d;"xPI4>ac(DE., Dp.)

}pl

=}pt
f· 00glzl2] t/J(DE2, DtrJ) - f [loglzI 2] t/J(DE EB DEa , Dpl EB Dp3) .
}pt
I

10

Classical Bott-Chern

Hut the last term does not change if we change z to z-l, hence it ia zero. Therefore, wehave

1. ~oglzI2]
pt

= I

)pt

dMxPI dMxP' tPBc(DE., Dp.)~oglzI2] d,d~4JBC(De" Dp.)

= i04JBC(DE.,Dp.) - i~rPBC(DE.,Dp.)
=

1.

pi

loglzl2 4J(DE2 , Dp2)'

Here, in the last atep, we use that fact that (DE., Dp.) ia aplit at infinity. So finally, by
axioms 2, 3, we know that the c1assical Bott-Chern seeondary eharacteristie form aBBociatedwith (E., p.) ia tbe one eonstructed above.

1.1.2.e. Properties of Classical Bott-ehern Secondary Characteristic FOrInSIn this subsectioD, we discuss the classical Bott Chern seeondary eharacteristie forms
4

in more detail.

Theorem. (1) Let 4J 1, 4>2 be two symmetrie power series in C [[Tl, ... ,TnJ] and let
E. : 0 - EI - E2 -Ea - 0be a short exaet sequence of vector sheaves on a complex manifold with hermitian
metrics pj on Ej for j = 1,2,3. Then

(4)1 + rP2)BC(E.,p.) =rPIBC(E.,p.) + rP2BC(E.,p.);(4)14J2)BC(E., p.) =I/JI Bc(E., p.)4>2(E2,P2) + q,l (EI $ Ea , PI $ Pa)4>2 Bc(E., p.)=4JI Bc(E., p.)4>2(E I 6;) Ea, PI 6;) P3) + 4Jl(E2 , P2)1/J2BC(E., p.).

(2) Let rP be a symmetrie power series in n variables, and let q,a(TI , ..• , Tn1 ) and<Pa(Tn1 +1l .•• , Tn ) be the symmetrie power series defined by

er

Let

&.,1 : 0 - E1,; - E2,; - E3,. - 0
be a ahort exact sequenee of vector sheaves on a complex manifold with hermitian
metrics Pj,i on Ej,i and rk(E2il = ni I for j = 1,2,3, i = 1,2. ThenrPBC(&'1 $ &',2, P.,l 6;) P.,2)= L[q,a Bc(f,I, P.,t}<t'a(E2,21 p2,2) + tPa(Et,l $ E3,t, Pt,l EIl P3,t}<t'er BC(E.,2, P.,2)]'erChapter 1.1.

11(3) LetC. : 0 - [1 - [2 - [3 - 0
be a ahort exact sequence of vector aheaves on a eomplex manifold with hermitian
1,2,3, and rk( (2) "1. Let (.1', T) be a hermitian veetor
metrics Pj on [j for jsheaf with rk(.1')
n2, and tP a symmetrie power series in "1"2 variables. Define
the symmetrie pdwer series tP{J, <P{J in "1, "2 variables, respeetively, by

=

=

=

4J(T1 + U1 , ••• , Tnl + Ul, ... ,Tl + Un2 , · · · , Tn1 + Un ,)=:L 4J{J(T

1 , . •• ,Tnt )<p{J(U1 , • .. ,Un ,).

fJ

Tben

tPBc(E. 0.1', p. ® T) =I: tPß BC(l'., p·)<pfJ(.1', T).
fJ(4) (Nine Diagram) Let

0
0

0

0

0

0

!

!

!

[11

[12

[13

!

!

!

[21

[22

[23

!

!

1

E31

[32

[33

1

!

1

0

0

0

- - - - - - - - - -000

be a eommutative diagram of veetor sheaves with lines cj. and columns E.i exact.
Let Pii be hermitian metrics on Eji for i, j
1,2,3; and let tP be a symmetrie power
series in n = rk(E,,) variables. Then=

Proof. (1) By the facts that, for any hermitia.n vettor sheaf (E, p),(<PI + 4>,)(E,p) =<Pl(E,p) + <P2(E,p),(4> ItP2)( E, p) =t/J 1(E , p) 4>2 (E, p),
we know that both sides of tbe equalities satisfy tbe axioms for classical Bott-ehern secondary eharaeteristic forms. By uniqueness, we have the assertion.
(2) Defineq,Bc(1,2) := :L:[tPo Bc(E.,I' P.,t}<Po(E2 ,2, 1>2,2) + q,o(EI,1 $ E3,1' PI,I $ P3,t}<po BC(&'", P.,,)]·o12Classical Bott-Chern

With tbe same notation as in the construction of the previous subsection, we know thatin Ä(M). In fact, the above expression is invariant on replacing z by 1/z. Furthermore, by
the fact that•

4>(E2,1 e E2,2. P2,1 EB P2,2) ==L 4>a(E2, P2,x}l,l'a(E2,2. p2,2),
11

Cl

we know thatdlFtPBC(l, 2)=~:::)tPa(E2,1l P2, t}l,l'a (E2,2. P2,2) + f/l:a(EI,1 e E3 PI,I E9 P3,t}CPa(El,2 EB E

3,2, Pl,2,1,

e P3,2)]aTherefore, we bavetPBc(E.,l e E.,2, P.,l EB P.,2)== f [loglzj2] tP(DE2,1 e DE2,2. DP2,1 e DP2,2)}pl== f Ooglzl2] dcftPBc(l, 2)}pl= f d!f[loglzI2] tPBc(Dl, D2)}pl=i;tPBc(Dl, D2) - i~tPBC(Dl, D2)==tPBc(l,2) - 0 == tPBc(l, 2).
The results in 3 and 4 are direct consequences of the construction stated in the last
subsection. We leave the verification to the reader. In particular, we have the following

Corollary. Wi th tbe same notation as above, we have
1. Ir (.c, T) is a bermitian line sheaf,
4JBc(E,(lJ.c , p. ~ T) ==L 4Ji Bc(E., p.)cd.c, T)i.
i~O

Here t/Ji ia defined by the relation:t/J(T1 + T, ... ,T n + T) ==:L: 4>i(T

1 , ... , Tn)r.

i

Chapter 1.1.

13

§I.l.3. Superconnections
In the above discuBSion, the key point is that for a given hermitian (holomorphic) vector
bundle (E, p), there exists a unique canonical connection V t,P' From this, we obtain the8880ciated curvature form. By tbe local invariant property under conjugation, we construct
the characteristic forms, which are global differential forms on M. In order to go further , itis very important for us to-find what is the main ingredient in the definition of connections.
It has only recently been discovered by Quillen that a key point in the definition of connections is that the connection is a special odd endomorphism of A( M ,E) wbich satisfies theLeibniz rule. Thus we may generalize definitions from connections to these for euperconnections. This procese ie not difficult to understand formally, but it ie very powerful andmathematicians have spent several decades to discover it.

=

Let E E+ $ E- be a superbundle on a complex manifold M, Le. it is a Z2-graded
vector sbeaf. Let A(M, E) be the 8pace of E-valued differential forms on M. This space hasa natural Z-grading given by the degree of differential fonna. Usually, we will denote tbe
degree i component of a differential form 0 by O[i]. Also if let

A±(M,E):=

EA (M,E±) e EA + (M,ET),
2i 1

2i

i

i

we have tbe total Z2-grading on A(M, E) as follows:

A(M,E)=A+(M,E) $ A-(M,E).

By definition, a superconnection A on a. supervector sheaf E is an odd first-orderdifferential operator
which satisfies the Leibniz rule in the Z2-graded sense: If Q E A(M) aod 8 E A(M,E), thenA(a 1\ 9)=do 1\ 9 + (:.... 1)1 1 1\ A8.
00

Here 10'1 denotes tbe degree of Q.
As usual, we also define tbe curvature of a supercoonection A to be tbe operator A:1
00 A(M,E).

lt is not diflicult to"prove the following

Proposition 1. Let E be a supervector sheaf on a complex manifold M. Let A be asuperconnection on E. Tben(a) The operator A{1] is a ~ovariant derivative on E which preserves the sub-shelWe9
E+ and E-. Tbe operators A[I' for i :f 1 are given by tbe action of differential
forms W[ll E A(M, End(E» on A(M,E), where W[i] E A(M, End-(f» if i is even,14

Classical Bott-Chern
and W[i] E A(M, End+(&)) if i is odd. We caU A[l] the covariaDt derivative
component of A.
(b) The space of superconnections on & ia an affine space modelIed on tbe veetor space
A-(M, End(&)).
(c) Ir A, is a smooth one-parameter family of superconnectiona on &, dA 8 /ds lies in
A-(M, End(&) .•In or~er to uae auperconnections to deHne characteristic forms, we need tbe supertrace
mapping on the space A(M, End(&). By definition, the supertrace map

Tr, : A(M, End(&)) - A(M)ia defined locally as foilows:
On eacb fiber of End(&) on %, Tr"S" : End(&)S" -C ia defined byb) := Tr(a) - Tr(d).

a·Tr"S" ( c dSince the algebra A(M) ia aupercommutative, this map vanishes on the supercommutatorsand preserves the Z2-gradings.
Now let rP be apower series as in section La. For any superconnection A on a. supervector sheaf &, we know that 4J( -A2) ia in A+(M, End(&), since A 2 is in A+(M, End(&».
Thus Tr 8 [4J(-A 2 )] is an element in A+(M). We denote this element by Tr,[rP(&,A)], andcall this differential form the complex characteristic form of A with respect to thepower series q,. Here, we need to make the following remark. Classically, as we cODaider
the problem in the integral cohomology theory, for a covariant derivative 'V, we define thecharacteristic form by Tr,q,(- ~'V2). But when we consider the superconnection formalism
in the sense of index theorem, it is not quite natural, since now we may meet certain scalar
factors for appropriated degrees. Ta deal with this difference, we may usually introduce thefollowing operator on forrns (ar on currents):
Let M be a complex manifold. Define [2ri] as an operator on EBAP,P(M) such that[2ri] LW(P] := L) 2~i)Pw(P],Ppfor any element L p w(P] E EBpAP'P(M) with w[p] E AP,P(M).
We list the most important properties of such a differential form in the following
Proposition 2. Let A be a superconnection of a supervector sheaf & on a complex
manifold M. Then
1. The characteristic form Tr 8 [rP( E, A)] ia a closed differential form of even degree.
2. (Transgression Formula) Ir At is a differentiable one-parameter family of superconnections on f,

Chapter 1.1.

15

3. Ir AI and A2 are two superconnections on E, then the differential forms

Tr.[tP(E, A)), Tr.[tP(l', A)]
lie in the same de Rham cohomology dass.
For the proof of all tFlese properties, we only need to know that Tr. vanishes on the
supercommutators and for any Cl' E A( M I End( E)) I d(Tr 6 Cl' )Tr. ([A, Cl']). Later I we will seethat superconnections are very powerful.

=

16

I. Axioms

Chapter 1.2
Relative Bott-Chern Secondary Characteristic Forms
For Smooth Morphisms I: Axioms
From tbe previous chapter, we see ~hat the classical Bott-ehern secondary characteristic
form measures the change of cbaracteriatic forms with respect to the change ofmetrics. Whatshould be tbe corresponding objects in the relative case? To explain this in more detail, werecall the classical Grothendieck-Riemann·Rech theorem in algebraic geometry.
Let f : X - Y be a smooth merphism cf regular algebraic varieties. Theo for any
vecter sheaf E 00 X, we ffiay define the push-out morphism of E in the sense of K -theory:
fK(E) := L/-lY Ir f.(E). Then, at the cohomology class level, we have tbe following
Grothendieck-Riemann-Roch tb.eorem:
JCH( ch(E) td(TJ))= ch(fK(E»,where fCH ia the natural push-out morphiam of algebraic cycles, td is the Todd characteristic
cl ass, and TJ is the relative tangent sheaf of f.
Tbe first observation towards the relative Bott-ehern secondary characteristic forms
with respect to amooth morphisms comes from the following fact: At the level of differential
forms, the aimilar equalityfCH( ch(E) td(TJ))=ch(fK(E))

00 longer holds in general. More precisely, now we may assume that E ia f-acyclic, that is,
the higher direct images of E with respect to f vaniah, i.e. Ri f.E
0 for i > O. Then f.E
is a vector sheaf 00 Y. Furthermore, with respect to a hermitian metric PJ on the relative
tangent aheaf of I, if P is a hermitiao metric on E, we may naturally define the push-out
matric f.p on I.E. In this way, we may get two differential forms on Y:=

ch(f.E, f.p)
aod

J.(ch(E,p) td(TJ,pJ)).
As an easily corollary of the Grothendieck-Riemann-Roch theorem, at the level of de Rham
cohomology classes, Le. modulo the d-exact forms, we know that they are just the same.

Chapter 1.2.

17

However, if we eODsider the problem at tbe level of differential forms, we see that the differenee of these two differential forms ia usually not zero. So, as in the classieal (absolute)situation, we may ask how one can measure the difference of these two differential forms
on Y. Aa one may imagine, the aupposed concept about relative Bott-Chern secondary
characteristic forms are distributed to measure thia difference. In this sense, a key axiom of the relative Bott-Chern secondary characteristic form for (E, P; I, P/), denoted as
chsc(E, Pi I, PI)' should be tbe following equation:=

dycfycbac(E, P; I, P/)
1.( ch(E, p)td(7j, P/)) - cb(/.E, I.p)·
Another way to tbink ofthe above problem is that in tbe bermitian K·theory, the direct
image (/.E, I.p) is not a good definition for IK(E, p), when E ia I-acyclic, since by checking
certain concrete examples Buch as Riemann surfaces or projective spaces, the metric I.p is
not tbe right one. So we need to introduce a new metrie PI,RR on f.E, the Riemann-Rochmetric associated to (E,p;J,PI), 80 tbat it ia compatible with the refined Riemann-Rochtheorem at the level of differential forms, and it should also induce tbe Quillen metric on thecorresponding determinant line sheaf. Suppose such a metric does exist, we now may thinkof tbe reletive Bott-Chern secondary characteristic forms with respect to smooth forms asa measure for the change of ehern forms from the Riemann-Roch metric to tbe push-out
matric on f.E.
This chapter consists of two sections. In the first one, we introduce the axioms for the
relative Bott-.Chern secondary characteristic forms with respect to smooth morphisms. In
the second, we give the existence theorem for them.

§I.2.1. Axioms or Bott-ehern Secondary Characteristic Forms
With Respect To Smooth MorphislD8

1.2.1.a. DOWDstairs Rule
Let I : M -.. N be a smooth morphism of Kähler manifolds witb a hermitian metricPlan tbe relative tangent sheaf T/. Let (E, p) be an 1-acyclic vector sheaf on M. Then by

tbe Grothendieck·ßjem&lln-Roch theorem, we have the formula:

I. (ch(E) td(7j)) = ch (J.E).
This formula only holels at the level of cohomology classes. Usually, if we consider thesituation at the level of differential forms, the difference of the two differential forrnsf.(ch(E, p) td(T, PI» - eh J.(E,p)
is not O. It is natural to ask how we can measure such a difference. In general, motivated by
the Quillen metric on the determinant line sheaf, we also want to introduce a good metric18

I. Axioms

on !.E, when E is f-acyclic. In this sense, the relative Bott-Chern secondary characteristic
form measures the change of the Cbern characteristic fonns with respect to the new metrie
and the natural L 2_ metric on !.E.
From the remark above, we are led to coinsider the following axiom for tbe relative
Bott-Chern secondary characteristic form ChBC(E, P, /, PI) on N:.

Axiom 1. (Downstairs Rule) Let! : M - N be a smooth morphism of Kähler manifold~fwith a hermitian metric PIon the relative tangent sheaf 7j. Suppose (E, p) is an/ -acyclic hermi tian vector sheaf on M, then there exists an element ch BC (E, P, /, PI) E
Ä(N), such thatdNdNchBC(E, P, /, PI) =J.(ch(E,p) td(7j ,PI)) - eh (/.E,I.p).

1.2.1.b. Functorial Property
For the classical Bott-Chern secondary eharacteristic form, we have an axiom in tbe
sense of functors. Now we should have a similar axiom. Since the relative Bott-Chern secandaey eharacteristic form ia defined over N, it is enough for UB to consider ita behavior
under the base change. But for a most general base change, we know that even tbe cohomology groups da not behave very weil. Thus we 88Sume that our base change is a special one,say, a fiat morphism. We know that in this CaBe, everything works weil. (See Proposition
111.9.3 of (H 77]).

Axiom 2. (Base Change Rule) For auy Hat baBe change 9 : 1\1' - N, we have

Here gl denotes the induced morphism of 9 with respeet to J, aod similarly for /9'
That is, we have the following cammutative diagram:

.!!-

M

1f
..!.....

N.

Also here PI. is the natural metric induced by the fiat base change 9 from PI'

I.2.l.c. UniquenesB
The next axiom for the classical Bott-ehern secondary characteristic form is the uniqueness rule. This rule represents tbe initial condition in the present context. In fact, theChapter 1.2.

19classical Bott-Chern secondary characteristic form involves the triangle relation .in the category of hermitian sheaves. What is the situation when the original triangle degenerates?
Usually this kind of initiative condition will determine the solution uniquely among generalsolutions.
Classically, there is only one triangle relation, i.e., the one for hermitian veetor sbeaves.
But now, there are two triangle relations: one is for hermitian vector sheaves, while theother ia for smooth morphisms. Now we give the triangle relations in a gerenal situation;
later we will show that they are equivalent to BOrne other degenerate triangle relations.

Axiom 3. (Uniqueness With Respect Ta Vector Sheaves) For any short exact
sequence of I-acyclic vector sheaves

with hermitian metrics Pi on Ei for j = 1,2,3, let

1.&· :
be the direct image of E. with 8880ciated hermitian metrics I.Pi on I.Ei for j = 1,2,3.
Thenchac(E:;I, P'l; I, PI) - ChBC(E 1 , PI; I, PI) - ChBc(E3 , Pa;!, PI)= !.(cbBC(E., p.) td(T/,P/)) - ChBC(!.&,,!.P.).
Axiom 4. (Uniqueness With Respect Ta Morphisms) Let! : M - N and
Q be two smooth morphisms of Kähler manifolds. Let (&, p) be an f -acyclic
hermitian vector sheaf on M Buch that f.& ia g-acyclie. Then

9 :N -chBC(E,p;g 0 I, P,o/) - chBc(/.E, I.p; g, pg) - g.(ehBc(E, p; I, PI) td(~, P,))= (g 0 !).(ch(E, p) tdBC(!,g)).
Here tdBC(!, g) denotes the classical Bott-Chern secondary characteristic form associated with tbe following short exact sequence of tbe relative hermitian tangent sheaves:

From above, we know tbat the axioms here for the relative Bott-ehern secondary ehar·
acteristie form bave a similar pattern aB those for the classical Bott-ehern secondary cbaracteristic form. They are eomposed by tbe following aspeets: Downstairs Rule, Base Change
Rule, and Uniqueness Rule. Now taking Proposition 1.3 into the eonsideration, we knowthat one ean also define ternary ob'jects aod 80 on. In this way, we get a special kind of
hierarchy for characteristic forms.

20

I. Axioms§I.2.2 Existence Theorem
Far Relative BoU-Chern Secondary Characteristic FormSWith Respect Ta Smooth Marphisms
Note that we have set up the axioms for the relative Bott-Chern seeondary eharacteristie
forms with respect to emooth morphiems, it ie a natural question to ask whether they exist
or not. Ir they exiat, are tliey unique? For anawering these questions, we have the following

Existence Theorem Gf Relative Bott-ehern Secondary Characteristic Forms
With Respect To Smooth Morphisms.
Let 1 : M - N be a smooth morphism of Kähler manifolds with a hermitian metrie PI
on the relative tangent aheaf Tl' Then for any 1-acyclic hermitian veetor sheaf (t:, p) I
there exists a uniq ue element ehBe (t: , P, I, PI) in Ä( N), whieh satisfies the axioms in
tbe last Beetion.
Tbe proof of this existenee theorem has the same style as the one for the classieal Bottehern secondary characteristie form. That is, we first introduce a new one-dimensionalparameter for the connections. But in the case now, we do not have the canonical connection
and its natural generalization ofthe Chern characteristic form. Inatead, we have to introduce
the so-called Biamut supereoonectioo aod the heat kernels associated with the generalized
Laplaeian. As a eonsequence, technically, instead of integrating over pI! we will use theMellio transform to integrate our family over R~o. All of this will be done in the following
chapters.

Chapter 1.3.

21

Chapter 1.3
Existence Of Beat KerneIs
The beat kerneis were used in the proof of the Riemann-Roch theorem is slightly strange
but very powerful. Soon after Hirzebruch found his remarkable Riemann-Roch formula,
Grothendieck and Atiyah-Singer generalized the formula to more general eontexts. For
Grothendieck, in algebraic geometry, the Riemann-Roch theorem means the foUowing commutstive diagrarn:

Ko(M) ch(~Tr) CH(M)Q.
fK !

Ko(N)

~

! fCH
CH(N)QFor Atiyah·Singer, they noted that, by Bodge theory, the eohomology groupe for a vector
sheafare nothing but the kernela of certain elliptic operators. Henee they ean study tbe index
of an elliptic operator over more general manifolcls, say spin-manifolds. Technically, the first
proof.ofthe Hirzebruch-Riemann-Roch formula or the Atiyah-Singer index theorem ia in thestyle of the cobordiBm theory. Later, Grothendieek and Atiyah-Singer gave the proofs of
their theorems using algebraic K -theory. It was only after Patodi that mathematicians
realized that the use of heat kerneIs, foUowing Seeley and others, has greater ßexibility. It
is in this way that the local family index theorem ean be proved at the level of differential
forms. (The ordinary family index "theorem may be thought of es an integration form ofthia local version.)In this ehapter, foUowing sorne classical methods, we will prove that for a generalized
Lapla.cian, tbere exist beat kernels.
Partially because tbe heat kernel technique is hard to understand for most of algebraic
geometef8 and is the eore of our method here, we will devote it this ehapter with details.
The references here are [BGV 92], [Gi 84].

§I.3.1. Sobolev Spaces
In this sedion, we introduce a basic taol for the study of heat kernela. We first da
everything locaUy, i.e. we study the situation over Euclidean apaces snd then note that

Existence Of Heat Kerneis

22

since aur manifold ia compact, we may choose a finite open covering and extend the Iocal
discussion giobally by using a partition of unity.

1.3.1.a. The Situation For Euclidean Spaces

For aoy Euclidean space R m , there is a natural scalar product aod beoce ametrie: For
any two veetors x := (Xl,"" ZOrn) aod y := (Yl"'" Ym) in R m ,m< X,Y >:= LXjYj, aod lxI :=< x,x >1/'1.
j=1

Let 0'= (0'11 ... , O'm) E Z~o be a multi-index, we letm

j=1

Definerr.. . ·m.... .- ~Q
L.., j, ;"1·...... 1""'1'-'-41 1.,m

x Ol :=

n

01"

Xj] .

j=1

j=1

{)CJt
80..
~ := aXI . " 8x 'TI

D~:= (_l)ICJI~.

We also fix a volume form dx on R m , which comes from the usual Lebesgue meaaure on
R m , but with an additional normalizing faetor (21r)-m l '1. Tben we have

L'1(Rm ) := tbe space of aB complex valued functions f on a m such that, under thenatural L'1 inner product (h,g) Ja" h(x)g(x)dz,(f,f) ia finite;=

COO(Rm ) := the space of aU smooth complex valued functions on Rmj
Cö'(Rm ) := the space of all functions in COO(Rm ) with compact supports. This spaceis adense subset of L 2 (Rm ).
CI:(Rm ) := the space of continuouB functions on R m with continuous partial derivatives
up to the order k.
We consider a special dass of smooth complex valued functions on R m , the Schwartz
class S. Hy definition,

S := the space of the functions in Goo(Rm) such that for each pair of multi-index a, ß,
there is a constant CCJ,fJ such that

It ia a basic fact that Gr(Rm ) eS

IzO IY; fl :5 Cn,ß'c L2(Rm ). Thus, S is dense in L 2 (Rm).

The next key concept is that of the Fourier transform on S. Hy definition, for any f in
S, the Fourier transform j is defined byi(o:= fJa.. .

e-i<r,~> f(x)dz.

Chapter 1.3.

23

The fundamental dual properties for the Fourier transform are stated as folIows:

In particular, the Fourier transform definea a map from S to itself. Also there ia a fixed pointof the Fourier transform, namely, the Gaussian distribution exp(-~lxl:2). Moreover, theFourier transform is a bijection of S, since we have/(x) = j( -x).
There are two natural ring structures on S: one is defined by the ordinary pointwise
product, while the other is defined by the following convolution: for any /,g ES,(/.g)(x):=JfR .. I(x - y) g(y) dy.

The Fourier transform gives a homomorphism of these two rings. That is, we have=

Finally, since S is dense in L:2(Rm ) and (j, g)(I, g) for any I, 9 E S, we know that theFourier transform may be extended to a unitary map

This last result is UBually called the Plancherel theorem.
With the above construction, there is tbe Sobolev space H, (Rm ) which is a measure of
the L:2 derivatives. For auy tJ E R and fES, we let

Tben tbe Sobolev space H,(Rm ) is the completion of S witb respect to tbe norm 1'1,. Ina certam sense, the subscript tJ counts the number of L'J derivatives: Ir s = n ja a positive
integer I we define the norm 1.ln' by

Obviously, this iB an equivaient norm for Hn(Rm). With this interpretation in our mind, it
is not surprising tbat lolth L 2-derivatives are lost, when we extend D~ to B,. That iSt D~
defines a continuous map

Existence Of Heat Kernels

24(In fact, this comes from the following very simple estimation:

for certain constant C wbicb depends on the parameter (8, a) but not on I.) It also follows
from the Plancherel theorem tbat H I (Rm) is isomorphic to the L:2 space wi th the measure(1 + 1~12)'/2ete.
In the following, we displaya few relations among the various kinds of norms. First we
introduce a new norm on 8: For aoy Je E Z~o, and I E 8,

1/100,.1: := sUPrER" L ID~ft·
lalS;i:Obviously, tbe completion of 8 with respect to this norm ia a subset of C"(Rm ).

°Sobolev Lemma. Let Je ~ be a: positive integer and 8 areal number with 8 > k+ T'
Ir I EH!!, then f E cA:(Rm ) and

IJloo,.I: :5 CI/I,
for some constant C.

Proof. First for Je= 0, since f R ... (1 + 1e1 2 )-'ete is bounded forB> T and J ES, webave

1/(z)1 2 = II(z)1 2=

11. {e j(~)(1 + 1~12)'/2}{(1 + 1~12)'/:2}eteI2
ir

'{

R'"

:5 CI/I:·
Therefore,

1/100,0 :5 CI/I!!·
Since elements of B, are tbe limits of elements in S with respect to the I'I,-norm, and since
the uniform limit of continuous function is continuous, so the elements of B, are continuousand the same norm estimate extends to B,. [n general, if k > 0, we may use the following
statement to obtain similar assertions: If lai :5 k and 5 - Je > T' we haveTbis completes the praaf.
Next, we consider tbe relation between 1'1. and 1'11 for different 8 and t. Since far s > t,

we know that the identity map on S is actually an injection of B, c...... BI which is norm
non-increasing. -Furthermore, if we restrict the supports of our elements, this injection iscompact. That is, we have the following

25

Chapter 1.3.

Rellich Lemma. Let {In} C S be a sequence of functions with supports in a compact
subset [(. Ir there is a constant C such that Ilnl, :5 C for aH n, then for any s > t,there exists a subsequence {In •..} which converges in H t .

Proof. Let gE C(j(Rm ) be a functian such that g = 1 in a neighborhood of I<. Thengin = In aod

1/~(z)1 = 11. in (€){}(z - €)del
R"

:5 (

JR"

lin(€)lIo9(z - €)Id€~ I/nl,{1.

lo9(z - €)I:l(1 + l€l:l)-'d€]t

R":5 Ch(z),where h is a continuous fundion of z. Similar estimates hold for all derivatives of in(z). So
there is a 8ubsequenee In,. of In, such that in. converges uniformlyon ea.ch compact subset
of Rn. Thus for aoy r> 0,
I/nj - In.l~ ==1. I/nj - In,. 1 + lei) d€ + 1. t-.I/nj - In,. 1 + leI) de
~

-:2

IEISr

~

~

:2t

(1

~:2

lEI~r
:2

:5 C,.m8.XJEISrlfnj -In. 1 + 2C(1 + r)

~:l

(1

:2t

.

Others are trivial.
Another very useful estimation is the following

Lemma 1. Ir 8 > t > u, and E > 0, there ia a conatant C(E) such that

Illt :5 EI/I. + C(E)l/lu'
Proof. This inequality is a direet eonsequenee of the following

We end the discussion of Sobolev spaces with the following

Lemma 2. The L:2 pairing on Sextends to a perfect pairing of B, x H _I we may identify H_. with H;.

Proof. By the Cauehy·Schw:artz inequality, we know that for aoy 1,9 ES,

C. Thus

26

Existence Of Beat Kerneis

Thua, for any 1 E S, let 9/ E S be the function defined by

1(/,9)1

111, =sUPgES,g;tO-I-j-'
9-J

So we have the lemma.

I.3.1.b. Pseudo-Differential Operators On R m
By definition, a linear partial differential operator of order d is B polynomial
expression P
p(z, D)
Llal<d aa(z)D~, where the components of G a (%) are smooth.
The symbol of P, u P p, is denned by
__

=

=
=

uP

=p(z,~) = L: aa(z)~a,
lalSd

which is a polynomial of degree d in the dual variable €. It may be helpful to regard (x,O
as defining a point of the cotangent space l(Rm ). Tbe leading symbol ULP of P ia tbe
part of p( z,~) of the highest degree.
Now we go slight furtber. Note that, for aoy 1 ES,

Pf(z) =

f e,<z'(>p(x,€)f(€)cJe,
Ja.. .

so, we may use tbis formalisffi to define tbe action of pseudo-differential operators for awider dass of symbols p(x,~) tban polynomials. That is, we say that p(z,€) is a symbolof order d and write pE Sd if p(x,~) is smooth in (z,€) E R m x R m with a compact z
support and for aU (a,ß), there are constants Ca,fJ such that

For such a symbol, we define the associated operator P( %, D) aB the linear operator map
S - S given by

Pf(x):= [J

eir·ep(:c,€)f(~)cJe.

R-

Note that since for aoy dER aod f E C~(Rm), l(x)(l + 1€1 2)d/2 E 3 d , we see thatthe order of a symbol needs not .be an integer. Usually we refer to such an operator aB a
pseudo-difFerential operator, or shortly a \liDO. Ir for aB d, f E Sd, we denote this by
pE S-CQ and say that p is infinitely smoothing.

Chapter 1.3.

27

We have the following

Lemma 1. For a p E Sd, there ia a constant C such that for I E S IP II,-d :5 CI/I,.
It follows that P may be extended to a continuous map P : H, - B,_d for all s.
Proof. For our purpose, letq((,€):=

f e-i<z'(>p(z,€)dz
JR'"

aod define

K(C€) := q( - €,€)(l + I{I)-'(l + I(I),-d.
Theo, by the Cauchy-Schwartz inequality, we haveI(P I, g)1 =1

L..

:5{ f

JR"

x

K(, {)j({)(l + I€I)' g()(l + 1(1)d-'d(d{1IK(,e)I1i(e)1 2 (1 + 1€1)2'd(de}l/2{L..

IK(,e)llg()I2(1 + 1(1)2d-2'd(d€}1/2.

Now the lemma is a consequence of the following two easy estimationa

Next we discuss the sffioothiog approximation for a ~DO. We first introduce an equivalence relation on the symbols by defining p -- q if p-q E 8- 00 • Also for aoy giveo symbols
P and Pi E Sd; , we say that P is approximated by 2:~1 Pi, and denote it as
00

P- L:Pi';=1

if di -. -00 and for aoy d, there is an integer k(d) such that k 2: k(d) implies thatP-

2:;=1 Pi E 3 d •For the application to Riemannian manifolds, in the following discussion, we will needto restrict the domain and the range of our operators. Let U be an open subset of R m witha compact cl08ure. Let 'l'd(U) be the space of these operators P for which the associatedsymbol p(z,e) E 5 d haB support in Uj thus P may be thought of as P : CQ(U) - C(f(U).
We also let'i(U) :=

U'1'd(U), Woo(U) := n9"d(U),ddMore generally, we may consider the matrix valued symbol, which lead to little additional difficulty.

28

Existence Of Heat Kerneis
Lemma 2. Let r( z ~ y) be a matrix val ued symbol which is smooth in (z ~ y) and has
compact x support inside U. Suppose that for each multi-index (er, ß, "y) there existB a
constant Ca,ß;r such that
I

I

I

,

We have(a) If f is vector valued with compact support in U ,Ris defined by

Rj(z):=r

Ja - xa "ei<Z:-II'(>r(zl~, y)j(y)dycJe.

Then the operator R is in Wd( U) and(1

R(z,~) ......

{L deD: r/Q!}Iz:=~·a(b) Ir d < -m - k andcethen K isin z, y and Rj(z) = Ja.. K(z, y)j(y)dy.
(c) Ir the z-support of r ia disjoint from the y-support of r. Then R is infinitely
smoothing and is repreaented by a smooth kernel function K(x, y).

Proof. (a) First we may assume that the support of r in y is also compact. This may
be done by multiplying r by a cut-off function in y with a compact support and witb value
1 on U. Now let q( z, ~ ,() be the Fourier transform of r in Y then by an easy estimation
we know that
I

Thus , if we letthen

Rj(z) =

r ei<Z:'(>p(x,()i«)d(Ja..

is a wDO.
But by the Taylor expression of q on the middle variable, we haveI

Chapter 1.3.

29

Note that since 91: decays to arbitrarily high order in (~, () and gives a symbol in Sd-I: after
integration, we haveHence we have (a).
(b) Hy our condition, we know tbat K is well-defined. Now the result is a consequenceof Fubini's theorem.
(c) In tbis case, we can not deHne K as in (b), since the integration here does notconverge. But tbere is Motber definition for K. In fact, from our 885umption, we know tbat
on tbe support of T, Iz - Yl ;::: , > O. Define tbe Laplacian Äe := Lv
Thus formallyDi",

So we may deHne

for sufficiently large k. Now the assertion is a consequence of the fact that Ä~r decays toarbitrarily bigh order inThis completes the proof.

e.

From the proof above, and becall8e of the presence of the terms Iz - YI- 21 , we know
that, in general, K(z, y) becomes singular when z
y. But if K(z, y) is a smooth matrixvalued function with a compact z-support, we may define an operator P(K) E W-00 as
follows: For any I with a compact support in U,=

P(K)(f)(z):= IJa""

K(z, y)/(y)dy.

Thus P(K) defines a continuous operator P(K) : H. --+ H t for aoy s, t. Let IPI." denotethe operator norm, theo, for any fES,

IP fit :5 IPI•.,I/I.
and if k E Z2: 0 ,IKloo,1: :5 C(k)IPI-k,k'

30

Existence Of Reat Kerneis
Finally, we give a result on the smoothing approximation for a wDG.
Lemma 3. Let Pj E Sdi(U) with dj > dj+ll dj - -00. Then tbere exists a symbolpE Sdo such that P -- Lj Pi' Moreover, P is unique modulo S-oo.
Proof. We only need to prove the existence. For thia, we introduce a cut-off function<p: 0 :5 tP :5 1 , and t/J(€) ~ 0 (resp. 1) for I€I :5 1 (resp. 2: 2.) Thus we may cut away tbe
support near € = O. Let
P(Z,€) := ~ <P(ti€)Pi(z,€),j

Now note that since Pj - t/J(tj€)Pi E S-oo, by using a diagonalization argument, we havethe assertion.

I.3.l.c. Situation For Manifolds
In this subsection, we extend the above discusaion globally by using a partition of unity.
Let M be a smooth compact rn-dimensional Riemannian manifold witbout boundary.
Let dlJ denote tbe Riemannian meaaure on M. Let C oo (M) be tbe space of smooth functions
on M.
A linear operator P : CCO(M) - CCO(M) is called a "DO of order d if for every open
chart U on M aad every t/J,'P E Cr(U), t/JP'P E Wd(U), denoted by PE .d(M). Let~(M) =U"d(M), w-co(M):= n'Pd(M).
dd

In any coordinate system, we define cr(P) as one for t/JP4> with 4> = 1 near the point inquestion; this is unique modulo S-co. In the same way, we may define the leading term. It
is 0 bvious that tbe leading symbol is invariantly defined on tbe cotangent bundle T- (M).
We define L'J(M) using the natural L'J inner procluct, that is , L'J(M) is the completionof CCO(M) with respect to tbe L'l norm. For.P : CCO(M) - CCO(M), we let p. be definedby (P /, g) = (1, p. g), if such a p. exists.
Now we use a partition ofunity to define the Sobolev space H.,(M): Take a cover of M
by a finite number of coordinate charts Ui ; for each i there is a diffeomorphism h i : Gi - Ui ,where Gi ia an open subset of R m with compact closure. For / E Cr(U,), define

Ifl~i) := Ih; 11,·Let {<p.} be a partition of unity associated to this covering. Then define

111, := 2: It/J./I~i) .

•Obviously, 11. is a well-defined norm. In this way, we have the Sobolev space for M. Since
M is a compact manifold, it is not difficult to generalize a11 the results in the previous
subsection to the similar results for M.

Chapter 1.3.

31

§I.3.2. Elliptic Operators And Fredholm Operators
1.3.2.a. Elliptic Operators
Let M be a compact rn-dimensional Riemannian manifold. Take a finite covering of
M by coordinate charts {U,}. For simplicity, we may think of U = U, as an open subset of

Rm .

•

Let pE Sd(U) be a square matrix and U 1 an o·pen subset with Ü1 C U. We say tbat p is
elliptic on U1 if there exists an open subset U2 with Ü1 C U2 C Ü2 C U and if there exists
q E S-d such that pq - I E S-oo and qp- I ES-co over U2 . (Recall that if rES-co then forevery f/J E
rf/J E 8- 00 .) For example, for any 4J(z) E
the symbol 4J(x)(l + leI 2)d/2 I
is an elliptic symbol of order d wbenever <p(x) 1:- O. Globally on M, we say that P is elliptic
iffor any <P,'P E Cö'(U), <pPtp is elliptic whenever <P'P(z)::F O.

er,

er,

Tbe basic properties of elliptic operators are contained in the following
Lemma. Let P E ~ d be elliptic. Then
(1) There exists Q E ~-d such that

PQ - I E ",,-00'

QP - I E 'V-oo.

(2) P is hyperelliptic. Tbat is, I E H, and PI ia smootb implies / ia amooth.
(3) (Gärding's Inequality) Tbere exists a constant G sucb that for any f E Gü(M),1/141 :5 G(I/lo + IP110)'co-

Proof. Hy using a partition of unity associated with a finite covering of M byordinate charts, we know that it ia enough to prove the statements locally. Let U bean open subset of R m. First note that if P E 'V d, Q E q, e, then PQ E Wd+e andu(PQ) ~ Lo dePD;q/o:!. Thus by tbe recursion, we may letq" := -qLd'tp D't9i/ Q !

101+;=.I:,j<1'
and qo = q. From tbis, we know tbat if Q has the symbol qo + ql + ..., tben we have theconclusion in (1).
(2) is trivial.
Finally for (3) we cboose a cut-off function 4J E

er witb value Ion Ul. Then if J E er,

IJI4 = l4J/ld :5 1<p(I - QP)Jld + It/lQPlld.
Since f/J(I - QP) ja an infinite smoothing operator, 14J(I - QP)/! ~ Cl/lo. Now (3) comesfrom tbe fact that 4JQ ja a bounded map from L 2 to H d.

Remark. From thiB lemma, we know that for d> 0, one may define Hd by uBing thenorm 1/10 + IP/lo and define H-d by tbe dual of Hd.

Existence Of Heat Kerneis

32

I.3.2.b. Fredholm Operators
From above, we know that an elliptic -PDO is invertible modulo compact operators.
Takeing this fact out, we introduce the concept for Fredholm operators.
Let 1t be a Hilbert space and End(1t) the space of all bounded linear maps 1t With tbe natural norm •

1i.

ITzl
ITI =: sUPrE'H-{O} [;I'End(1i) is a Banach space. Let GL(1i) be the subset of End(1i) consisting of maps T which
are bijections. Hy the inverse boundedncsa theorem, we know that GL(1t) ia an open aubset
of End(1-l) aod is a topological group.
Hy definition, T E End(1i) is compact if {x n } is a bounded sequence, i.e. IX n I :5 C for
any n, there is a su bsequence {zn.} SO that Tz n • - Y for same y E 1-l. Let Cpt(1l) denotethe set of all compact mapa. An easy statement is that Cpt(1-l) is a closed 2-aided ideal of
End(?-l).
Let 1i1 and 1t2 be two Hilbert spaces, theo Hom(1iI,1i 2 ) is the Danach space of aU
bounded linear map from 1t1 to 1i2 with the operator norm. For any T E Hom(1l I ,1l2), let

N(T) := {e E 111 : T(e)=O} and R(E):= {f E 1i1,: f =T(e) for some e E 1l1}'

If ..L denotes the operation of taking orthogonal complement, then R(T).l = N(r). For
any T, N(T) ia closed.
Let Fred(1iI,1i2) be defined by the element T E Hom(1i I ,1l2) so that there is S E
Hom(1l l , 11. 2) such that ST - I E Cpt(1-l 1) T S - I E Cpt(1-l 2). An element in this space ia
called a Fredholm operator.
I

Lemma 1. Tbe element T E Hom(1i 1, 1l 2 ) is a Fredbolm operator if and only if T ia
such that R(T), R(r) are closed, anddimN(T) < 00, dimN(T') < 00.

Thus, if T ia Fredbolm, sa is T*; if Tl, T 2 are Fredholm, so ia T 2 0 Tl.

Proof. Let TE Hom('H I ,'H2) and let zn E N(T) be auch that Ixnl

= 1. Thenwith C being compact. So we have a convergent subsequence. Hence the unit sphere in
N(T) lS compact. AB a consequence, N(T) is finite dimensional. Next, we prove that R(T)is closed. For thiB, let Yn = TZ n and Yn - y. Without lass of generality, we mayaiso
assurne that X n E N(T).l. There are two possibilities.
.
(1) {x n } is a bounded sequence. Then by the fact thatZn

=StYn + (1- StT)x n , SIYn - StY,

Chapter 1.3.
we may further assume that X n -33

x. Hence,

which is in the range of T .

.

(2) lxnl- 00. For this CaBe, let x~ := xn/lxnl. ThenTx~ -Yn/lxnl- O.

Using the same argument B8 above, we may B8Sume that x~ - x with Tx = 0, lxi = 1 andx E N(T)1.. This is impossible.
Tbll8, R(T) is closed. Hy duality, we easily have the same assertion for r.
Conversely, suppose N(T) and N(r) are finite dimensional and R{T) is closed. Webave tbe decomp08ition

Now tbe assertion ia a coosequence of the fact that
T: N(T)1. - R(T)is a bijection.
For any Fredholm operator T, we define tbe index of T by
Iod (T) := dimN(T) - dimN(T').
Obviously, Ind (T) = -Iod ('r) and Ind (T2 0 Tl) = Ind (T2 ) + Ind (Tl)' In particular, forelliptic lPDOes, we bave tbe following

Lemma 2. Let P : COO(M) - COO(M) be an elliptic wDO of order d over a compact
manifold witbout boundary. Then(a) Tbe dimension of N(P) is finite.
(b) P : H, (M) -+ H,_d(M) is a Fredholm operator and Ind (P) does not depend on
8.

(c) Ind (P) onIy depends on the homotopy type of t1'LP.

Proof. Ooly (b) oeeds to be proved. Note that since there exists an elliptic "DO Q oforder -d such tbat Q P - I and PQ - I are infinitely sffiootbing operators, P : H, - H d-,and Q : Hd-, -+ B, are cootinuouB. Thus QP - land PQ - I are continuous, hence they
are compact.

34

Existence Of Heat Kerneis
1.3.2.c. Examples

Let E be a graded vector bundle {Ej liez such tbat Ej f. 0 for only finite numberof Fs. Let P = {Pj} be a collection of operators such that Pj is lfh order \lJDO with
Pj : CCO(Cj) - CCO(Ej+t}. We say that (E, P) is a complex if

Pj + 1 0 Pj

=0 and lJ'LP +j

1 0 lJ'LPj=O.

A complex (E, P) is called an eUiptic complex iffor € # o. As usual, we define the cohomology by

Ir tbese cohomology groupe are all finite dimensional, we define
Ind(P):= I)-1)idimHj(V, P)
jas the Euler characteristic of the complex (E, P).
Cboose a fixed bermitian inner product on the fibers of C. We may define L 2 (E). Let p.
be tbe adjoint of P with respect to this structure. Ir (E, P) ia elliptic, we have a self~adjoint
Laplaciantl j := (p. P)j :=Pj + Pj-1Pj·_ 1.

PlFor Pj := lJ'LPj, we have O"L{tl j ) = pjPj + Pi-1Pj-l' Hy definition, we know that (E,P) isan elliptic cfh order partial differential complex if and only if tlj is an elliptic operator oforder 2d for all j. The most important result in this direction is the following
Hodge DecompositioD Theorem. Let (C, P) be an elliptic lfh order \lfDO complex.
Tben(1) L 2 (Ej )N(tl j ) e R{Pj-d e R(pr) aB an orthogonal direet sumo(2) N(il j ) is a finite dimensional vector space and there ia a natural i80morphism ofHi(E,P) ~ N(tlj). The elements in N(tli) are smooth sections of Ei'=

Proof. Think of a j aB an operator a j : Hu(Ej ) know that
(1) N (aj) consists of smooth Beetions of Ej;(2) N(aJ) is finite dimensional;(3) R(aj) ia c1osed;

L2 (Ej ). Since this is elliptic, we(4) L2 (Ej) = N(tl j ) $ R{tlj ).
Sißce PjPi - t=0, R(Pi-d and R(pr) are orthogonal. But for f E N(ßj),

35

Chapter 1.3.

=

so N(Aj) N(Pj )nN(Pl_ d. Thus R(ßj) contains the span of R(Pj-d and R(Pl). Hence
we have (1).
The natural ind usion of N (ß j) into N (Pj) defines a map N (D.j) -H j (&, P). If

1 E CfXJ(&j) and Pj 1 = 0, there is a 10 E N(ßj) such that 1 = 10 + ßh with ft E COO(Ej).
Note that since Pj ßj h = 0, we haveo = (Pj / 1, PjPl Pj !1 + Pj Pj - 1Pj-- 1/1) = (Pj- Pj!t, Pl Pj!d·Thus Pl Pjft = 0, so ßjft = Pj - 1Pj-- 1h E R(Pj-d. Therefore the mapping N(ßj) Bj (E, P) is Burjective. This completes the proof.

Remark. As a more concrete example, consider the de Rham complex: Here Ej :=COO(l\j (7- M)), Pj := d the total derivative. Obviously, if for any ~ E T- M, let ext(~) :" j (T- M) - 1\;+1 (T- M) be the exteriour multiplication, 80 that ext(e) w :=1\ w, then0'L (d)( z,~)i ext(~). Now we know that the de Rham complex is an elliptic complex.
Thus if we denote the adjoint of d as 6, .6. 6d + d6 (d + 6)2 with O'L.6.
lel 2. Hence, by
Hodge decomposition theorem, we know that

e

=

=

=

=

is tbe space of harmonie p-forms. AB a consequence, we know that Ind (d) = X(M), theEular-Poineare characteristic of M. Actually, if we let
I\P(T- M) - I\m-"(T- M) be the
Hodge star operator, defined by w 1\ *W := (w,w) diJ, then by Stokes' theorem,*:•• = (-l)p(m-p ),6 = (_l)mp +m+l. d •.

Tberefore, we furtber have the Poincare duality: * : N(.6. p ) ~ N(ll m _ p ).

§I.3.3. Existence oe Heat Kernels: I
In this section, we will prove tbe existence of heat kerneis associated with a self·adjoint
elliptic ~DO.
We begin with the spectral theory. Let TE Cpt(1i) be a self-adjoint c.ompact operator
on the Hilbert space 1(.. Let
Spec(T) := {A E C : T - AI t1. GL(1l)}.
It is an easy exercise to show that Spec(T) is a dosed subset of C which ia contained in
the dosed interval [-ITI,ITI]. For aoy A E [-ITL ITIL let E(A) := {z E 1l : Tz
Az} bethe eigen.space. If A #: 0, Tz
Az irnplies that the unit disk in E(A) is compact. Hence
E(A) ia finite dimeDBional. On the other hand, dirn {E( -ITI) e E(ITI)} is not O. Indeed,suppose T #:- 0, we choose Zn 80 that IZnl = 1 snd ITznl--+ ITI. We msy 8B8ume {zn} iB the
Bubsequence for. which TZ n - y. Then.

=

IT2 z n -JT12zn12== IT 2z n j2 + IITI2znl2 - 2ITI 2(T2z n , Zn) :5 21TI 4 - 21TI 21Tzn l2 - O.

Existence Of Heat KerneIs

36
Therefore, %n -%:= Ty/ITI 2 "I O. Hence IT 2 x - ITI 2 z1 = 0, i.e.
(T - ITI)(T + jT1)z = O.

From thie relation, we easily have the assertion.
In this way, we have-a non-trivial decomposition '}{ = E( -ITl) EIl E( IT1) EIl'H 1 . Let
Tl := Tb", I , so that ITII :5 [TI. Then we may decompose 1i1 in the same manner. Continuing
thie process inductively, we may find Tn so that jTnl :5 ITn- 1 1.

Claim. ITn 1 -O.

In fact, for any ", choose %n so that T%n = ±ITn I%n and I%n I = 1. Thus lXi - Xj I = ..[i.
On the other.hand, since T ie compact, we may choose a convergent subsequence lTn IX n - y.
From here, we know ITn I- O. That ie, we have the following

Lemma 1. Let TE Cpt('H) be self-adjoint. Then'H =EB E(>'I:) EIl E(O).
Je

Hente we can find a complete orthonormal system for '}{ coneieting of eigenvectore ofT.

AB an application, we have the following
Lemma 2. Let P : CCO(l') - COO(l') be an elliptic self-adjoint wOO of order d > O.
Then(80) There existB a complete orthonormal basis {tPn} for L 2 (E) so that PtPn = >'ntPn.
(b) tPn ia emooth and lillln_oo I>'n I = 00.
(c) Ir we let the An be such that lAd :5 1>'21 :5 ... , then there exists constants C,6 > 0such that I>'n I :2: Cn 6 for eufficiently large n.

Proof. Note that since P : Hd(E) c........ L2 (&) is a Fredholm operator, by Gärding's
inequality, we know that P : N(P)l. H d (&) - N(P)l. L 2(&) ie a bijection. Oefine theGreen operator G = G( P) as the inverse of this map aod extend G to be the zero on
N(P). Since H d (&) c........ L 2 (&) is compact, G ie a compact self-adjoint operator. Let {tPn}be a complete orthonormal basis of eigenvectors of G with GtPn
JJntPn. Note that, since
N(G)
N(P), if J.Jn "I 0, then PtPn = JJ~ltP. Thue I>'nl - 00. Also if k is an integer sothat dk > I, then pI: - A~ I ie elliptic. Thus by hyperelliptic, (pi! - ..\~ I)tPn = 0 implies
tPn E COO(E).

n

n

=

=

=

For (c), without loss of generality, we may assurne that E
M x C and d > T' Let
F(a) be the space spaooed by the tPj with I>'j I :5 a. Oenote n(a) as dirn F(a). Then on
F(a), we have
SUPrEM

1/(;z:) I :5 Cllld :5 C(lP 110 + 1110) = C(l + a)l/lo.

Chapter 1.3.

37

Therefore"(0)

L ~j (Z)~n(Z) :5 C 1( 1 + a)1.

;=1lntegrating this estimate over M, we haveThat iswhich completes the proor.
Now let P : COO(E) - COO(E) be a self-adjoint elliptic wno of order d > O. We saythat P has positive definite leading symbol if there exists p(z,~) : 7-(M) - End(E)such that P(ZI'~) is a positive definite hermitian matrix for # 0 sod erP - P E 5 d - 1 inany coordinate system. We claim that there is a constant G such that for every such P,
Spee(P) C (-G,oo). In fact, we may construct a 00 with the leading symbol JP sod if we
let 0 := QoQo, then P - Q E 5 d - 1 • Thereforee(PI, I) = (QI, I) + «P - Q)/, I) 2: (Qo/, Qo/) -I«P - Q)/,/)I~ (Qo/, 001) - Gl/ldnl(P - Q)/ldn~ (Qo/, 001) - Cl/l d/ 2 1/Idn-1~ (Qo/, Qo/) - Cl/ldn(!IP Ild/1 - ~(e)l/lo)

2: (Qo/, Qo/) - 2C!IQo/l~ - C(!)I/I~~ -G(i)I/I~,
with 2Ce :5 1.
With above, we may introduce the heat kerneis. Fix a P as above, 80 that Pisa
self-adjoint elliptic 'lJOO of order d > O. Then the associated heat equation ia defined by
the system:(1t+P)/(Z,t)=O, fort;:::O{ l(z,O)=/(z).
Form all y, it has a solution I(z, t)
Fourier series and= e- tP I(z): Let I(z) = Lnwhich is usual called the heat kerne! of P. Then

n

CnlPn

be the generaJized

38

Existence Of Heat Kernela

and

e-

tP /(z) = iM K(z, t y)/(y)dJJ(Y)l=L e-t>··~n(Z) { /(y)4Jn(y)dp(y).
1MnHence we have

Now we justify the above formal proCe8S. In fact, if jd > Je +!:fl

Thua, by the fact that Spec P C (-c, 00), without 1088 of generalitYI we may assurne that a11,\ > O. Note that since eU.,\i :5 t- j C(j)e-t>.·/~ I we haven

On the other hand, by Lemma 2 above, we know that for n big enough, ,\ ~ Cn 6 with 6 > 0,so K(z, t, y) ia an infinitely amooth function of (t, z, y) for t > O.
Thus if (f, P) is an elliptic complex, ·e-t~j is in '1'00 with a smooth kernel function.
Let= {4J E L (Ei) : ßi 4J = A.p}.
2

Ei (A) :Then Pi: Ei(A) --+ Ei+l(A) defines an acyclic complex if'\"1 0, so that L:(-l)idimEi(A)o for ..\ "I O. Therefore, we have the following=

Lemma 3. With the notation as above,
Ind (P)= I) _l)iTr(e-t~j).
i§I.3.4. The Existence Of Heat Kernels: 11
In the previoUB section, we used the following facts to prove the existence of the heat
kerneis for an elliptic self-adjoint 'lrDO P of degree d > 0 with a positive leading symbol:
The spectrum {An} is a bounded below subset of Rj there exist positive numbers 6 snd C
such that for sufficient large n, '\n ~ Cn D• Since P is also a Fredholm operator, we knowthat the index of P ia well-defined. AB a consequence of this theory, if (E, Q) ia an elliptic
complex, then
Ind(Q)_1)iTr(e- t6 ;),

=l:)
i

39

Chapter 1.3.
where ß is the Laplacian aBSOciated with Q.

In this section, we consider another aBpect of the heat kernels, that is, the local asymptotic expansion of heat kernels, by using the geometry of the vector bundle E and the
manifold M itself. Hence we find a deep relation between tbe index of certain elliptic
operators and the geometry of objects in question.
From now on, we only consider the objects which come from geometry. Let M bea manifold, and E be a vector bundle over M. The algebra of differential operators
on E, denoted by V(M,&), is the subalgebra of End(COO(M,&)), generated by elements of
CCO (M I End (&)) and the covariant derivatives V x, where V is any connection on E and
X ranges over all vector fields on M. If V' is another connection on E, V
V X is in
COO(M, End(&», V(M, &) is independent of the choice of V.

x-

There is a natural filtration on 1>(M, &) defined byVi(M,E):= CX'(M,EndE)Span{Vx , ... V x ; : j:5 i}.
We call an element ofVi an i-th order differential operator. Thus the symbol morphismis the natural morphismco1) - grV := LVk/Vk-l .

.t=o
Hy Leihniz's rule, we know that, B8 an associated graded algebra, grV is isomorphic to the&pace of sections of tbe bundle S(TM) 0 End(E), where S denotes tbe symmetrie product.
Moreover, the isomorphism

may be given by the following formula: ie D E VI:, then for z: E M and ~ E Ts:M,

O'.t(D)(z,~) = liIDt_coCk(e- itJ DeitJ)(z) E End(Es:),e.

=where / E COO(M) such that dj(z) =(We may check this with D -i d~ to get a good
illustration.) Also, locally, any differential operator D may be written uniquely aBn

L L aa(z)D;
j=O lal=jover any trivialized open aubset wi th coordinates (z: I, .•• , Zm ) . Witb thiB, we know thata differential operator D of order k ia elliptic if the sec ti on O'.t(D) E CCO(T- M,1f-End(E))
over tbe cotangent bundle spa.ce is invertible over the open subset {(c,~) : ~ #; O}. Here 1f
denotes the natural projection.
Now we fix a Riemannian metric gon M. By definition, a differential operator H on E
ia a generaIized Laplacian if locally H has the form - LiJ gi J BiBj I up to a differential

Existence Of Heat Kerneis

40

operator of order :5 1. Here 8i denotes Ir; and gi J stands for g( dZi, dz j )Id E • Thus that H
is a generalized Laplacian means that H is a second-order differential operator such thatcr2(H)(z,~)
1~'Fl. Or equivalently, for any f E CCO(M),=[[B, f], J)= -21dfI 2 .

Example: Let f be a vettor bundle on a Riemannian manifold, with a connection '\JE.
Let '\J be· the Levi-Civita connection on AI. Then the usual Laplacian u E on COO (M, E) isdefined by~E S := _Tr('\J T • M0t Y't s).

Here we denote by Tr SECCO (M, E) the contraction of an element SECCO (M, 'r M '9

r- M 0 E) with the metric 9 E COO(M, TM 0 TM). Thua, for any two vector fields X, Y,
we have
Therefore, if ei is a local orthonormal frame of TJ.\1,

uE =- L)'\7;,'\7;. - Y'~.;eJ.
i

On the other hand, with respect to the frame 8/8z i , defined by a coordinate system arounda point in M, we havertrt

Here the Christoffel symbols j are defined by 'V 8/8:: J)/ 8zj : = LI: j 8/8z 1:. So the Laplaeian ~E ia a generalized Laplacian. (Usually, this formula ia called the Weitzenböck fürmula. Later we will give its generalization, tbe Lichnerowicz formula.) It is not difficult toshow that any generalized Laplacian is of the form ~E + F, with F a seetion of the bundle
End (E). Similarly, for superconnections, we mayaiso introduce the associated Laplacians.
The next aim is to prove that for aoy generalized Laplaeian over a compact manifold

M, there exists a unique heat kernel. For this purpose, we make the following
Definition. A heat kernel for a generalized Laplacian H is a family of seetions
p(x, t, y) of E. 0 f; depending on t E R~o, such that the following conditions hold:(1) The action ia C oo at (z, t, y) E M x R>o x M.
(2) For every y, (8. + Hr)p(z,t,y) O. (3) For any continuous section B of E with compact support, with respect to the supre-=

mum norm,, 1fM p(z, t, y)s(y) dp(y) = s(x).

1imt_O

Here we may choose any metric on E.

Chapter 1.3.

41

Remark. If p(z, t y) exists, we may define an operatorlP, : cgo(M,E) -by P,s(x) :=COO(A1,E)

IM p(x, t, y)s(y) djj(y). Thus (2) may be rewritten a.s(at + H)Pt = 0,

w hieb is the heat equation associated with H. Also (3) is j ust an initial condition for the
first order differential equation
lilIlt_oP,
Id.

=

Thus, formally, we may think of Pt aB e- tH .
Theorem. For any generalized Laplacian over a compact manifold M, there exista aunique beat kernel.
The existence of heat kerneis for a generalized Laplacian H follows from the results inthe lBBt sedion. But in that approach, we need to know about the spectrum of Hand tbe
eigenvectors. Now we UBe an approximation to construct the heat kerneis. Nevertheless, tbe
agreement of the heat kernela obtained by these different methods is a direct consequenceof tbe following
Uniqueness Lemma. Suppose there exist heat kerneis for aB generalized Laplacians,
then tbe beat kernel for a fixed generalized Laplacian H ia unique.
Proof. Let <I >: E~ xE; -C be a natural pairing defined by< s, U >:= 1M < 8("'), U("') >% dl'("').
Tben for any given H, H- is such that < Hs, u >=:< 8 1 H-u > . By an easy computatioll,
we know that H- is a generalized Laplacian on E- too. Suppose H (resp. H-) has theassociated operator Pt (reep. Pt) aB above. Obviously, if < Pt81U >=< 81PtU >, by tbe
duality, we have our assertion.
To prove the last relation, considere

for 0 << t. Difierentiating with respect to 0, we see that f is a CODstant.
lim,_o/(O) = lim,_t/(O), which completeB tbe proof.

Hence

We next prove the existence of heat kerneis by an approximation procees from thefollowing four steps:(1) The existence for H = - L. 8? on Rm.
(2) The existence of a formal solution.
(3) Tbe existence of an approximate solution.

42

Existence Of Heat Kernels(4) Construction of an exact solution from an approximate one by aperturbation process.
Proof of the theorem. (1) Over R m , let [, be the trivialline bundle R m x C. Then
- Li
is the standard Laplacian. In this case, we may precisely let

8lp(x, t, y) := (41ti)-m/OI e-I;r-yl'J /4'.
Hy an easy calculation, it follows that p(x, t, y) is the heat kernel of -Li er(2) The aim in this step is to construct a formal solution for H on a compact manifold
M by using (1) and the Strcalled normal coordinate system.
Over a Riemannian manifold M, we say that a smootb path 2: : [0, 1] - M is a geodesie1
if it minimizes the function L(x) := J0 Iz(t)ldt. This leads to an Euler-Lagrange equation,
which is of order 2. From this, for any y E M and ~ E T y M, there exists locally a uniquegeodesie x with initial conditions z(o) y, z(o) ~, which enable us to define exp ~ := x( 1).
Hence for a suflicient small " we have a diffeomorphism==

exp : {e E T,M : lei< e} - an open neighborhood of y E M.
Identifying T"M with R m , we get coord~nates near y. Such coodinates are called normal
coc;>rdinates. In tbe following, for any x E M near y, under the above identification, we
will also denote the point in R m as z.
Now, imitating the situation over Euclidean space, we let

Then' by a loeal calculation, we know that for any C oo - family of sections St in C oo (M, E),(8r + H;r)(q(z,t,Y)St(z))=((8t + t- 1 V

L (;8i + Idetgij 1 /1 20

Ho Idetgijr /2)s,(x)) q(x , t, y).
1

From this, by a formal solution of the heat equation (8t + H:r) p( z , t, y) ::: 0, we meao
an element q( x, t, y) ~(z, t, y) such that near y, the section 4'( x, t, y) of &;r 0 [,; satisfies theequation

Existence of Formal Solutions. There exist unique sections <l>i(X, y, H) of E;r @ E;
for all i ~ 0 such that(a) cl-o(y, y, H) Id.
(b) (8, + t- 1 V (;8; + IdetUij 11/ 2 0 H 0 Idetgij 1- 1/ 2 ) Li2=,O <l>i ti ::: O.

=

L

Proof. Suppose we have the assertion. Let fi(S) := s'~i(exp(sx), y, H). Then theconditions becomedds!i(s) ={O,.

-si- 1 ldetgijp/2 0 Hz 0 ldetgijl-l/Ol~i_l'

ifi=O;

ifi > O.

43

Chapter 1.3.

But this ia a system of differential equations of order one, which we may solve easily. Thusby the recurrence on i, we have the existence of formal solutions.

(3) In this step, we use a cut-off function and the formal solution above to obtain thefollowing approximations:oeExistence
Approximate Solutions. For every positive integer N, there exists asmooth family of smooth sections K N (z I t , y) of Sr 0 S;, such that for any integer d,we have(a) VT > 0, J(N (x, t, y) is uniformly bounded for I Id in the range :5 t .'5 T.
(b) For auy section 8 of S,°

with respect to lid.
(c) As t - 0,

Proof. Note that eince M is compact, we may find a E such that the normal coordinates
map exp ie well-defined for all point Y EMin a email ball I~I < E. (In fact, we may assume
that e is smaller than the injectivity radius, i.e. the radius of the largest ball in TrDM
such that the exponential map is a diffeomorphism from this ball Mound zero in T rD M to
the neighborhood of %0 in M: Geometrically, the injectivity radius is simply the largest
ball in the normal coordinates for which geodesics do not intersect.) Now define a cut-off
function<p : R2: 0 - {O, 1] by
1;'(8) = {I,0,~f 8 < !~/4;

If 8

>! .

Using this cut-off function, we may smooth our formal solution by definingNkN (x, t, y) := I;'(d(x, y)2)q(x, t, y)(E ti<l>i(X, y, H)).
.

i=O

Here d(x, y) denotes the dietance of z and y in M. Now by a local estimation, we have tbe
assertion.
(4) Tbe last step is to construct the exact solution from the approximate 8Olutions. For
this, we need to use the Volterra series: aperturbation process.
We illustrate the situation by an analogue for a finite dimensional space V, i.e. when
M becomes a point. Suppose that there exists a fundion K t : R~o --+ End (V) such that

Rt := d~, + H K t = O(t a ),=for same Q ~ 0 and K o 1. Here H is a linear endomorphism. We introduce the following
perturbation process: Let
R2: o - End (V) be defined by

Q~:= {}Qr :4~ :={(h ,'" ,t. ):OStl St~ S ·"St. St}

K t - t • Rt.-t._ t ..• Rt~-tt Rt 1 dtl ... dtJ;.

44

Existence Of Heat Kerneis

Then Q~ = K t and tbe surn of the convergent series Lk>O( _1)kQ~ is equal to Pt = e- tH .
Thus the fact that the volume of-ia ~, which decaya rapidi)', we have Pt= K t + O(t +a ).
1

The ,situation in general ia very aimilar. We fix an N ~notation. Define r(z, t, y) := (at + H. )k(z, t, y). Let

rr aod omit tbe N in our

and

mi

d
k
Suppose that N >• Sy the existence of approximate 8OlutioDS, we know that ql:, r are
Cd with respect 10 z and y. Moreover,andfor certain constants A, B ~ O.
Thus, put all above together, we have the following

Existence oe Hest KerneI!. Let p(x, t, y) := Lk>O( _l)k qk(z, t, y). Then
(1) The series converges absolutely.
(2) Pt is Cd with respect to z, y and satisfies-(3) With respect to the sup-norm,(4)

lp(x, t, y) - k(x, t, y)ld $ O(tN-m/2-d/2).

Chapter 1.3.

45

§I.3.5. Clifford Algebras And Dirac Operators
In the previouB sections, we have seen two methods to construct the heat kerneis. The
first is rather neat and is very convenient for us to study axiomatically. On the other hand,the approximation method is rather eomplicated, and does yield eertain geometrie properties
direetly. In order to diseutJS this later aspect , we reeall in this seetion certain eoneepts andresults eoncerning Clifford algebras and Dirac operators.
Ooee we have obtained a generalized Laplacian, a natural question is to ask what are
the first order operators D which have the generalized Laplacians as their squares. Suppose
D may be expressed locally asD=

L: aa(z)8a + b(z),awith aa, b the sections of End(E).
Hom(T" M, End(E)),

Hy an easy calculation, we see that BB a section ofThus the square of D is

"D"J = ~ L:(a i (z)a1(z) + a (x)a i (z))8i8j + first order operator.

jijThus D"J is a geoeralized Laplacian if aod ooly if for any {, 7] E T- M, we have< a(z), { ><' a(z), 7] > + < a(z),7] >< a(z), € >= -2(€, 7])~,
where (., ) is the metric on 1; M.
Thus locally, we may proceed as fallows. Let V be areal vector space with a quadratic
from Q. The Cüfford algebra of (V, Q), denoted by C(V, Q), is the algebra over R
generated by V with the relation

vw + wv = -2Q(v,w)
for all VI w E V. Ir Q ja fixed, we may write C(V) for C(V, Q) and (v, w) for Q(v, w).
Since C(Q, V) ie a quotient of the tensor algebra T(V) := EBi @i V, and is a superalgebrawith the generators contained in the eveoly graded subalgebra of T(V), we know that C(V)ie a superalgebra too. We say that a auperspace E ia a Cüfford module, ·if there is asuper-action of C( V) on E. We will denote by c(a) the action of an element a of C( V) on
E. Let a ...... a- be the anti-automorphism of T(V) such that v E V is sent to -v. Sinceleaves the quotient relations unchanged, we obtain an anti-automorphism a ...... a- on C(V).
We say a Clifford module E of C(V) with an inner-product is self.adjoint if Q is positivec(a)*. We also denote Endc(v)(E) the algebra of endomorphism of E
definite aod c(a-)
supercommuting with the action of C(V).

=46

Existence Of Heat Kerneis
C(V) has a natural inereasing filtration

Ci (V):= the apan of elements of the form VI ••• Vl: with Vj E V and k :5 i.

o bvioualy the MSOciated graded algebra gr C( V) ia naturaHy isomorphic to the exterior
algebra "V. Thua we may define a Clifford module action of C( V) on "Vasfollows:
For any v E V, let E(vL t(v) be the exterior product, the eontraction via Q, respectively, ofv. Then the Clifford action ia defined byc(v) := E(V) - L(V).
Moreover if Q is positive-definite, L ia the adjoint of E, so that 1\ V ia self-adjoint. Usually,
we call the isomorphism (T : C( V) -+ 1\ V defined by 0'( a) := c(a) 1 the symbol map. Its
inverse c is called the quantization map. Sinee C(O, V) is just 1\ V, we mayaiso think of
I\V as adeformation of Clifford algebras.
The most important result in tbe local situation is tbe following

Lemma. Let V be a 2m-dimensional oriented Euclidean vector spaee. Then there is aunique Z~·graded Clifford module S = S+ (fJ S-, called tbe Spinor module, such that
C(V) 0 C ::::: End(S).
Proof. Let {Cj} be an oriented orthonormal basis of V and let P be the span of elements
C~j-l - ie~j with 1 ~ i ~ m. Then we know that V 0 C
P Ei) P. Now let 5
I\P,and define a Clifford actien as fellows: If w E P, c( w)s := J2E( w) s; if w E P ::::: p.,c(w)s := -J2L(W)". Since the algeb.ra of matriees is simple, it has a unique irreducible==

module. Hence by eomparing the dimensions, we have
C(V) 0 C ::::: End(S).

Now we eonaider the global situation. Let M be a Riemannian manifold. The Clifford
bundle C(M) is the bundle of the Clifford algebra over M whose fiber at z E M is the
Clifford algebra C(T; M) of tbe Euelidean spaee
M. There ,is a natural symbol map(T : C(M) -+ I\T· M defined hy tbe loeal symbol mapa (T%: : C(T; M) I\T; M. A Clifford
module f. on an even-dimensional Riemannian manifold M ia a Z2-graded bundle f. on M
with a graded action of C(M) on it. If E ia a Clifford module with metrie p, for which E+ and
E- are orthogonal, we say that the Clifford module is aelf-adjoint if the Clifford action
is self-adjoint at each point. For any vector bundle E', the twisted Clifford module
obtained from E by twisting with E' is the bundle E' 0 E, with Clifford action 1 ~ c. If
a E A(M, C(M)) is a Clifford algebra-valued differential form on M, we may define anoperator c(a) as folIows: If a, ß are differential forms on M, a ia a Clifford algebra section,and 8 is a section of E, aB homogeneous with respeet to the Z~-grading, thenr;(c(a'9 a))(ß 0 s) := (_l)laIIß1(a 1\ ß) 0 (c(a)s).

47

Chapter 1.3.
Define a Dirac operator Don a super-vector bundle
operator on [,e aa a first-order differential odd

D: CCO(M,e±) _ CCO(M,e T ),such that D 2 is a generalized Laplacian. Sy a local calculation, we know that if D is a
Dirac operator, then the action of r M on e defined by [D, fl := c(df) is a Clifford action.
c(df) for a11 f E CCO(M) is
Conversely, aoy differential operator D which satisfies [D, flactually a Dirac operator. Thus the collection of all Dirac operators on a Clifford module
is an affine space modelIed on CCO(M, End- ([)). In order to sharpen this identification, we
consider a special kind of connection:=

A connection "1[. on a Clifford module [ ia called a CliHord connection if for any
a E COO(M, C(M)) aod X E CCO(M, TM),

[Vk,c(a)] = c(Vx a ),with V the Levi Civita connection extended to C(M). A superconnection A on a Clifford
bundle [ ia called a ClifI'ord superconnection if for any a E COO(M, C(M)),
4[A, c(a)] = c(Va),with Va the Levi-Civita connection at 0, which is an element of Al (M, C(M)). Since locally
we may decompoBe aB Endc(M)(S,[)~Sfor a certain Clifford module S, we know thattbere mosts a Clifford auperconnection on any Clifford module by using a partition of unity.

e

Now let A be a Clifford superconnection on a Clifford module e. We may define a firstorder differential operator on COO(M, [), denoted aB DA, by comp08ing the superconnection
witb tbe Clifford multiplication:

There is an orthonormal frame {fi} of tbe tangent bundle sucb that with respect to thislocal coordinate system we bave

E

A=Ldzi~8i+c(el)~Al,

lC{1, ... ,2m}
with AI being sections of End(E). HenceDA= Lc(dz )8i +iLc(e1)Al.

lC{l .... ,2m}

In particular, if \jE ia a Clifford connection, then the associated first-order differential operator D ia a Dirac operator. Locally, we have.
E
D = '""~c(dZ')'V8i'

48

Existence Of Heat Kerneis

Moreover we know that the map sending A to DA ia a one-to-one correspondence between
Clifford supereonnections and Dirac operators compatible with the given Clifford action on&, i.e.
Indeed, this is a consequenee of the fact that the difference of two Dirac operators is asection of End-(&).
Proposition 1. If A is a Clifford superconneetion on &, then the curvature of A
deeomposes under the isomorphismEnd(&) ~ C(M) ® Endc(M)(&)
as folIows:A2=If +rls .

Here RE E A2(M, C(M)) C A2(M, End(&)) ia the action of the Riemannian eurvatureof M 00 the bundle & given by the formula

aod FEls E A(M, Endc(M)(&)) is an invariant of A, called the twisting curvatureof the Clifford module E.
Proof. Let FelS be the difference A 2 - ~. We have to show that c(Fel s ) commuteswith the operator c(a) for any a E COO(M, 1" M) C CfXJ(M, C(M)). But this is a directeonsequence of the eondition that A is a Clifford superconnection. In fact, we have

[A 2 ,c(a)] = [A,[A,c(a)]] = [A,c(V(a))] = c(V 2a) = c(Ra) = [nt,c(a)].
Now, by applying the existence of the heat kernels for a generalized Laplacian, we havethe following
Proposition 2. Let D be a Dirac operator on a eompact manifold, then D has a finite
dimensional kernel, aod D2 , aeting on CfXJ(M, &), has smooth heat kernels

We defioe the index space of a self-adjoint Dirac operator

to be its kernel

49

Chapter 1.3.
Then the index of D ia tbe dimension of the aupetllpace Ker(D),Ind(D) := dim(Ker(D+) - dim(Ker(D-».
Since for a self-adjoint operator 1 we have D- = (D+)*, bence
Ind(D) = dim(Ker(D+)) - dim(Coker(D+)),
which is tbe classical definition.

McKean-Singer Theorem. Let D be a self-adjoint Dirac operator on a compaet
manüold M. Let< zle- tD ' IY > be the heat kernei! of tbe generalized Laplacian D'2.
Then for aoy t > 0,
Ind(D) = Tr.(e- lD ') =1M Tr.( < ,*-lD'lx > )dx.

It follows tbat tbe index is an invariant of M and E.

Proof. Tbis result has already been proved by using the spectral theorem for D'2, aod
this is c10sely related to our first method for the existence of heat kerneis. Here we give
another proof by using tbe approximation process.
Let a(t) := Tr.(e- tD ') and let Pt := 1 - Po be tbe projection onto tbe orthogonal
complement of Ker(D). Then, using tbe approximate solutions for heat kerneis, we knowthat for t big enough,with ..\t the smallest non-zero eigenvalue of D'2. Thus, by the exponential decay,a(oo) := lim,_oo Tr.[e- CD '] = Ind(D).
Now the a.ssertion is a consequence of the fact that a( t) = a(00) for aB t.
differentiation with respect 10 t, we bave

In fact, by

From this result, we see that it is possible to express the index in geometrie terms which
ooly involve M aod E: This is what the usual index theorem means.

50

Existenee Of Heat Kernela§I.3.6. Local Index Theorem
In this section, we give a proof of a loeal verston of the index theorem. Hy the MeKeanSinger theorem above, we know that the usual index theorem ia an integration of this loeal
result: The loeal index theorem is 8t the level of differential forms.
Let M be a compact oriented Riemannian manifold of dimension 2m. Let D be a
Dirac operator on a Clifford module & on M 88SOeiated with a Clifford connection '\Je.
Let k(z, y) =< xle-tD::IIY > be the heat kernel associated with the generalized Laplacian
D'J. From the MeKean-Singer formula, it ia enough for UB to know the behavior of therestriction of the heat kernel to tbe diagonal, that is k(z, t, z). Hut for a general t, there isno satisfactory expression for k(z, t, x). However, if t is small enough, our system is in thesituation at tbe very beginning. So, tbe change of the heat ftow ia rather regular. Henee wecan control it. More precisely, in our situation, we may go as follows:i,

Think of tbe heat kernel k(z, t, z)' as a section of the bundle of filtered algebraEnd(t:) ~ C(M) @ Endc(M)(&),
where the filtration is induced by the filtration of C(M) := C(T- M) and tbe elementsof Endc(M)(E) are given degree O. Denote by Ci(M) tbe subbundle of C(M) of Clifford
elements of degree less tban or equal to i. The associated graded algebra is the bundle
AT" M @ EndC(M)(E).

Local Index Theorem. Let M be a eompaet oriented Riemannian manifold of dimension 2m. Let D be a Dirac operator of a Clifford module & on M associated with a
Clifford connection V€. Also let k(z, t, y) =< xle-tD::Ilv > be the heat kernel aBBociatedwi th the generalized Laplacian D'J. Then for t -+- 0+, the restrietion of k( x, t, y) to the
diagonal haa the asymptotie expansionk(x,t,x).-v(4111)-m

L t ki(x) = (ko(x)1
km- (x) 1km(.x))+ ... + ()-+ (4) + ....
oo

i

1

411" -m t mi=O47l'" -m t

7l'" -mFurther we have(1) The coeflicients /ci E COO(M, C'Ji(M) @ Endc(M)(&»)'(2) If u(k) := L:~o U2i(ki), then

u(k)

R/2
eis
p (-F
=det 1/'J (sinh(R/2))ex),

where R is the Riemannian curvature of M and FelS = ('\Je):! - R€ is the twisting
curvature. Usually, we caU det / ( !lln~~/2» the li-genus form of the manifold
M with respect to the Riemannian curvature R.
1 2

Proof. The basic idea for proving this theorem is that first by using the normal
coordinates, one may reduce the problem to a loeal one; then by Lichnerowicz's formula,Chapter 1.3.

51

the problem becomes the one about beat kerneis for the harmonie oseillator on Euclidean
spaces. In this process, naturally, we may use the rescaling technique to make everything
go through clearly.
Basically, the proof may be divided into the following four steps.
(a) Ta reduce the situatiQn to the local one by using the normal coordinates.
(b) Lichnerowicz's formula.
(c) Heat kerneis for harmonie oscillators over Euclidean spaces.
(d) The expansion of the heat equation.
(a) Fix Zo E M and trivialize the vector bundle f in a neighborhood of Xo by a paralleltransport &long geodesics. More precisely, let V := TroM, E := Ero and U := {~ E Vll~1 <!L where e is smaller than the injectivity radius of M at Xo. By the exponential map~ ........ eXPro~' we identify U with a neighborbood of Xo in M. For:r = exPro~' the fiber Er
and E are identified by the parallel transport map T( XO, x) : Er - E· along the geodesicx, := eXProse. Thu8 the 8pace COO(U,E) of sections of [; over U is identified with thespace of E-valued COO-functions on V, defined in the neighborhood U. We also identify
CCO(U, End(E)) with COO(U, End(E)). Hence D =aa(~)8{, with aa(~) E End(E). On
the other hand, by the i80morphism t7, we bave

La

In this way, for simplicity, if we introduce a rescaling on the space of functions on R~o x Uwith values in "V· 0 Endc(v.)(E) by the formula .
m(oua)(t,€) :=L u- a(ut, u /2€hij.
i 2
/

1

i=O

Tben the loeal index theorem is equivalent to saying that, withlilTlu_o( um Ou k) I( t ,e )=( 1,0)=(41r)-mdetl/2( . R/2 ) exp( _FelS) ..
smh(R/2)
The reason we ehoose tbe rescaling operator as above ia that for t, the heat equation onIy
contains first order derivatives, but for x, it contains the second order derivatives. HiatoricaUy, this technique W88 introduced by Getzler [Ge 86].

ou

The rescaling operator
introduces a filtration on the algebra of differential operators
action on COO (R~o X U,"V· 0 Endc( v.) (E»): An operator D has the filtration degree d52

Existenee Of Heat Kerneis

6u if;(e)6;;1= if;(u l / 2e), Vif; E CCO(U);6u 8t 6;;1 = u-1ßt ;1 - u- 1/2 a··6u 8.61 U
1,=U- 1/2 g(a), Va E V·;
6u L(a)6;;1 = u- 1/2L(a),- 6u E(a)6t7 1

we know that a polynomial p(e) has degree -deg(P), that a polynomial P(t) has degree-2deg(P), that a derivative alße' has degree one, that a derivative alm has degree two,tbat an exterior multiplieation operator ! i has degree one, and that an interior multiplieation
operator Li has degree -1. In particular, in (b), we will show that D 2 haB degree two, Le.
up to an operator of lower order, 2 may be identified with a harmonie oscillator withdifferential form coefficients.

n(b) Here we want to unclerstand the generalized Laplacian D2 associated with the Clifford eonnection '\1E. Sinee now we do everything loeally, we may find a spin deeompositioo
E = S 181 W with W := HOInc(V-)(S, E) so that
End(E) ~ End(S) ® End(W) ~ C(V·) ® End(W).

a

Let i be the orthonormal basis of V with' its dual basis d(i of V·. Denote ci aa c(d~i) E
End(E). Let ei be the local orthonormal frame obtained by parallel transport along geodesics
from the orthonormal basis ß, of T~o M, and let ei be the dual frame of T· M. Thus by thefact that for the radical vector field 'Tl := Li x,ßi ,e '1['\1 R, c( e)]= c('\1 Re'), =0,

we know that the End(E)-valued function c(e')( is the constant endomorphism ci.

Theorem. (i) (Lichnerowicz's formula) Let A be a Clifford supereonnection of a
Clifford module E. Denote the Laplaeian with respect to A by Ll A and let rM be thesealar curvature of M. Then=

Here c(FEIS) Li<i FElS (ei, ej) c(e i ) c(e i ).
ii Let L be the differential operator 00 U c V, with coeffieients in C(V·) 181 End(W),defined by1
L := ßC + c(FEls) + -rM.
4

=u6 L6;; as the rescaling operator of L. Then for u - 0+ ,Deoote L( u) :u

1

Chapter 1.3.

53

Here Rij := (Rz:oBi, Bj) = L.k<'( Rro)jiklEkel, and F is the element .of A 2 V· 0End(W) obtained by evaluating the twisting curvature FEls at the point zoo

Proof. (i) Locally, we have DA
the direction 8/Bi. Thus

= L. c( dz i )Ai wi th Ai the covariant differentiation in

Di = ~2 l:lc(dzi)Ailc(dxi)A j ]..
I}=~ L)c(dz i ), c(dzi)}AiAj + l : c(dzi)[Ai' c(dx-1 )]Aj
ij

ij

+ 4L c(dzi)c(dxj)[Ai. Aj]
ij=-

L gij(AiAj + L r~jAk) + L c(dz )c(dx-1)[Ai. Aj]
i

ij

i<j

k

"
i
.
;2
=LlA + 'L,...c(e)c(el)A(e"ej).
i<j

On the other hand, by Proposition 5.1, we know that

L c(ei)c(ej)A2(eil ej) = L FEls (ei, ej) c(ei)c(ei) - ~ L llij#:/ c(ei)c(ei)c(ei)c(e l ).
i<j

i<j

ij J:lSinee the antisymrnetrization of llijil over ijl: vanishcs , we have
" RijJ:l c(el)c(eJ)c(e" . J : )c(e I ), 'L,...

=- L.." c(el)c(e ) R;Uj + L,... c(eJ)c(e ) Ri'ij
""

I

,",,'

ijl

ijil

=2

L c(ei)c(e') L Riijb
ij

=2

1

ijl
k

L L c(ej)c(e i ) Rikjk
kij=- 2 L RtkiJ: = -2rM.
ik
So we have (i).

(ii) First consider the local expression of the operator
curvature of 'iJl I tben

FE''Vl. Hy definition, if F l is the= ~ L «R(8i,8j)eJ:,el)ckcl~i AdE. + FIS(B 8j)dE. i Adei"·jiI

i<j;i<lSo by a loeal ealeulation ,
I",,·
"
'iJE
8;
Bi + 4 L,... RkUj€J, Ck CI + "L.."
fikl (€) Ck cI + gi (~)
~ I
jii</
11:<1

=

54

Existenee Of Heat Kerneis

where the error terms are

Henee

T'7C ,U._ l/'h T'7C c-lvB; .-U
Uu., V 8i u U=8i + ~

E Rklii€i ut.k)(!f - ut.
+ u-1/'l L fiJ:f(u1/'l€)(!1: - ut.I:)(!' - ut. + u 1/ 2gi(u 1/ 2€).
1)(!.l: -

ij.l:<f1)

1:<,
Therefore

That is,

Now

L(u)

=- LCv~;,U)2 + E FC/S(ei,ei)(ul/2Ü(e:i - ut.i)(!i - ut.i )i<iiTake the limits for these four terms: we know that the first term has the limit- L)V~;o)'l = - L(8i - ~ L(R)ji€i)'l;ijthat the seeond term beeomes F, while others are zero.
We end this step by the foUowing observation: L(O) = K is a generalized harmonie
oscillator over the Euclidean spaee V. Therefore, we may prove the loeal index theorem by
eonsidering the situation on Euclidean spaees.

(e) Mehler formula.
Now we eonsider tbe situation for the harmonie oseillator on Euelidean spaee V. Let R
be an 2m x 2m antisymmetrie matrix and let F be an N x N matrix, both with eoeffieientB ina eommutative algebra A. Hy definition, a differential operator H aeting on A 0 End(C N )valued funetions on V is ealled a generalized harmonie oseillator if

H= -(L.,... V"d + F = - ~L)8i + -1~
L.... ~jZi) '1 + F.
.
4.
"

2

I

.

J

J

Chapter 1.3.

55=

Obviously, H is the Laplacian on V associated with the non-trivial connection Vd+~ Lij Rtj;r;j d;r;i. Thus if let 'R : Li;r; i 8i be the radieal vector field on V, and 8t a A ®
End(CN)-valued smooth function on V, tben=

(8 c + H)qtlJt = qt(Ot + i-ln + H)St.
In this case, aB in the proof for the existenee of heat kernela, for a formal power series ~t(;r;)
in t, whose coefficients are smooth A@ End(CN)-valued funetions defined in a neighborhoodof 0 in V, we Bay qt~t ia a formal solution of the heat equation(8 t + H)Pr

if
(8 t + t- 1

=0

n + H)~t = O.

Before we introduee the most important result concerning with the generalized harmonie
oscillator, we need the foUowing notation:Let
jv(R) := det(e R /2 _ e- R /2
R/2
).

=

Siace jv(O) 1, j;l / \tR) ia well-deflned for t small. Similarly, we know that the A-valued
quadratie fonntRtR

< ;r;1 2 coth( 2 )Iz >is well-defined for .small t. With this, we may state our result as

Theorem (i)(Mehler's formula) The kernel pt{z, R, F), taking values in A 0 End(C N )and defined for small t by the formula:t(4rt)-mj;1/2(tR)e~p(- < zlt: coth(t:)lz »exp(-tF)
is a solution of the heat equation(8, + H r ) Pt(z)=O.

(ii) For any 00 E A 0 End(C N ), there exists a unique formal solution ';,(z, R, F , ao) of
the heat equation

with

00

Pt(z) = q,(z}such that ~o(O)formula

= 00.

_

.-1/2

(4,")

m Jv

L t 1<1>J:(z)J:=O
Furthermore, the function Pt(z, R, F,oo) ia given by the
1tR
tR(tR)exp(- 4t < zIT coth (2)lz »exp(-tF)oo·56

Existenee Of Reat Kerneis

Proof. (i) may be eheeked by a direet ealculation. Classically, we only eonsider the
harmonie oseillator+ r 2 on the real line. In that very simple situation, Mehler firstoffered his famous solution.

-6

For (ii), we have to salve
00(8 t + t-1n + Hz;)L t'1:~l(X) = O.
1:=0Therefore we should have

'R.<I>o = 0('R. + k)<I>l= -H:~l-l if k > O.

FrOfi here, by reeurrenee, we have the" proof of the theorem.

(d) The Expansion of the Reat Equation.
Form (b), we know that L(u) has a limit K when u goes to O. On the otber hand, if
P(z:, t, xo) is the heat kernels of the operator D2 , we may letk(t, €) := r(xo, x) p(z, t, zo).
Obviously, the "V· 0 End(W)-valued function k(t, €) satisfies the differential equation(8, + L) k(t,€}= O.

Also if we letthen the loeal index theorem is equivalent to saying that.

()I(-m det 1/2( sinh(R/2)
R/2) exp ( -F t/s) .

hrTlu_or u,t,€ (t,()=(l,O) = 41r)

Thus by the fact that r( u, t, e) satisfies the cl ifferen tial equation

if we ean prove eertain results which make UB expanse the heat equation, we may only
need to eonsider the formal solutions for the harmonie oseillator on an Euelidean space toeomplete the proof of the local index theorem. For this, we have the following
Lemma. There exist "V· 0 End ( W)- val ued polynomials
for every integer N, the function r N (u, t, ~) := qt(€)
r( u, t, e) in the following sense:,i(t, €)

on R2: 0 x V, such thatl:::-2m u i / 2;i(t, €) approximatesChapter 1.3.

57

To N > i + Inl/2, there is a eonstant C(N,i, o:) such thatfor°< :s 1uand (t, €) E (0,1) x U. Furthermore "'Ii (0, 0)°= if i i: 0, while "'10(0,0) = 1.

Suppose that this lemma holde, then expanding the equation(at + L(u»r(u,t,!) =°and r(u, t,!) in u 1/2, we have00

r(u,t,!) ...... qt(!)

EU

i / 2 1'i(t,!)

i=-2mand tbe leading term satisfies the heat equation

Sinee tbe formal solution of tbe heat equation for tbe harmonie oscillator is uniquely determined by 1-1(0,0), and 1-1(0,0)
for I > 0, we see that "'I-I = unless I = 0. In
partieular, we see that there is no pole in the Laurent expansion of r in U 1 / 2 • Also we knowthat the leading term of the expansion of r( u, t, e), i.e. r(0, t, €) = qt (€) 10 (t, €) satisfies theheat equation for the operator L(O) K with the initial eondition "'{0(0, 0) 1. ThuB finallyby the expression of K and the situation for the generalized harmonie OBeillator, we havethe following

=°

°

=

=

Theorem.. The limit li~_or(u,t,e) exists, and is given by(41ft)

-m

det

1/2

tR/2
1
tR
tR
(sinhtR/2) exp( - 4t < €1 2 coth("2)!e » exp( -tF).

In particular, we have the local index theorem by letting (t,e)= (1,0).

Tbus, we ooly need to give tbe following

proor of the lemma. By the proof of the existence of the heat kernei, we know thattbere exist functions tPi E Coo (U, End( E», wi th tPo (0) 1, such that for any ! EU,=Nllk(t, €) - qt(e)L t .pi(€)l1 :s C(N)ti

i=O

Note that sinee'

N

-

m

•

58

Existence Of Heat Kernels

we may replace <Pi(€) by its Taylor expansion lpi(€) of order 2(N - i):N

Ilk(t,O - qr(€) Lti~i(€)ll :5 C'(N)t N - m •i=OThus, we. haveN

Ilk(u,t,€) - qr(€) I)ut/~i(ul/2€)II:5 C'(N)u N
i=O

for (t, €) in (0,1) x U and 0 < u :5 1. We mayaiso make a similar estimate for the derivativesof k(u,t,€). Therefore, the function k(u,t,€) has an asymptotic expansion in u 1/ 2 , u small,of the form
00k(u,t,€) ..... qr(€) LWi(U,t,€),j=O

where 'l'i(€) is an End(E)-valued polynomial so that Wi(€) on V and ~o(O) = 1. Moreover
this expansion is uniform for (t, €) lying in compact su bsets of (0, 1) x U, and the asymptotic
expansions for the derivatives 8tat(k( u, t,€» may be obtained by differentiation the above
estirnation. Thus considering at tbe p-th term, we haveN

Ilk( u, t, €)lP] - u-P/'Jqe(€)

L( ut)i Wi ( u €)(p)11 :5 C( N)u -p/2t
N

l 2
/

N

-m.

i=O

Ir lj(t,€)(p] is the coefficient of ui/ 2 in tbe surnCi +p)/'J

u-p/'J

L (ut)i'Pi(UI/'J€)(p],i=Otben lj(t,€)(p] is a polynomial on R~o x V with values in I\PV· 0 End(W). It is clear thatthe surn I j (t, €) := L:~:o I'j (t, €){p] satisfies Cu 1 j = uj/'J I' j. And l j (t, €)[p] = 0 far j < - p.
Hence lj(t,€) 0 for j < -2m. In particular,=00

10(0,0) =L u 1i(O,O) = (6 '1')(0,0) = 1.
i 2
/

u

i=-m

Similarly, we have tbe statement far the derivates. This campletes the whole praof.

Chapter 1.3.

59

§I.3.7. Applicaiions oe Local Index Theorem]n this seetion, we give an application of the local index theorem itself and an applieationof the proof of the loeal index theorem.

1.3.7.a An Application oe The Local Index Theorem
By the McKean-Singer formula, for every t > 0, we have
Ind(D)=1M Tr,k(z, t, z)dlJ.

The loeal index theorem implies that if D is associated with a Clifford eonnection, then tbe
integrand itself has a limit when t tends to zero. Now as the supertrace vanishes on aB
elements of tbe Clifford filtration strietly less than 2m dim( M), the first part of the loeal
index theorem implies that=

Tr, c(z,t, z) - (4rl)-m

L tiTr, ki(z).
i~mRenee there are no poles in the asymptotic exp&Il8ion of Tr, k(z, t, z). Furthermore, as theleft hand side ]nd(D) of tbe Mckean-Singer formula is independent of t, we necessarily have

while the integrals of aB other terms

IM Tr, kj(z) dJl vanisbes for j # m.

To identify the term Tr, cm(x) as a cbaracteristic form on M, ~e need certain morenotation. Let fE COO(M, C(M)) be the chirality operator, i.e. locallYI if Via a Euelidean
space witb {ei} an oriented, orthonormal basis, then==n/2 if n is even, and p(n + 1)/2 if n iB odd. We know that as an element
where pin C(V) ® C, r does not depend on tbe basis of V used above, and, further, r satisfies= -vr if n iB even, wbile=
if n is odd. Also r 2
1. Locally, if V is an evendimensional real EuelideM spate, then every finite dimensional euper-Clifford module E ofC(V) ie isomorphie to W 0 S with S tbe spinal spaee and W
Homc(v)(S. E). In thisC88e, by a direct calculatioD, we have Tr"s(f) = 2n/2 , and tbe supertraee over W of anelement
Fe End(W) ~ Endc(v)(E)

rv

rv vr

=

=

is given by tbe formula

60

Existence Of Beat Kerneis

Motivated by this, we may define the relative supertrace of a E CCQ(M, Endc(M)(&)) by

Then we extend tbe relative aupertrace to a linear map.Tr"e/s : A(M, Endc(M)(f)) - A(M).
Thua if b E CCQ (M, C( M)), the point-wise supertrace of the sectionb ~ a E COO(M, C(M) ~ Endc(M)(&)) ~ CCQ(M, End(f))
is equal to the Berezin integral: This may be described aa folIows:

Tr"e(b(z) 0 a(z)) = (-2i)m U2m (b(z))Tr"E/s(a(z)).
HenceTr"E km(z)= (-2i)mTr"E/s[un (km (z))].

Thus we have tbe foUowing
Theorem. (Patodi, Gilkey) Let M be a compact oriented Riemannian manifold ofeven dimension 2m, with Clifford module f aod Clifford connection \JE, let D be
the associated Dirac operator. Ir k( z, t, z) is tbe restriction of the heat kernel of thegeneralized Laplacian D'J to the diagonal, then limf_oTr,(k(z, t, z)) IdlJl exists and is
the volume form on M obtained by taking the 2m-form piece of

R/2)
[(
EIS)]
( 271"1')-md et 1/2( sinh(R/2) Tr, ,E I5 exp - F.

L3.7.b An Application Of The Proof Of The Local Index Theorem
In this subsection, we will give another construetion for the classical Bott-Chern secondary characteristic form via superconnections by using the perturbation method in the
proof of tbe local index theorem, especially the Duhamel's formula. (Even through thisformula ia very important, we will not give a precise formulation, as there are tao many
variations for it. But still, the reader may get a good feeling from section 3, the use of theVolterra series.) Later we will use a similar method when we discuss the relative Bott-Chern
secondary cbaracteristic objects associated with Chern forms.
Let

f.:

O-&n~"'~&l~fo-O

be an exact sequence of veetor sheaves on a complex manifold M. Put hermitian metrics
Pj on &j for j
0, ... ,n. Thus we construct a supervector sbeaf & on M as fallows: Set=

Chapter 1.3.

61

The metries Pi give us canonical connections \7± on E±. Also v acta aB an odd endmorphism of E. With respect to the metrics Pi, there is an adjoint v· far v. Hence we havethe euperconnection A := \7+ + \7- + v + v· on S.
Next we introduce a new parameter in pI. First, we extend (E., p.) naturally to acomplex on M xe. Let

88

A z := v+ + V- + dz 8z + di Bi + zv + zv·
be a superconnection on E over M x pI.

Theorem. With the same notation aa above, we have

Proof. By the construction, it is enough to prove that axiom 1 ie satisfied by'7:= l(loglzl'j Tr. [exp(E,A,)).
But this may be proved as follows.
First, we tonsider the convergence of 11 when Izl- 00. Hy definition, we know that

A~ = Izl 2 ß + R z ,.

where ß := (vv· + v·v), R z = \72 + vdz + v·di + zV(v) + iY'(v·) has the degree ;::: 1 on
M x pi with V := V+ + V-, 80 it ie nilpotent. Thus by Duhamel's formula , we have thefinite sum expression:

Note that since .ä. has the smallest eigenvalue ,;\, > 0, we havefor all j, and for at least ooe j,

Therefore, we havelIexp( -.A;)II :5 C(1 + Izl m) :5 e- ~lzl:Jwith C a uniform constaot with respect to the M·coordinates. That is , we have the exponential decay.

62

Existence Of Heat Kerneis
Similar esti mates hold for the derivatives of exp( - A ~) wi th respect to z, Z.

Thus, we may extend a(z) := exp( -A~) in a smooth way anto M x pI by declaringthat a(z) is 0 on Mx {oo}. Also it is not difficult to show that a(z) E $pAP,P(M X pI). Soa(z) is also cfMXP1-closed. In this way, we haved~cfMrt = dMdM}pl( [loglzI2] a(z)=}plr dMxPldMxP1WoglzI2]a(z))=}plr dtfCOoglzl a(z)2]

= i~a(z) - i~a(z).
So we have our theorem.

63

Chapter 1.4.

Chapter 1.4
The Mellin TransformIn this chapter, we introduce the Mellin transform. The technique comes from classicalmathematica, but 8B the existence of the Bott-ehern secondary characteristic objectB dependB heavily on thiB teehnique, we will give a1l the detailB of it. The references are [T 30]
and [WG 89].
This chapter consists of tbe foUowing sections. In section one, we recall the basic
properties of Bernoulli polynomials and Bernoulli numbers. In Beetion two, we recall certain
properties of Gamma functions. In section three, we recall tbe properties of tbe Riemannzeta function. In section four, we disCUBS the Mellin tranBform in general. Finally, as anapplication of tbe Mellio transform, we give aootber coostruction for tbe classical Bott-ehern
secoodary characteristic fOrIns followiog [BGS 88].

§I.4.1. Bernoulli Polynomials and Bernoulli Numbers
The n th Bernoulli polynomial, denoted by Bn(z), are defined by using a generating
functioo 8B folIows:
te~t
co tn
- t - = '"' - I Bn(z).

e -1

(1)

L.... n.

n=O

The series iB convergent for It I < 211", since tbe Dearest singularities of tbe generating function
to t = 0 are ±21ri. When z = 0, we bave

(2)
Usually, tbis formula is expressed as
1

1

t et / 2 + e- t / 2

co

t 2n

t(et _ 1 + 2') = 2"(et/2 _ e-tn) = 1 + ?;(_l)"-l (2n)!B n •Thus, we see that

(3)

64

The Mellin Transform

for n == 1,2, .... As usual, we eall B n the Bernoulli numbers.
Sinee

t ~t
e

00

Ln

00

_ '"'" I';

t1

00

tn

n

f:'o

()

I
'"'"
'"'"
n
n _ .I:e' - 1 - ~ kfBk 0) ~ li x = ~ n!
Je Bk(O)z('"'"I

we have the following relation of Bernoulli numbers and Bernoulli polynomials(4)
Also by_' et _ 1 00 tJ:_
1- - t- {; Je,Bk(O) -t'-1 00 t k

E. T {;00_ 00 n-l n-l Bk(O)kl 81'(0) - : ; t{ ; /cl(n _ /c)!'

we get the recurrence formula for Bernoulli numbers:n-I

8 0 (0)8,1:(0)= 1, L k!(n _ /c)! = 0

(5)

1'=0

for n ~ 2. Symbolieally, we may write the above formula as

Bn(x) = (8(0) + x)n
for n == 0,1, ... aod(B(O) + 1)n - Bn(O) = 0

for n = 2,3, .... Here it is understood that, after the binomial expsnsions have been developed, the symbols 8.1:(0) for powers are to be replaced by B,I:(O).
We list the properties of Bernoulli polynomials and Bernoulli numbers as follows:
Properties. 1 (Derivatives)

dP
n!
-d
Bn(z) = ()I Bn_p(x).
Pxn - p .

2. (Difference Relations)

Bo(x + 1) =80 {x),8 1 (x + 1) =8 1 (x) + 1,

Bn(x + 1) =Bn(x) + nzn-I, for n ~ 2.
3. (Funetional Equation)

8 n (1- x) = (-I)nBn(x).

Chapter ].4.

65

4. (Addition Formula)

5. (Summation Formula)1

mLS" = --1
n+ [B + (m + 1) - B + (O)]
n 1

.=1

n 1

for n ~ 1.
We end this sedion by using the Bernoulii polynomials to obtain the asymptotie expansion of an analytie function, which ean then be used to deduee the famous Stirling formulafor nLLet f(z) be an analytic function along the straight line from a point a to z. Then forany polynomial <p(t) of degree n, and 0 :5 t :5 1, we havedn

- """ (_1)m(z - a)m<p(n-m)(t).t<m)(a + (z - a)t)
dt L.J
m=l(z - a)q,(n)(t)!,(a + (z - a)t) + (-l)n(z - a)n+lq,(t)f(n+l)(a + (z - a)t).

=-

Hence by integration, we have the Darboux formula:4l(n)(o)(/(z) - /(a))
n=L (_l)m-l(z - a)m[4l(n-m)(l)/(m)(z) - 4l(n-m)(O)f(m)(a)]m=l+(-l)n(z - at+ 1
Now let tP(t)].1.p(t)/(n+l)(a + (z _ a)t)dt.

= Bn(t) and replace n by 2n. Note that sinte(2 n.,)1 B(2n-m)() _ (2n)! B ( )
2n
;r: I
m;r:

(2n)(O) B 2n
-

m.

I

we havej(z)- /(a)
n

(

)2k

=: - a(f'(z) + I'(a)] + """(_l)i Z - a
2
L.J
(2k)1

i=l

+

(z

-

a)2n+1].1.

(2n)!

Bi (fCU)(z) _ jC2k)(a)]'tP2n(t) 1'2n+l)(a + (z - a)t)dt.

0

66

The Mellin Transforrn

Let F(r) = j1(z), write h for z - a, take tbe summation of the integration of the abave
formula with respect to [a, a + hL [a + h, a + 2h], ... , (a + (m - l)h, a + mh], we haver + F(x)dx
a

mh

Ja

F(a)
F(a + mh)=h[-2-+ F (a+h)+ ... +F(a+(m-l)h)+2]+ ~ (-li~~~h" [FC't-l)(a + mh) - Ft't-l l (a)J + Rn,
withh 2n +1

r

1

rn-I

Rn := (2n)! Jo B2n (t) ~ F(2n)(a + hs + ht)dt.
We call this formula the Euler formula. We can further simplify the formula by introducing
periodic functions Pn(t) with the period 1 as folIows:

Since

we have

Rn =h'n+1 im P'n(t)Ft'n)(a + ht)dt=- h'n+' im P'n+l(t)FC'n+1)(a + ht)dt.
A natural question is to ask how fast the errar term Rn goes ta O. Ta answer this, let UB
cansider the period function Pn(t). Since it is a function with the period 1, aodfor 0 ::5 t < I, we see that P2n(t) is an even function. SocoP2n(t) =L

G.I: cos(2k1rt),

.1:=0

with

1
1

ao =

P'n(t)dt = P'n+I(l) - P'n+1(O) = 0,

at =21 P'n(t) cos(2brt) dt = (_I)n+1 (2k~)'n'1

Bi

Chapter 104.
for k ~ 1. We may do the same for P:l o+l (t). In summary, we have
2n(t) =(_1)0+1 ~ 2 cos(2k1Tt) "In >_ 1
L- (2k1r Fo

2.n+l(t) =(_1)0+1 ~ 2sin(2,brt) "In >_ O.
L- (2br)2n+lPP.i::::l.i::::lEspecially, we have the estimates(6)

Example. Let F(x) = eh:, a = 0, m = I, h = I, we have

Thus

t

et _ 1t= 1- '2 - E
Tl(-1).i:BJ: 2t 2n +2

t .i: + et _ 1(2.1=)!

1
1

0

P2n+l(s)e"ds,(7).i::::lwhich iB the fini te Taylor expansion of the fundion t/ (e t -1). Note that there is no restriction
on t now (compare this with the equation (2».

§I.4.2. Gamma Function
Here we list the basic properties of the Gamma function which are needed for our own
interests.
The Gamma function r( 8) is the function which iB given for Re( z) > 0 by the formula

( 1)
Since

r(z + 1) =/0'" e-'t'dt=[-e-'t'J:~:i" + z /0'" e-'t,-ldt = zf(z),we have that

1

r(z)

= (z)n f(z + n),

(2)

68

The Mellin Transformwhere (z)n := z(z + 1) ... (z + n - I), andf(l) =

loo

e-'dt = 1.

(Here, the reader should know that the Gamma function is a natural generalization of n1.)
Thus r(z) is a meromorphic function with the simple poles at z
0,-1,-2, ... , -n, ... I
and wher~ the corresponding residue at z
-n is

=

=

Resz=_n(f).
f(z+n+l)(-lt= hmz__ n(z + n)f(z) =(z )Iz=-n = - 1 - '
n.

(3)

n

Using the fact thate -f = I'IITln-oo (1 - -t )n ,n
we have, for large n,

1

() 1

f z -

n
n( 1- -t )" t z -dt =[e- I - ( 1- -t )n] t z -dt +
o
n
ton
t

1

00

n

dt.
e- t t z t

Obviously, the Betond term has the limit 0. For the first, let us look at e- l - (1- ~)n. Since
for 0 :5 y < I,Thuat

(1 + _)-n ~ e- t ~ (1n

t
-t.
n

Now by et ~ (1 + t/n)n,t)" =e -t[ l-e t( 1-t)"] :5e -t[ 1- (1 -(.I)"]
O:5e _, - (1 - - .
nnn2Hence by the fact that (1 -at ~ 1 - nn for a E [0,1], we have2t" :5-e-.
tt
ü:5e- t -(1--)
nn

In particular, the first limit goes toThat ia
with

( 1"(

An z) :=o1 - -t)n tzdt-.
ntChapter 1.4.

69

On the other hand, if we let t = nT ,=n.ln(n - 1) ... 2.1[I ~+n-ldT
z(z+1) ... (z+n-1)Jon!
z=(Z)(n+l) n.
Hence by the fact that

(resp.

1

o

n.l

n

= e.l 1nn =exp{z(ln n - E m)] II ez / m ,)
m=1

m=1

we bave the following

Euler's Infinite Product Expression.

.f(z) = ! ft[(1 + !-)-1(1 + .!.)z]
z

n

n=1

n

(resp. Weierstr88s' Infinite Product Expression1-- = ze"Y.I
r(z)

rr +
<Xl

[(1

0=1.

!- )e-':/"]nwhere 7 := lilIln_<Xl {~=l ~ - Inn} is the Euler cODstant.)

Remark. The right hand side of Euler's product formuIa ie well-defined for all z in C,r(z).

80 we may take it 88 tbe general definition for

As a consequence of Weierstrass' infinite product formula, we know that

Hence

1T

r(z)r(l - z) = . (sm 1t'Z)

70

The Mellin Transform

andAnother very irnportant property of the r function is the following

MultiplicatioD formula.
1
2
n - 1
r(z)f(z + -)r(l + -) ... f(z + - )nnn= (211")(n-1 )/'2 n l/'2-nz f( nz).
In particular, when n=2, we have

Proof. For this, letnnzeP(z):= n r( nz )rr + -).
r

n-1

r(z

n

r:::O

Then, by the limit formula, we obtain
1

ljJ(z) =nnz- lilI1m_oo[(rn - l),]nrnnJ+t(n-l)nnm(n~ _ l)!(nm)nz.
[(m - 1)!]n m Hn-1)nnm-1=hlI1m_oo(nm - I)!
Thus tJl(z) is independent of z. Now let z = t, we have
n-1

tJl =

r+1

r:::O

So
</>'2

n-1

r

n-l

r

II r ( -n ) = TI r(-)n = TI r(1- -).
n
r=1

r:::l

r
r = 1I"n-1 II (am-)-I.
. 1l"r= II r(-)r(l-)
n
n
n
n-I

n-I

r=1

r:l

Hut bywe have

n-l

n =

0-1

n-1

II (1 - r 27fir /") = II r'lfri/n( -2isin 1T:) = 2n - II sin 11":,
1

r:::l

r:::1

r:1

Chapter 1.4.

71

which completes the proof.
We end this section by deriving the Stirling formula.
Let

r/(z)

d

rp(z)

= dt 1n r(z) = r(z)'Then , by the functional equation, we know that1

rp(z + 1) = rp(z) + -.
z

Therefore1n-llP(z + n) = lP(z) +

I: .
z+r
r=O

On the other hand, by Weierstraas' infinite product formuIa, we havelP(z) =- l' -

1

00

1

Z

n=1 n

1

-+ L(- --)
z

+n

1

m

1

Z

n=1 Z

+n

=- - + liIIlm_oo{lnm - L: --}.
So r' (1) = lP( 1) = -1. Furthermore, by
In m =

100

dt(e- t _ e- mf ) _ ,towe have

Now sinGe the integrations of the terms containing e- mf go to 0, we have -fOO e- fe-,flP(z) = Ja (-t- - 1 _ e- f )dt.
When z= 1, we have

1

00

1

and

1

= a {1- e

1-f --]e-tdt ,t

72

The Mellio Transforrn

Hut!p(z) =10 Z +00 1

1(-_)e-ztdt
o t
1 - er11= In Z - - 1 +
2z

00
(

0

-21 + -t1 - -1 --1)et e _, t dt.

Thus, integrating with respect to z from 1 to z, we obtain1

In r(z) =(z - 2") In z - z + 1+ [00 (~+ ~ _1 )[e- t _ e-~tldt.
1 - e- t
t

1
2

dt
1
)e- t - = - In (21r) - 1.
t
2

Jo

2

t

1 +- -

1

Now we claim that
I :=In fact, let z =

00

o

1

(-

t. 1 - e- tt and we have
1
1
I - J = - In 1r - -

2

where

2'

.-1 - + - OO

J .-

o

(1
2

1

1 )e
_t/'Jdt-.

1 - e- t

t

t

Also

so

I - J =

00 1
-t/'J
dt
(_ _ e)e-t/'J_.
t

1
o

Thus

J=

1 - e-

t

t

CO 1

1
o

1
e-t/'J dt(_e- t + _e- t - - - ) - .
2

t

t

t

Integrating by parts with respect to the last two terms, we have

- - eJ -

t

t 2

e- /

-

11

Ioo
0 --

2

t

0

00
[-'

e

-e -1/'J] -dt
t

1 1=- -21 - -In-.
2 2
Therefore,Iandlnr(z) =[(z -= 2"1 In (21r) - 1
1

1

2") loz - z + 2"10(211')]_ [CO ( ! + ! _

Ja

2

t

1 )e- lt dt .
1 - e- tt

Chapter 1.4.

73

Furtherrnore, by the resuit at the end of 4.1, we know that

Hut

aod

1
1

I o PJn+l(z)e-tz

4dzl :5 (2 1rPn +1

1
1

0

e

-tz

4
e -t - 1
dz = (211" )2 n +1
-t .

So, we have1
1
10 r(z) =(z - 2) Inz - z + 21n (21r)

+L
n

(

- l),.-IB,. z-2,.+1 + o(z-Jn-l).

,.=1 2r(2r - 1)
Then, by putting z = n, aod using r(n + 1) = n!, we have
Stirling's formulll1

1

In (n!) - n In n - n + 21n n + 2'ln (2/f) + 0(n- 1 ).

§I.4.3. Riemann Zeta Function
In thia section, we consider the famous Riernann zeta fuoction. Tbe reference here is[T 51].
Tbe Riemann zeta functiou, (Q(s), ia defined by
00

1(Q(s) := '"'L...J n'n=l

for Re( s) > 1. Since each integer n bas a unique factorization as a product of prirnes, weknow that.

(Q(s) =

II(1 - .!. )-1.
Pp'

The restriction to Re(s) > 1 in tbe definition depends on the fact that, for any 6 > I, theinfinite surn and product are absolutely convergent when Re(,,) > 6. On the other hand, we
have the following

74

The Mellin Transform

Theorem. (Q(s) haa a meromorphic continuation on the whole complex plane. On the=complex plane, this function is regular for aB 8 except 8
I, where there is a simple
pole with residue 1. Furtbermore, it satisfies the functional equation€(s) = €(1- s).
Here

Proof. (Riemann) (I) The meromorphic eontinuation: This is baaed on the foBowing
fundamental formula
1 (CO x' dx(Q(s) = r(s) Ja e~ - 1 -;for Re(s) > 1. In fact, for Re(s) > 1,

/00 x'e- nzdx =....!... /co y'e- lIdy = f(8).
x
n' Jo
y
n'

Ja

Rence, by the absolute convergence for Re(s) > I, we have

100

f(s)(Q(S)= E a z'e00

n=1

nz dx
_
l:

1 E
00

x' 00 e- nz _dx
a
l:
n=O

=

Now consider the integral[(S):= {:,-1

Je e - 1dz

1

00

z' dx
= a -z--e - 1l:,where the contour C starts at infinity on the positive real axis, encircles the origin onee inthe positive direction, exeluding the points ±271"i, ±471"i, ... and returns to positive infinity.
Here z,-1 is taken aB e(,-l)log.l' when the logarithm is real at the beginning of the contour.
(Thus Im (log z) varies from 0 to 211" round the contour.)Take C as the real axis from 00 to p, the cirele Izi
with 0 < P < 211". On the eirele,=p, and the real axis from p to00

Iz,-11 = e(Re(" )-1 )10 8I.1'I-c arg" .s; IzlRe(, )-1 e 2:.- j 'I,
andle" - 11 > Alzl·
Rence the integral round this circle tends to 0 with p if Re(s) > 1. But if p - 0, we haveI(ß)=-

1

00

z"

dx

1

--+ a
er - 1 Xo=(e 2ri , _ 1)f(8)(Q(s)
21ri e J'i'=[(1- s) (Q(S).

00(e 2:'- i ), -1 z" dz-xer - 1

Chapter 1.4.
Hence

75

(Q(S) = e-~Ür(~ - 8) 1(8)211"1for Re(s) > 1. On tbe other hand, the integral l(s) ia uniformly convergent in any finite
region of the s-plane, and BQ defines an integral fundion of 8. Hence the formula provides
the analytic continuation "f (Q(8) over the whole 8-plane. In this way, we see that the only
possible singularities are the pole of r(l - s), say, s
1,2,3, .... But we know that (Q(s)
ia regular at 8 = 2,3, .... So tbe onIy possible singularity is a simple pole at 8 = 1. Since

=

1(1)=
and

r(l - s)tbe reaidue at 8r

Je ~=21ri
e - 1

1= ---1+ the regular part,s-= 1 is 1.

(11) Tbe functional equation. To deduce tbe functional equation, take tbe integral along
the contour en consisting ofthe positive real axis from infinity to (2n+ 1)1r, tben round thesquare witb corners (2n + 1)1r(±1 ± i), and finally back to infinity along the positive realaxis. In tbe region between tbe contours C and Cn tbe integrand has poles at the points±2'1l"i, ... ,±2n1fi. The correaponding residties at 2m1ri and -2m7l"i areSo, by the residue theorem,

Now let Re(s) < 0 and n - 00. Tbe function l/(e' - 1) is bounded on en , and z·-l =O(lzl Re (')-l). Hence tbe integral round en tends to 0, and we obtain

From here, by tbe properties cf the Gamma fUDction, we easily bave tbe assert.ion.
With above, we may also obtain the va.lues cf the Riemann zeta function on tbe positive
integers. In fact, if n is an integer, the integrand in I(n) is one valued, and I(n) ean beevaluated by tbe residue theorem. Therefore

76

The Mellin Transformfor each positive integer m. Now by the functional equation of the Riemann zeta function,
let s == 1 - 2m and we have

( (2m)
Q= 22m-11r2m (2m)!'
Bm

Another minor applicatioQ ia as follows: By tbe functional equation, we know that-(q(1- s)11f/(8) (q(s)== -log21T - -1I"tan-511" + - - +-(Q(I- 8)22f(s)(Q(8)'

But in the neighborhood of S == 1,111211" tao 281'1'" == - (8 _ 1) + 0(18 - 11),r'( 8)

f'(I)f(8) = f(l) + ... = -..,. + ... Iand-1/(8 - 1)2 + k + ...

(q(S)

1

(Q(ß) = 1/(8 - 1) +..,. + k(s - 1) + ...
where k ia a constant. Hence, making 8 -=- 8 - 1 +..,. + ...

I

1, we have

We end tbis seetion with the following remark. The Riemann zeta funetion is very
important in number theory, complex analysis, etc. One reason is that there are two expresaiona for it: One ia as a sum, while the other ia aB a product. Especially, one may
study primes by using the Riemann zeta funetion. There are many conjectures related to
the Riemann zeta function. The most famous one ia the following

Riemann Hypothesis. All the non-real complex zeros of (Q(s) lie on the line Re(s) ==12 •§I.4.4. Mellin Transforrn
The Mellio transform comes originally from the Mellin inversion formula, which coonecta two functiona f( z) and :F( 8) by the relations:F(8) =

00

10o

dx

f(x)x J - ,X

1f(x) = ~
1rt

l

a ooi

+

F(s)z-'ds,

a-ooi

where a ia given areal number. The simplest example of this isf(x) = e-1:, :F(5) == f(5).

77

Chapter 1.4.
From an earlier section, we also have the example with

f(x)

1
=--,
e
1
Z

F(8)

-

= r(s)(Q(s).

The importance of the above formula is that after such a. transform, we may make acontinuation of a function, which is originally ooly defined in a restricted region.
Let /(t) be a continuoUB function of t > 0, Buch that
Mt. For t -- 00, f(t) decaye exponentially. That ie, /(t) is O(e- ct ) with cE R>o.
M2. There is n, for t -- 0, t n /(t) ie Coo. That ia, /(t) has an asymptotic expansiono/(t) =

l: Get e + O(t) .
.c=-nThen the Mellin tra.nsform of f(t), denoted as M[J)(s), is defined to be the complex
function
1 [00dtr(s) Jaf(t)t' t'

The basic properties of M[f](s) are in the following

Proposition 1. (a) M[fJ(s) converges for Re(8) > n.
.
(b) There ie a meromorphic continuation of M[fJ(s) to the whole complex s-plane.
(c) M[IJ(s) is holomorphic at 0, and hence it makes sense to talk about M[IJ'(O).
Proof. We divide the integration into three parts: [0, 6], [6, N], and [N,ooJ for 0 < 6 <N < 00. The fact that the integrate for each part is convergent just cornea from the above
conditions. Hence we have (a). For the proof of (b), we UBe the same method as in lastsection by using the contour C. In this way, we get the expression that

Hence we have (b). Now (c) ie a direct consequence of the above expression. Sinte the ooly
possible singularities are that s = 1,2, .... Therefore s 0 ie a regular point for M[j](z).

=Example. Let f(t)= e->.t, then we have M[e->.t](s) = ..\-, and
00

10o

dt
e->.t - = -Iog..\,twhich is equal to M[e->.t]'(O). Motivated by this fact I we will also denote M[jJ'(O) by.r; f(t)df, even through the integrate may not really exist as what stands.
Now we give a precise expression for M[IJ'(O).

78

The Mellin Transform

Proposition 2. With the same condition and notation aB above, if

f(t) =

L aJ:tJ: + po(t),
J:~O

then

Pronf. In fact, by our condition, we haveM[f]( ) _ 1 [I () ,dt '"' an (I n+,dt
s - f(8) Jo Po t t T + ~ f(s) Jo t
T1

+ f(5)

M[fl'(O)Joodt
1· f(t)t' T·=[Po(t)(r~:»~=o ~t +~(r(.);:+.»~=o

j oo

t'

dt

+ 1 f(t)( f(8) )8=0 T'
Now the conclusion comes from tbe following facta:t'(r(8) )~=o = 1,1)1(
1,
f' (1)'(( f(s)s,=0 = r(8 + 1) )'=0 = - f(1)2 = -f 1),
1( f(s)(n1 + 8) )',=0 = ;;.
Now we turn to applications of tbe above idea. Thc final aim is to deal with Laplacians.
By Lemma 3.2, we ean show that the eigenvalues of a Laplacian over a eompaet manifold
are non-negative numbers and they are discrete.
Recall the following

Fact. Let A be an endomorphism of an n-dimensional Euclidean vector space V withreal strictly positive eigenvalues

Then

79

Chapter 1.4.

But for infinite many positive numbers Aj as the produet may not be eonvergent, how
we ean offer a reasonable definition for
Aj? To do so, let us look at Stirling's formula. So
we may take the finite part of the expression of n! a.s the definition for 001. That isn001 := v2'1r

I= exp( -(Q(O)).

With this in mind, we introduee the folIows:
For an inereasing sequenee of positive real numbera

the zeta function assoeiated with this sequenee is defined by(>.(8) := LA;;'.
02:,1

Suppose we have the following eonditioDS:

Zl. (>.(s) eonverge8 for Res ::> o.
Z2. (>. (s) has a meromorphie continuation to the whole s-plane.
Z3~ (>.(s) haa DO pole at s0.

=

By definition, we set

rr00

An := exp( -(~ (O»~n=1

With an iocreasing sequence as above, we define tbe associated theta function asco8>.(t) :=L e->.·t.
n=1

Lemma. Suppose AisBuch that
81. 8>.(t) converges for t > 0;
82. For t --+ 00, 8>.(t) decays exponentially,
83. For t --+ 0, t n 8>.(t) is Coo.
Then we havesatisfiea Zl, Z2, aod Z3. In particular,r;lj(~(O) = Ja

dt
6>.(t)T'80

The MeHio Transform§I.1.4.5. Another Construction oC
Classical Bott-Chern Secondary Characterisiic Forms

1.4.5.a The Double Transgression Formuls
Let B be a eonnected eomplex manifold and let J E End (TB) be the eomplex strue tureof B. Let·o- Eo ~ ... ~ Ern - 0
I

be a holomorphic ehaio eomplex of finite dimensional eomplex holomorphie vector bundles
on B with hermitian metrics Pj on Ei for 0 ~ j :5 m. Set· EE· E- .-lI:>.
odd EJ'
E+l1:'lEE + .-l1:'l.
'-WJ even JI
'-WJ
W
.

Let N be the number operator on E which defines the Z-grading of E, Le. N is themultiplication by j on E j • Similarly, let T be the number operator defining the Z2-gradingof E, i.e. T = ±1 on E±. Also let v· be the adjoint of v. For a E C, set va := av + äv· andV := Vi. Theo, if V = V' + V" is the canonical eonnection of (E,p := EBPi) with V', V"
the holomorphie and antiholomorphic parts of V, V + va is a supercoonection on E.
In tbe pr.oof later, we have to use the following properties of the number operator N,
which may be easily ehecked:[V, N] = 0,[v, N] = -v, [v·, N] = v·, [ä, N] = -B, [ä·, N] = -ä·.

c

Let P be the su bspace of the smooth sections of I\T B, w hieh are sums of the differen tial
fonns of complex type (p,p). Let pt be the set of G, o-exact smooth forms in P, i.e. thesethat cao be wri tten as 8 B TJ+8 B TJ' wi th TJ, TJ' amooth forms on B. Then if Adenotes the vector
subspace of I\7CB@EodE generated by smooth seetions of I\P,fT B@Hom(Ej , Ei+p-q) forall p,q,j ~ 0, A ia an algebra and Tr,TJ E P for any TJ E A.

c

Theorem.. (1) For any a E C, the differen tial forrIlB

are in P aad ooly depend on la I.
(2) Tr [exp(-(V + va )2)] ia closed.
(3) (Double Transgression Formula.)
j

:0
:ä

Tr,[exp(-(V + V a )2») = - aBTr,[vexp(-(V + V a )2)];
Tr,[exp(-(V + V a )2») = - aBTrj[v·exp(-(V + V a )2)Ji

Tr, [au exp( -(V + V a )2») = - 8 B Tr [N exp( -(V + V a )2)];
Tr, [au· exp( -(V + V a )2)] =a BTr [N exp( -(V + V a )2)].
j8

Chapter 104.

81

Therefore,

Proof. 1. From the definitions, we havefor 0 E R. Thus Tr,{e.xp( -(\7 + VO)2)) and Tr,{N exp( -(\7 + VO)2)) are radical functionsof a. On tbe other hand, by a direct calculation,

So, by definition, we have (1).
(2) That Tr, {exp( -(\7 + VO)2)] is closed is a standard result in the superconnection
formalism, say Prop. 1.3.2.

(3) Tbe last two relations are direct consequences of the formal relations. For the firsttwo equalities, we proceed as follows: On B xe, tbe form Tr J {exp(- (\7 +da +da +V O )2))
is closed. It ie equal to Tr, [exp( -(V + VOfl - da v - da v· )]. Thus by Duhamel's formula,it becomes

iä

:lJ

Tr, {e.xp( -(V + V lI )2)] - Tr, [v exp( -(\7 + V ll )2)] da- Tr,[v·exp(-(\7 + VO)2)] da + erdada,
witb er a differential form on B, since Tr, vanishes"on supercommutators. Now our first two
relations may be deduced from the fact that this last combination of form is closed under
8 B + dala and fJB +Finally, a simple degree counting argument of the following
relations will complete tbe proof:

dälä.

dBTr, [N exp( -(\7 + VO)2)]= Tr,{\7 + V lI , N exp( -(V + V lI )2))
= Tr,([V lI , N] exp(-(\7 + VlI )2)J
= Tr, [( -av + av·) exp(-(\7 + VO)2)].
Here we use tbe properties of number operators listed befare tbe theorem.

82

The Mellio Transform
I.4.5.b. Asymptotic Expansion

For any tl ~ 0, let Au := "V + y'UV. Then, a.s in the Mellin transform, if we have theright asymptotic expansion, then
1 1+~~
(E'I/',(8) := r(s) 0tl'Tr~ [N exp( -A~)]-;;-

is a well-defined element in P for 3 E C, Re(s) > 0, and <E.,p.(S) has a unique meromorphic
continua~ion on the whole complex plane such that it is holomorphic at O. Thus by theabove theorem, we know that10+ Tr, [V exp( _A~)]ul/2 duu = _(IIB - äB)(e,p. (0)00Tr, [exp( - '1 2 )] = _aB aB <~"p. (0).
Hence, if we define

ChBC(E.,p.) := [2ri](k"p,(0),axiom 1 is valid. Axiom 2 is trivial. And axiom '3 is a consequence of tbe fact tbat, in the
splitting Ca&e, v commutes with "V aod vv· + v·v = Id.
Now we have to show that <E"p.(S) makes sense as in the Mellio transform. For this,we check the condition for the Mellio transform. First, look at the asymptotic expansionof Tr,[Nexp(-A~)] when tl - 0+. Even through at this stage, we may affer a simple
method to deal with it, but in order to explain tbe basic idea of tbe proof for the infinite
dimensional case later, we prove it by a certain concrete calculation. Also, we will go widely.
More precisely, we have tbe following
Theorem.. When u - 0+ ,J:

Tr,[Nexp( -A~)]= L: #Jj ~ + o(ui:)
j;.-l

with
#Jo = Tr,(Nexp( - V 2 ).

Here 0 is uniform for aoy compact subset of B.
We da this by introduce a new parameter b. Tbe key point for this is the following easy
generalization of the theorem in subsection a:

Proposition 1. For aoy a, b E C, we have

8 B Tr J [exp( -(V' + V II )2 + bN)]
=bä Tr,[v·exp( -('\7 + V lJ )2 + bN)];
äBTr, [exp( -("V + VlJ )2 + bN)]=- baTr,[vexp(-(V + V )2 + bN)].
II

Chapter 1.4.

83

Proof. Indeed, we may deduce the reault from the degree counting of the following
obviouae relation:dBTr, (exp( -("\7 + V a )2 + bN))

=Tr, (["\7 + va 1 exp(-("\7 + V a)2 + bN)])

c:bTr,([V a , N]exp(-("\7 + V a )2 + bN)])=bTr,« -av + äv·)exp( -("\7 + V a )2 + bN)).

Now let O'I(a), 0'2(a), aod 0'3(a) denote the differential fOrIns on B defined by

a

a

a

Tr, [N exp( -("\7 + da Ba + dä Ba + V )2)]
=Tr, [N exp(-("\7 + vayz)] + 0'1 da + 0'2 dä + 0'3 da daaccording the Grassmannian degree on C. (In general, if 11 = '10 + 111 da + '72 da + Tf3 da dä
ie a decompoaition of 11 according to tbe Graasmannian degree of C, we mayaiso denote Tf3
by [11l da dll.) On the other hand, eince
Tr,[exp(-("\7 + V a )2 + bN)]

is a sffiooth function of 1al2, then by Duhamel's formula, there exists a smooth form
ßl(x,a,b) for (x,a,b) E B X C X C, which depends smoothly on lal 2 and is such that
Tr 8 [exp( -(V' + V a )2 + bN)] = Tr, [exp(- "\72 + bN)] + lal 2 ßl (x, a, b).

Proposition 2. With the eame notation as above, we have

Proof. By Duhamel's formula, we know that in the expression of the Grassmannian
degree on C,

Tr,[exp(-(V+ da

8
{)
G
+dä
aa
aä + V )2+bN)L

the form which appears as the coefficient of da is given by-Tr,[vexp(-(V + V G )2 + bN)],
l.e.

84

The Mellio Transformby the above proposition for ab # O. On the other hand, forrnally, by the fact that

we have

a" [
(
8
8
tl2
]
8b Tr, exp( - V + da 8a + da 80. + V ) + bN) "=0
1 {)2

[(

8

(

_8

tI

2

]

= 2' {jb2 bTr, exp - V + da aa + da {ja + v ) + bN) 6=0a{j= Tr, [N exp( -(V + da Ba + da Ba + V tl )2)].
Hence,

aal(a) = aB ~ :22 Tr, [exp( -(V + V tI )2 + bN)]b=O.
b
Let a = 0 and we have-B 1 {j2{j22' {)2b Tr, [exp( - V + bN)]b=O = O.

So
{)2
[(
tI 2
2]
aal(a) = aoB 2'1 {j2b(Tr,exp -(V + V ) + bN) - e.xp(-V" + bN) )0=0.

N ow by the definition of PI (:r I a, b) I we have the first relation. The proof for a2(a) is similar.

Proposition 3. For any (al b) E C X C,{)

{ja (Tr,[exp( -(V + V )2 + blal N)]) = _dB (Tr" [v exp( -(V + V tI )2 + blal N)])
2

2

tI

+ bä Tr,[exp( -(V + V )2 + (blal 2 + da dä)N) - ada v - 0. da V·)]dtldll.
tI

In particular I

8a8a laI 2 (Tr,[Nexp( -(V + V )2)]) = _dB ab (Tr, [v exp( -(V + VO)2 + blal 2 N)])b=O+ o.Tr,[exp( -(V + V )2 + da da N - a da v - a da v· )]dtldll.
tI

tI

=

Proof. Differentiating the first relation at b 0, we easily have the second relation.
For the first, we know that the LHS is the coefficient of the da in

85

Chapter 104.
But, by the properties of the number operator N, we know that the last expression ia

8 8
8
2

Tr, ([da 8a' -("\7 + da 8a + VO)2 + bj a l N]exp( -("\7 + da 8a + VO)2 + blal 1 N))=Tr,((["\7 + VO, ("\7 + da :a + VO)2] + blal N)exp(-("\7 + da :a + vo)2 + blal N))
22= Tr,((["\7 + V 4 , (V + da :a + V 4 )2 - blal 2 N] + blaI 2 [V 4 , N] + bfl N da)aexp( -("\7 + da 8a + V 4 )2 + blal N)]
28= -dBTr,(exp( -(V + da 8a + VO)2 + blal 2 N))+Tr,(blaI 2 ( -au + flv·) + ba N da) exp( -("\7 + da!.... + V 4 )2 + blal 2 N)).

aa

Here, 88 before, we also use the properties of the number operator N. On the other hand,taking the factor of da in

8

.

8

Tr,([da 8a' -("\7 + da 8a + V
we have82

4{Ja Tr, [exp( -(V

)

8

1

+ blal N]exp( -("\7 + da 8a + V )2 + blal 1 N)),4

+ v4f2 + blal 2N)] =- dB (Tr,[v exp( -(V + V 4 )2 + blal 2 N)])+ ba Tr. [exp( -(\7 + V )2 + (blal 2 + da da)N)4- av da - (-av - av·) da + N-da da)]doda.
Thus the assertion comes from the facts thatav da + (-av + aV·) dä =av (da - da) + a v· da;(da - da)da =da da.
With above propositioDs, if we let a = u 2 for u ~ 0, we easily see the following

Corollary. With the same notation 88 above,:u[UTr,[NexP(-A~)]]u=o =Tr,[Nexp(-V 2 )].
Flom this we e88ily see that as u - 0+,
Tr,[Nexp( -A~)]
has an asymptotie expansion starting from u-I. Hence, we have the theorem stated at
the beginning of this subsection. Moreover, in the definition of (E, we do have the same
situation as in the Mellin transform. Indeed, for the purpose here, we only need to know theasymototie expansion for the 88BOeiated trace class. We will see later that for infinite dimensional cases the analogue still holds, because we will carefully choose tbe superconnection
and the number operator, which provide the right cancellation.

86

The Mellio Transform
1.4.5.e. A Construction

In this subseetion, we use the results above to give another eonstruction for the classicalBott-ehern secondary charaeteristic forms.
In order to use the Mellio transform, we still need to discuss the behavior of
Tr,[Nexp( -A~)]
when u - 00. For this, we introduce the basic assumption that (E, v) is acyclic. With
this asaumptioo, V = (v+v+)1 is self-adjoint aod positive definite. So we may use Duhamel's
formula to deduce the fact that aa u - +00, 'Ir,[exp(-A~)] and Tr,[Nexp(-A~)] decay
exponentially aod uniformlyon compact subsets of B. Therefore, we have tbe following
Proposition and Definition. Let (E., p.) be a complex of hermitian vector bundles
on B.
(1) For ß E C, Re(s) > 0, let
1 f+~~(Ct.,p.)(ß) =: r(s) Ja
u''Ir,[NexP(-A~)]-;;-.

Then (CE.,p.)(s) E P is well-defined.
(2) There exist5 a meromorphic extension of (Ct.,p.)(s) on the whole complex plane
which ia holomorphie at O.
'Obviously, we know that<u:.,p.)(O) =Tr, [N exp( - \71)]fl~«e.,p.)(O) = Ja [Tr, [N exp( -A~)] + Tr, [N exp( - \72)]-;;-

1

+~

du

Tr,[Nexp(-A~)]- - r'(1)'Ir,(Nexp(-Y'1)].
u

1Theorem.. Suppose E. ia acyclic, thenIn particular, we getchBC(E.,p.) = [21Ti]«e.,p.)(O)

Later we will give a similar construction for the relative Bott-ehern secondary characteristic objects. At that moment, since the situation is infinite dimensional, we have tochoase the corrett superconnection and number operator in order to have certain cancellation, and henee, a suitable asymptotic expansion.

87

Chapter 1.5.

Chapter 1.5
Local Family Index Theorem
In order to prove the existenee of relative Bott-Chern secondary eharacteristie fonns
with respect to a smooth morphiBm, we need to study the Ioeal family index theorem:
Roughly speaking, the construetion of relative Bott-Chern seeondary characteristie fonns
for smooth morpbiBms iB very similar to the oue in the final seetion of the last ehapter.
Tbere we used the local index theorem in tbe absolute situation by a discussion about
the assoeiated heat kerneIs. Here we will UBe the heat kernel associated with a suitable
generalized Laplacian.
To do so, we now meet certain problema. Tbe most important one is about the eonvergence: For the loeal index theorem in the absolute situation, we know that the heat kerneis
assotiated with a generalized Laplacian haB an asymptotic expansion when time goes to
emaIl; in the relative situation, if we choose a 'natural' connection, the associated seeondorder differential operator has a kernei, aod usually, tbere is no good asymptotie expansion
for it wben the time is email. So we need to modify this eonnection. Now, by the proof ofthe loeal index theorem in the absolute situation, we see that the Lichnerowiez formula isa crucial point. Therefore, we know basieally how to make this modifieation. (In practice,
we introduce the Bismut superconneetion.) The reference here is [BGV 92].

1.5.1. The Bismut Superconnection
Let 1f' : M -+ B be a family of oriented Riemannian manifolds (M~ Iz E B) with a
Riem8llnian metric 9M/B on each fiber M~, and let E be a vector sheaf on M such that
E~ = EIM, is a Clifford module for each z E B. We denote by T(M/B) the bundle ofvertieal tangent vectOf8. We 8BSume tbat the bundle M / B possesses the following additional
structure: a splitting TM
THM EIl T(M/B), so that the Bubbundle THM is isomorphie
to the vector bundle 7t·TBj and a eonnection V M / S on T(M/B). Let P be t~e projection
operator P : TM - T( M / B) with kernel the chosen horizontal tangent space T H M. For X
a vector field on B, denote by XH the horizontal lift on M. Choose a Riemannian metric
9B on the base B and puB it up to TH M by means of the identifieation TH M ~ 1r-TB, wethen obtain an inner product on the bundle TH M, whieh we eall a horizontal metric. (Weusually make use of a loeal frame eO, of the vertical tangent bundle, and a local frame J° of
TB, with dual framee e' and JO.) We form the total metrie 9 9s EIl 9M/B on tbe tangent

=

=

88

Local Family Index Theorem

bundle TM of M. Let V g be the Levi-Civita connection on TM wi th respect to .this metric,
and define a connee tion V M / B on the bundle T( M / B) by projecting this eonneetion.

The first result is the following
Proposition 1. The eonnection VM/B on T(M/ B) is independent of the metrie 98
on TB used in the definition.
Proof. We first recall the following basic formula for the Levi-Civita connection

YL Z) - ([Y, Zl. X) + ([Z, XLV)+ X(Y, Z) + Y(Z, X) - Z(X, V).

2(V'~ Y, Z) = ([X,Thua, if X, Y aod Z are all vertieal, the right hand side redueea to the Levi-Civita conneetionon the fibers for the vertieal metrie gM/B. On the other hand, if X ia horizontal, but Y and
Z are vertieal, then [VI Z] ia vertieal, so that ([Y, Z], X) vanishes, and we see tbat

Prom this formula, it ia elear tbat ooly the vertical metric gM/B and the vertieal projection
P are used to define V~ / B for -X horizontal. Thia completes the proof.
Next we eonstruct a new eonnection on TM, that is

Note that if we replace 9 by tbe resealed metric ugB ® gM I B, wi th u > 0, then V' B does
not change, aod neither does \7$. Uaually, the eonnection V'$ has a non-vanishing torsion,even through it preserves the metric. The next proposition shows us the relation between
V'$ and v g .
Proposition 2. There exists a three·tensor w on M such that, for aH X, Y, Z E

CCO(M, TM),

(Vk Y Z)g = ('V~ Y, Z)g + w(X)(Y, Z).
J

Furthermore w E A l(M, ,,2T· M) is defined by the formulaw(X)(Y, Z) :=S(X, Z)(Y) - S(X, Y)(Z)1
1
1+ 2(fl(X, Z), Y) - 2(fl(X, Y), Z) + 2(O(Y, Z), X).
Here the tensor S, which is ealled the second fundamental form, ia the seetion of thebundle.
End(T(M/ B)) ~ THM ~ T-(M/ B) l3l T(M/ B) 0 TffM

89

Chapter 1.5.
defined by(S(X, 9), Z) :=< "V~/B X - P[Z, X]' 9 >for Z E COO(M,THM),X E CCO(M,T(M/B)) and (J E CCO(M,T*(M/B)), while thetensor S1 ia the section of the bundle Horn( 1\1TH M, T( M / B)) over M defined by

S1(X, Y) := -P[X, Y]
for X, Y E COO(M, TH M)l i.e., the negative vertical projection of [X, Y].

Proof. First, since for any 4J E COO(B), if Y is tbe horizontal lift of a vector field on
BI Y(7I"·4J) 1r·(7I".Y4J). Hence, for any vertieal vector X, [Xl Y] is also vertical.

=

Now observe tbat (V~ Y, Z) - ("V~ Yl Z) is antisymmetric in Y and Z, because eacb of
tbe connections "Vi and "VIEl preserves tbe total metric 9 on M. Hence we may check the
proposition case by case for the different situation in which X, Y and Z are horizontal orvertica1. AB an example, if X, Y aod Z are all horizontal lifts from the basel it is easy to
see tbat tbe difference is 0, while w(X)(Y, Z) vanishes. Other CaBeS wbich are very similar,
are left to the reader.
Let u E (0, 1], and let

be tbe metric on r M. Tben gO = lilllu_o9 U is tbe degenerate metric, wbich induces a
metric on r- M and vanishes in the boriwntal cotangent direction T( M/ B)J. C 'r M. Tbefamily of metrics gU ia a powerful tool to investigate tbe geometry of gO. Let gu be the dual
metric on TM so that
9u=U

-1

~

9Bw9M/B

which explodes in the bori2ontal direction as u -O. We call this process tbe blowing-upof the base metric.

=

Let Cu(M)C(r- M,gU) be tbe Clifford algebra bundle, and denote tbe canonicalquantizatioD map from A'r M to Cu(M) by Cu' Thus tbe Clifford bundle Co(M) is tbe
limit of tbe ODe parameter family of algebras bundle Cu (M). Let

be defined bywbere Q e A'Jr; M andee 1'; M. Tben

90

Local Family Index Theorem

Therefore, for the orthogonal frame of T- M of the form {ei} U {r~} I we have

~TU(eiei)e.l: =SiJ:eJ _ oi.l:e i ,2!TU ( ei JO)e? =6 ij Ja,
2,U(Jo fß)e i =0,!TU(eiJa)JP
2=- u6o Pei,~TU(Ja ffJ)r =u(6°'Y JP - 6P", Ja),
TU(eiei)Jo =0.
Taking the limit aB U - . 0+, we have!TO(eie?)e.l: =6 ilc e;i _ fJibei,
2!TO(e i JO)e;i =6 ij Ja,
2
TO(fO JP)e' =0,

TO(a)fO =0,
for all a E l\'J.r M. Hence, Ta vanishes on T" B.
Thus, if we denote the negative of the adjoint of TU (a) E End (T- M) by Tu (a) forU E [0,1], then12(Tu (O)X, Y),h =< 0, X 1\ Y >,rfor X, Y E CCO(M, TM) and () E 1\ 2 M. Hence if yrM,u, U > 0, is the Levi~Civita connection on TM corresponding to the metric 9u, then we may restate the proposition aboveaB

Proposition 2': '\JM,u= yriP + ~Tu(W).

It follow! that the family of connections \7M,u has a well-defined limit as u - 0, which
we will denote by yrM,O. In particular,

Obviously, we have the following facts: The connection '\JM,u is torsion free. The projectionof yr M ,0 to the bundle T( M / B) equals yr MI B . The restriction of yr M ,0 to each fiber depends
only on the vertical metric gM I Band t he connection on the fiber bundle M / B. We let
yrT· M,u be the dual connection on T- M for u E [0,1]. Motivated by this, we introduce thefollowing discussion:

91

Chapter 1.5.

=

Now let E be a vector aheaf on M such that El
E!M. ia a Clifford module for each z
and suppose that there is a connection V C given on E: whose restriction to each bundle El ie
a Clifford connection. Denote by rr.E the infinite dimenaional bundle over B whose fiber at
z E B is the space COO (Ml , E:l ). Let D = (D l Iz E B) be a family of Dirae operators acting
on the fibers of 1f.E, constructed from tbe Clifford module structure and Clifford connection
on E. That ia, on the fiber Ml , Dl is the composition of the Clifford connection followedby the Clifford action. Introduce the vector bundle E over M by

E:= 1f·(A'r B) ~ E.
This bundle carries a natural action mo of the degenerate Clifford algebra Co(M): theClifford action of a horizontal cotangent vector a E COO (M, TH M) Ü!I given by exterior
multiplication mo(a) !(a) acting on the first Caetor ATiIM in E, while tbe Clifford actionof a vertical cotangent vector simply equala ita Clifford action on E. This Clifford module
will be the main tool in calculating the index of the family operator D.

=

In order to study E, we write it aB the limit of a family of Clifford modules for thebundles of Clifford algebras Cu(M), all constructed on the same underlying vector bundle
E: The Clifford action
mu : Cu(M) - End(E)is defined as follows: For a horizontal cotangent vector a E C oo (M, TH M),

mu(a) := c(a) - u,(o),acting on the first factor ATiIM in Ei while the Clifford action of a vertical cotangent vector
is aimply ite Clifford action on E.

r

There are connections VE,u and V E ,61 on the Clifford module E analogoue to those in
M: Tbe connection V E •6l on E ~ 11". 1\ r B 0 E ie defined by

The connection V E •u is defined by tbe following formula (inspired by the discussion for
VM,U)1
1
V E •u := V E ,$ + -mu(w).
2lL

Proposition 3. For all u E [0, the conneetion VE,u ia a Clifford connection for the
Clifford action m u of Cu(M) on E. In particular, the connection

ie a Clifford connection for tbe Clifford action mo of the Clifford algebra bundle Co( M)on E. The restriction of those connections to each fiber of the bundle M / B is independent of the choice of the horizontal metric 9B used in the definition.

92

Local Family Index Theorem

Proof. Note that since

we know that the connection V E ,$ is a Clifford connection with respect to the Clifford
action m u . Now the first ~wo statements are the consequences of the fact that

Otbers are trivial.
Next we define the Bismut superconnection on 7(.E. We consider that the spaceA(B, 7r.E) of differential forms on B with coefficients in 1r.E as the space ofsmooth sectionsof the Clifford module E over M. The Bismut superconnection B,
B : A(B, 7(.&) - A(B,1r.E)
ia the following Dirac operator for the Clifford module E - M:.- ~ma"E,O
B .L-, 0 Y a .
a

Here m: denotes mu(e i ) or mu(fO), V:,u' denotes V~'U or 'V~~u, and the summation ia
taken over all the orthonormal frames ei aod Ja.

Remark. Remember that the Dirac operator for the absolute situation is given by=

Let B B{o] + B[l] + B[21 + ... , be the decomp08ition of the Bismut superconnection
according to its degree. Then we have the following

Proposition 4. With the same notation as above, the restrietion of B to COO(M, E)
has the expression

Lci'Vf +i

E,O(V; + ~ I)S(ei,ei),!a)) - ~ E LE:°!ßci(n(!o,!p),ei)'a

In particular, B{o)

a<f3

i

i

= D and B[11 = V e . (For the precise definition of 'V.".e, see 1.5.3.)
7f•

Proof. From the definition, we see that
B = LciV~·$ + LcaV~,$ + ~ Lw(ea)(eh,ee)momgmo.
i

er

abe

93

Chapter 1.5.
On the otber band, by a direct calculation , we have

LW(Ca)(Cb' ee)m:mtm~ = 2 L
abc

L(S(ei' ci), Ja)m~ - ~ E(O(Ja, Jß), ei)m~mem~.

a

api

i

Thus we bave the first formula. Then, by comparing the degree , we bave the other two.
We saw that the Lichnerowicz formula is very important in the proof of the local
index theorem, which makes us to UBe the Mehler formula possible. Next we discuss thecorresponding formula for the Bismut superconnection.

Theorem. (Tbe super-Lichnerowicz formula) With the notation as above, we have

Here rMIB denotes tbe scalar curvature, and Fels E A(M, Endc(MIB)(E)) denotes tbe
twisted curvature of tbe Clifford module E.

Proof. By definition, we know thatB 2 -~ '"'[maVE,O mbVE,O]
-2L.J
a ,
b

°

ab

°

-!
'"'[mO mb]VE,OVE,O-2L...t 0' ° abd+ !2L...t"" mOmb[VE,O
VE,o] •
+ '"'L.J ma[VE,O° lmb]VE,O
Ob
0
0
a , b
0

ab

ab

Now we compute each term in the last equation. Since for any () E COO(M, r- M),

we know tbat for tbe orthonormal basis ei , Ja, the first term is equal to - L:i(V~,O)2.
By tbe fact tba.t VE,o is a Clifford connection, we know that tbe seeond term is

L: mömo(V:;- M,Oe·)V~'o = L: mömg < V~- M,Oe ee > V~,o
lt

ab

,

abe

1 "'"'

ae VE,a .

= - 2 L...t mam o [C.,CCl]

+ '"'
VE,o
~ v:-,IBei·.

GetHere we have ueed tbe facts that VM,o agrees with V M1B wben restricted to a fiber M~,
that tbe connection VM,o is torsion-free, and tbe adjunction formula

V~·M,Oeb =L: < V~-M,Oeb,ee > e = - L: < e V~,oee > ee.
b,

C

e

C

Local Family Index Theorem

94

On the other hand, by definition, we have

Therefore ,Furthermore, we know that['V'E,$,' mo(w)}=mo(VlIl w);[mo(w), mo(w)J = ~O([WIW]O);
Also, if, locally, for any A E End(VL we define

A(A) :=

L < ei, Ael >

1

!jL ,

j,l

globally, we get

So, by the fact that

we getLmöm~CvE.Or2(ea,eb)ab=L mömgA(RB(e al eh)) - ~ L m~m~mgmgRahcd + L mgm~FeIS(eal eh).
ab

ahcd

ab

Flom the fact that the antisymmetrization of RB over any three indices is zero, we knowthat the first term is zero. [n particular, we see that

95

Chapter 1.5.
Now the original formula is a consequence of the following standard calculation:
'"'"'
a
b e
d D
L..J mOmOmOmO~"abed

abed

a
bad D
+ '"""
ab b dR abbd
= '"""
L..J mOmOmOmO~"ahd
L..J mOmOmOm
O

abd

tJbd

=2L m gm gRoidi = -2LRtjij = -2rM/B'a~ij§I.5.2. Existence oe Beat Kernels In Relative Version
In this section, we prove that, 8B8Ociated with the Bismut superconnectioD, there exist
heat kerneis. For doing so, we put the problem in a relatively large content.
With the same situation as in the previous section, we have a family of manifolds,
together with the associated atructures. Let V(E) be the bundle of algebras over B whose
fiber at z ia the algebra of differential operators, aod whoee smooth sectioDB are families ofdifferential operators IY, with coefficients (in a local trivialization of M and E) depending
sffioothly on the coordinates in B. Let K(E) be the bundle of algebras whose fiber at z is
the algebra cf sffioothing families of smoothing operators K~. Since K:(E) is a bundle ofmodules for V(E), we may form an algebra from the sums of operators in V(E) and K(E).
We refer to the sffiootb Bectians of the bundle 'P( E) + K:( E) as the 'P-endomorphiBmsof the infinite-dimensional bundle 'Ir.E, and denote by End'P(?r.E) the spate of its smoothsections.

=

As an illustration, first, we cOßsider the situation when B is a point. Let A L~o Ai
be a finite-dimensional graded algebra with identity. Let M be the algebra P 0 A. Thereis a natural decreasing filtration of the algebra M with Mi := Ln:/P 0 Ai. Let:F := Ho + K + :F[+]be in M with Ho a generalized Laplacian, K E K. aad :F[+] E Mi. We define a heat kemel
for:F to be a continuou! map (t,z,y) 1-+ Pt(z,y) E Ez 0E; 0A which is'C i in t, C1. in z
and satisfies the equation

with the boundary condition that for every s E COO(M, E) 0 A,

limt_o1

yEM

uniformly in x e M.

Pt(X, y) s(y) = sex)

96

Loeal Family Index Theorem

Theorem 1. Let

F := Ho + K + F[+lbe an element of M with Ho a generalized Laplacian, !( E K. and F(+] E MI. (In thesequel, we call F an essential generalized Laplacian.) Then there exists a unique heat
kernel Pt(z, y) for F.

==

Proof. First, we assume that A C. Then Jl+} O. In this case, since we know that
Ho has a·smooth heat kernel satisfying eertain strong estimates, it is not aurprising that wecan prove the same things for H = Ho + J( by Duhamel's formula. To eonetruct the heat
kernel for H, we use a generalization of the Volterra series as follows:
Lemma 1. With respeet to any Cl-norm, I ;::: 0, the series
00

Qt :=

L:) _t).t1e-uotHo Ke-UltHo ... Ke-u.tHodu

~.

.c=o
converges to a kernel

q, E Coo(M x M,p~E 0 p;E·).

The SUfi is COO with respect to t and ia a solution of the heat equation

with the following boundary eondition at t = 0: Ir tP ia a Bectien of E, then with theuniform norm,

liITlt_oQt8 = s.
Hence qt(z, y) ia a beat kernel for H. Furthermore, the differenceL( -t).c ( e-uotHo Ke-U1'Ho ... Ke-ut.tHodu
00

e- tH _ e- tHo =.c=ltende to 0 when t -

J~.

O.

Proof. Sirrce K ia a effioothing operator , the operator e-tHoK haa a smooth kernel for
all t ~ 0, andfor some ceostant C(l) depending on l. It follows that , for k ;::: 1,

Thus the series L.c> I converges wi th respeet to the Cl-norm, uniformly for t > 0, wi thsimilar estimates for-the derivative with respect to t. Others are easily to check.

97

Chapter 1.5.

In general, the operators :F and H differ by an operator :Fi+] of positive degree in thefinite-dimensional graded algebra A. Thus by the heat kernels of H the Volterra series onee
more gives a candidate for the heat kernet of:F: For fixed t > 0, define the operator e- IF
byl

e- tF := e- tH +L( _t)1 /1;,

1;>0

where

11

= f e-O'otHo :F[+]eO'I tHo ... :Fl+Je-O'i1tHod<T.
Jt:t.1a

The Bum is finite, since 11; E MI;. aod for klarge, AI: = 0, hence MI: = O. Thus it ia
sufficient to make sense of each term in this finite sumo For this, we need the foJlowing
Lemma 2. Let D be a differential operator of order k. There exists a constant C > 0such that if K is a smoothing operator, for t E [0, T]. with T being a positive real
number, we haveliDe-tB Kill :5 CIIKlh:+1tH
IIKe- Dill :5 CIlKlIl:+l.
Proof. By the fact that there exists a eonstant C( I) such that for tP E r 1(M, E), one
has for t E [0, T],

lle-tH 4>lh :5 C(I)II4>II"the bound IIDe-tH KII, :5 C(/)IIKIII:+I follows e88ily. Using the adjoint, we have the otherinequality.
Now we can complete the proof of our theorem. Obviously, it is enough to show thateach term 11; h88 a smooth kernel. On the simplex .!l,b one of the <Ti must be greater than(k + 1)-1. Since for a fixed rr with (k + 1)-1 < rr :5 1 and a fixed t, the operator e-O'tH h88
a uniformly smooth kernei, it follows by iterated applications of the above lemma that the
op~a~r.
-r: e-O'ltHo-r:.-O'"tHo
e -O'otHo.r[+]' ... .r(+) ...

has a smooth kernel which depends continuously on (rro, ... ,rrl;) E .6.1;. Thus the integral
maIrea sense aB an operator with smooth kernet. The rest is trivial.
We now torne back to tbe relative situation. Let 11" : M -+ B be a family of manifolds
over a base B. Denote by M x .. M the fiber product which is a fiber bundle over B withfiber at z e B being M. x M•. Let E -+ M be a family of vector bundles. By definition, asmooth family of smoothing operator acting on the bundles E. - M. along the fibers
is a family of operatol'8 with kernelk E co:'(M Xli" M, piE 0 pi~).
When restricted to the fiber M. x M~, the kernel k may be viewed as a kernel k~ in
COO(M~ X M. , piE. ~p;E:). Let K(E) be a bundle over B, whose smooth sectiODS are givenby98

Loeal Family Index Theorem

As explained above, A:(E) ia a sub-bundle of Endp(E).

Theorem 2. Suppose we are given a smooth family of generalized Laplacians HZ along
the fibern of M - B, then the eorresponding heat kernel p,(x, y, z) defines a smoothfamily of smoothing operators, that is, a seetion in COO(B, A:(E».

Proof. As usual, arOJlnd any point Zo E B, we ean find a neighborhood on which thefamilies M and E are trivialized. Thus, we may replace B by a ball U C RP centered at 0;
M by the trivial bundle Mo x U; and E by the bundle E:o x U, where Eo is a bundle over
Mo. Since the change of coordinates, and its inverse, used to obtain this trivialization aresmooth, we see that the data used to define the family of generalized Laplaciana HZ gives asmooth family of one for defining generalized Laplacian on the bundle Eo, parameterized by
tbe ball U. Hence by tbe result above, with respect to the parameter, we have our assertion.
Similarly: by loealization, we have the following theorem, wbich shows that the heat
kernel exista for the Bismut superconnection B u .

Theorem 3. Let A be a bundle of finite-dimensional graded algebras with identity
over Band let M be the bundle of filtered algebras M = A ® Endp(l:). Let :F E
COO(B, A ® Endp(E)) be a smooth family of P-endomorphism with coefficienta in A,of the form
:F = Ho + K +:F{+l'where Ho E COO(B, V(E)) ia a smooth family of generalized Laplacians, K E COO(B, A:(E).
ia a smooth farnily of smootbing operators, and :F[+\ is an element of COO(B,A i @Endp(E». Then for t > 0, the kernel ofthe operator e- F is a smooth family ofamoothing operators with coefficients in A, that is, a smooth section in COO(B, A ® K:(E)).

§I.5.3. ehern Characteristic Forms In The Relative Situation
In this section, following Bismut, we will extend Quillen 's theory of superconnection
to the infinite dimensional bundle 1r.E - B, thereby obtaining a formula in terms of heat
kerneis for ehern characteristic fOrlns in the relative situation. As the space of sections of
1r. & is Coo (M, &) I it is natural to define the space of differen tial forms on B with val ues in
1r.& by A(B,1f.&) = COO(M, 7r·(I\T· B) ® &).
By definition, a differential operator on A(B, 1f'.E) is a differential operator on the space
COO(M,1r·(l\l B) ® E). Let

A(B, V(&)) := COO(B, Ar B ® V(E))
be the space of vertical differential operators with differential coefficients. Ir a differential
operator Don A(B,1r.E) is supercommutative with the action of A(B), then this operator
is given by the action of an element of A(B, V(E)). Similarly, we write

99

Chapter 1.5.

for the space of smooth families of smoothing operators !(: wi th differential form coefficients.
Denote by d B the exterior differential on B.
Let D be a smooth family of Dirac operators on E. A superconnection associated to
D is a differential operator A on A(B,7r.E) of odd parity such that
(1) (Leibniz's rule) For alt 11 E A(B) and <P E A(B , 1r.E),(2) A = D + L1~n;(B) A[i)' where A(i) : A·(B,1T.[) - A·+ i (B , 1r.E).
. It is easy to see that A(i] supercommutes with A(B) if i #; 1, aod heuce belongs toA'(B, V(E)).

Next we construct a superconnection associated with a family of Dirac operators D: It
is sufficient to define a connection V r • E on the bundle 7r.E, Le., a differential operator from
COO(B,1f'.E±) to A I(B, ?r.l'T) such thatfor all f E COO(B) and tP E COO(B,1r.E).
For doing so, assurne that the bundle MI B possesses a splitting TM = THMEBT(M/ B),
so that the subbundle THM ia isomorphie to tbe vector bundle 7f·TB. If X ie a vector fieldon the base B, denote by X H its horizontal lift on M, i.e. tbe vector field on M whieh ia a
section of TB M and whieh projeets to X under the pushforward 1r. : (TB M)z: - T 7t (z:)B.
Furthermore, let U8 suppose that the bundle [ over M is provided with a connection V E I
which ie compatible with its hermitian structure. We ean now define a canonieal linear
connection on the vertical tangent space T( M j B) using the projection operator
P:TM-T(MjB)

wbich has the chosen horizontal tangent space TJi M as its kernel.
Proposition and Definition. Let B E COO(M, [). For X a vector field on B, define
the action of V~E on B by the formula

Then, we have(1) This formula defines a eonnection on 7r.[ over B.
(2) The conneetion V r • E is compatible with the inner product on 1(.E.

=

Proof. To show that V r • t is a connection, we must show that Vjf
iT· fV~·E forf e COO(B). But that is a direct consequence of the definition. On the other hand, by thefact that V E is compatible with the hermitian metric, we also have (2).

100

Local Family Index Theorem

Tbus 8B8Ociated witb a. connection on the fiber bundle MI Band a connection on thebundle E, there is a naturalsuperconnection A := D+ \111:.& for the family of Dirac operators(D Z lz E B). The curvature.

of the superconnection A is a. vertieal differential operator with differential form eoefficients.
Also D'J is a smooth family of generalized Laplacians andis a smooth family of differential operators with differential form coefficients which raises
exterior degree in "T; B @ Ceo(Mz , E;,). Here, by the results in section 5.2, we obtain theexistence of a smooth family of heat kerneis for :F, which we denote by e-t:F E A(B, K(E)) :e-t:F = e-tD~+ I) -t)" fle,
Ie>O

with

Since f" vanishes for k > dim(B), the sum above is finite.
On the otber hand, for K = (KZlz E B) E A(B,K(E» a smooth family of smoothing
operators with coefficients in A(B), given by a kernel

there ia a supertrace on A::(E;,) over each tiber Mz of MI B, which gives a supertrace

When restrieted to the diagonal, the kernei< zlKzlz > is a smooth section of the bundlez
7r. " 1" B @ End(E) over M, and its pointwise supertrace Tr.,t < xlK Ix > is a seetioo
in COO(M,1I"· "'r B). Such a section can be integrated over the tibers, aod hence gives adifferential form on B. Thus the A(B)-valued supertraee Tr, : A(B,A::(E)) - A(B) of thefarnily of operators K is the differential form on Bz.-

1

Tr"E < xlKZ]z > .

MI

A loeal calculation shows thatdBTrJ(K)=Tr,([A, KJ) E A(B).

Chapter 1.5.

101

With this, we may define the ehern characteristic farms in the relative situation for a
supereonneetion A on the bundle 1r.e, associated with a family of Dirae operators D, to be
the differential form on B given by the formula

:2

=

•

Sinee e- A
e- F E A( B , K: (&)), it is weIl defi ned. As an immediate eonseq uenee, we havethe following
Theorem. Let A be a supereonnection on the bundle 1r.& for the family of Dirac operators D.
(1) The differential form eh(A) is cloaed.
(2) Ir A q is a one parameter family of supereonnections on the bundle 1r.E for a familyof Dirae operators D q , thenThus, the dass of eh(A q ) in de Rham cohomology is a homotopy invariant of thesupereonnection A.

AB we have seen above, one may study the Chern characteristie forms by using the heat
kerneis assoeiated with a supereonnection on 11:.&. In the absolute situation, by resealing
the superconnection, we show that the assoeiated heat kernels give us the information weneed. Unfortunately, in the relative situation, the problem is rather complicated. The main
reason is that w~en the time goes to smaIl, the associated heat kernel is not convergent.
Thus, the first thing we have to do is to modify the superconnection so that the limit of thesupertrace of the heat kerneion the diagonal exists whenever the time goes to infinity or
emaIl; this is why we need to UBe the Bismut superconnection.

§I.6.4. Local Family Index Theorem
In this section, 8B in the absolute situation, we prove the local family index theorem bystudying the behavior of the heat kernel 8880ciated with the square of Bismut superconnection B when the time goes to zero. (By the result of §1.5.1 , we know that these heat kerneIs
do exist.)
Let UB summarize the data that we are working with:
1) A relative dimension 2m fiber bundle ~ : M - B with a vertieal metrie 9M/B
and a splitting TM
THM e T(M/ B) with THM ::::: 7r-TB. FlOm tbis, ·we obtain aconnection V M1B on the vertical tangent bundle T(M/B).
2) A Clifford module & for the vertieal Clifford bundle C(M/ B) witb a Clifford connection V E wbich is compatible with V M / B .

=Using this data, we conatruet a family of twisted Dirac operators D = (lY Iz E B), andtbe Bismut superconnedion B, with B[o] = D. Hence the curvature :F := B4 of B acts on

102

Loeal Family Index Theorem

the space A z 0 (1f.E)J of sections of the bundle E = AlI"·(r 8)0E along the fiber Mz. HereA z denotes the finite dimensional algebra AT; B. With this, we may state the following

Local Family Index Theorem. (1) For eaeh t > 0, the heat operator e- tF aetingon CCO(M, E) has a kernelin the sense that if <p E COO (M, E), we havewhere dy is the Riemanoian volume form of the fiber Mz aod z( 2.) Wheo t - 0+, we have tbe asymptotic expansion= 7t"(;t).

cok(z,t,z).-..; (411i)-m Etiki(z)i=OBueh that(a) The eoefficient ki lies in LiSiA2i(M,Endc(MIB)(E)).
(b) The fuH symbol of k(x, t, z), defined by u( k) := L1~~(M)/2 U2i (ki ) is given by theformulaProof. Hy the result of seetion 1 aod seetion 2, we easily have (1). The proof of (2) issimilar to that in the absolute situation:
For z E B aod ;to E M z , let V = T2:o (M/ B) aod H = Tz B be the vertieal aod horizontal
tangent spaces at ZOo Then T := TraM = V ffi H. Let U := {! E V: Il!ll < !}, where~ is a positive number whieh is smaller than the injectivity radius of the fiber Mz. So we
may identify U with a neighborhood of Zo in Mz by tbe exponential map € H- expz:o~'Let ,MI B (zo, x) be the parallel transport map in the bundle T(M / B) alang tbe geodesie
from x ta xo, defined with respeet to tbe eonneetion 'ijMIB. Sinee we are working on asingle fiber M z , this eonnection is nothing but the Levi-Civita eonnection of Mz. Using
this map, we identify tbe fiber Tr(M/ B) with the spaee V l so that tbe spaee of differential
forms A(U) ia identified with COO(U,t\V·). Choose ao orthonormal basis dei of V·, andlet ei E c;co (U, 1" (M/ B))c;eo (U, V·) be the orthonormal frame of r (M / B) aver U
obtained by parallel transport of df;i along geodesies by the Levi-Civita eonnection on Mz.
We deoote by e G a local frame of r M on U eonsiating of the union of the cotangent framee i aod of a fixed basis Ja of r; B.

=

Let E := Ero be the fiber of the Clifford module E at ;to, let Sv be the spinor spaceof V·, and let W = Hornc( v.) (Sv, E) l so that E is nat urally isomorphie to Sv lZl W. Let
TE (zo, ;t) be the parallel transport map in the bundle A z lZl & along the geodesie from x to

103

Chapter 1.5.

xo, defined with respect to the Clifford connection VE,o. Using this map, we identify thefiber A z (9 Er of E at X with the space A H - (9 Sv (9 ~V, and the space Coo (U, A z (9 &r) with

COO(U,AH- (9 Sv (9 W).
Ir we let .0. z be the Laplacian on Mz associated with the connection VE,o, then by thesuper-Lichnerowicz formula, we haveHence we may transform this operator to the one on COO(U, AT- (9 End(W)) by using
the quantization mapi tbat iSt replace the Clifford action rno(eO) at € = 0 by tbe actionm i = gi _ l. i, mll: = gll:. In this way, we get the corresponding operator
E ,O)2"'(('nE,O)
1' " E/S ° b
L ---LJVi-VV;I!:; +4'rM.+L....JFo6 mm.
o<biNext we introduce the rescaling operator 6u on the space COO(U,I\'r (9 End(W)): Ira E COO(U, AiT- (9 End(W)), then

In tbe same way, if a E COO(U x R>o, Air- (9 End(W)), we define

Tbus if we letk(t,€) :=.,E(xo,x) < xJe-tFlxo >,where x = eXPt:o €, we know that k( t, €) is in Coo (U, AH· 0 End( Sv ) (9 End (W)), whicb, by
tbe symbol map, may be thought of aB a map fr?m u to Ar- iSl End(W) and satisfies thebeat equation(at + L) k(t,€) 0=

witb tbe initial caDdition
Hente first as in tbe absolute situation, if we rescale k(t,€) aanr( u, t, {) :=L
i=O

we haveu(1m-i)/2 k( ut, u 1/1 €)[i),104

Loeal Family Index Theorem

In this way, we see that the loeal family index theorem means thatNow we expand the e.Auation

with respect to u 1/2. For this, we need the following easy

Proposition. When u End(W» has a limit

0+, the differential operator uouLO;l on COO( U,I\'r l&l

Here Fis the element of 1\ 2 V· ~ End(W), obtained by evaluting the twisted eurvature
FEls at zoo
The proof of it is quite similar to tbe one in the absolute situation.
On the other hand, we also have the A'r ~ End(W)-valued polynomials ,j(t,~) on
R>o x V such that for N > i + 10:1/2, u e (0, 1], (t,~) e (0,1) x U,

'lN

lIa!a{(r(u,t,~) - qt(~)

L ui/'l'i(t,~)lI:S C(N,i,o:)u

N

.

i=-2m

Therefore, we have00

r(u,t,€).- qt(€)

E Ui/'l'i(t,€).
i=OHenee, by using the Mehler formula, we eomplete the proof.
We end this section with the following applieation of tbe loeal family index theorem.

Theorem. (1) Let B f := t 1/'l6fB(6f)-1 be the resealed Bismut superconnection, thench(B t ) =f of1(Tr"e~[k(x,t,x)])dz.
1M.

(2) When t - 0+, the section 6f (Tr" ,e.,[k( z, t, z)]) e COO (M, 1r. 1\ T· B) has a limit,
which is equal toHence, we have

Chapter 1.5.

105

Here T M / B : A - COO(M,1f-(I\T- B» is the map given by deeomposing the bundle
I\r- M as a tensor product I\T-(MJ B) 0 1f-(I\T- B) and applying the Berezin integral
to the first factor; that is, projeeting onto I\nT-(MJ B) 0 1r-(I\r- B) and then dividing
by the vertieal Riemannian volume form.

.

Proof. (1) Note tbat sinee F, = töf F( öf) - 1, we haveeh(B I} =Tr,[e- Fl ]=Tr,[ö~(e-IF)]=öf (Tr, [e- tF ]).
Thus by

we bave (1).
For (2), we define tbe bigrading on

I\p,qT;M := L:A~ 0 N'T;(M/B).
P,i

Thus for aoy a E Aß @ End(tz ),Tr"t:~ (a) = (-21ri)m

L Tr"t:/s [11[p,nl (a)].
pTherefore, we haveöf(Tr"eJc(z, t, :r)]) - (21Ti)-m

L ti - m-p 2/Tr"e/s [l1[p,n] (c; (:r»].

j,p=Since for 2j :5 n + p, Tr, [11[p,n] (k; )]0, we see that tbere is no singular term in theasymptotic exp8JlBion of ö~(Tre.,.[k(z,t,z)]) as t - O. Henee, we have tbe result.

§I.5.5. The Situation At Inftnity
In this aection, we will study the behavior of the heat kerneis associated with any
superconnection A when t - +00. This is a result of Berline and Vergne. Since the proof
for this result has tbe same structure as the one for the finite dimensional Ca5e, we will first
explain such a finite dimensional result.

106

Local Family Index Theorem

1.5.5.a. The Situation In Finite Dimension
Let E = E+ EI;) E- - M be a hermitian super-vector bundle over a manifold M. Let
D be an odd endomorphism of E with components D± : E± - E~, such that ker D hasconstant rank. Then the family of su perspaces {Ker( D z) : z E M} fonns a su perbundle, theindex bundle of D, over M. Suppose that E has a hermitian structure so that the adjointof D- is D+. Then D is ~lf-adjoint. So, if let Eo C E be the superbundle Ker D graded by
E±, let Po be the orthogonal projection of E on Eo, aod let P l := 1 - Po be tbe orthogonal
projection of E to E1 , where El = Im (D) C E is the image of the operator D, then there aredecomp08itions

E± = Et $Et
with

Eg: := Ker(D±) and the endomorphism D+ gives an isomorphism between tbe bundles

Et and E1 ·-Let A := A{o) + A[l] + A(2) + ... be a superconnection of E, with curvature :F := A 2 E
A(M, End(E)). Tben,

in de Rbam cohomology, is equal to the difference of tbe Chern characters of the bundles

E+. and E-. Furthermore, if A[o) = D, thench(A) :=ch(E+) - ch(E-):=ch(Et) + ch(Et) - ch(E;) - ch(E1 ):=ch(Et) - ch(E;) := ch(Ker (D)).
Now we may aay that our final result in this section ia a refined version of tbe similar resultat the level of differential forms.
Let Ä be the superconnection

which preserves the 8paces A(M,Eo ) and A(M,Ed C A(M,E). We need to UBe the following
notation: Ir K E A(M, End(E)), we write

which aimply means thatwith n E r(M, End(Eo)), etc.

Chapter 1.5.

107

Since Ä cornmutes with Po, we see that ite eurvature has the form

~

~2

:F=A

= (R0 S0)

with R being obtained as follows: Denote by V' 0 the eonnection on the bundle Eo given by
the projection of the connection A(l) onto the bundle Eo:

We filter the algebra M := A(M, End(&)) by the 8ubspaces

Mi := LAi(M, End(E)).
i~i

Then we have

Lemma 1. The differential form R lies in M 2 , and the curvature of the connectioo
V' 0 equals Rr2]'
Proof. This comes from the fact that the eupereonnection A{o]&0 has its curvature A~ = R, andAo= PoAPo on the bundle= V'o + LPoA[i]PO

i~2

as PoA[o]Po= PoDPo =0.

Now, for t > 0, let 6t be the automorphism of A(M,&) which aete on Ai(M,E) by
multiplication by t- i / 2 • Then
tl/2~
A~-l
A , ..'Vev,

ie agam a 8uperconnection on & aod the deeomposition of At ioto homogeoeous components
with respect to the exterior degree is given by tbe formulaA, =t l/'JA [0] + A (1) + t -l/'JA [2] + ....

The curvature:F, := A~ of A, is tbe operator t6,:FfJ,l, and the eohomology cJBBS of ch(A t ) =(2ri] Tr.(e- Ft ] is independent of t: It is equal to the difference of the Chern charactersch(Et) - ch{Eö ) for al] t > 0. Next, we study the limit of ch{A,) aB t -+ +00. It isremarkable that the following stronger result holds.

=

Theorem. Let & E+ $ E- be an hermitian super-vector sheaf and let D be an odd
endomorphism of E whose kernel has constant rank. Let A be a superconnection of Ewith zero-degree term being D. For t > 0, let
A , = t 1/2~0, A~-l
Vt=t l / 2 n + A [1] + t-I/"lA (2] + ...

108

Local Family Index Theorem
be the rescaled superconnection, with curvature :Ft • Then for t large enough,uniformlyon compact subsets of M.

Proof. We begin with tbe following lemma.
Lemma 2. (a.) Under the decomposition &aa=&0$&11 the curvature:F may be written

F=(; nE(~: ~~).

(b) Tbe endomorphism 1(0) E r(M, End(&d) is equal to P1D'J PI and is positive definite.
(c) Denote the inverse of 1{O] on &1 by G. The curvature Rr2] of the connection \J 0 on
[0 is given by

Proof. Let A= A + w withThen

:F = j + [Ä,w] +w I\w,
so

Thus if we write

we see that

Next we give a key technicallemma.

Lemma 3. There exists an invertible matrix 9 with 9 - 1 E MI such that
9 :F9

-1

(U 0)
=9 (XZ Y)
T 9 = 0 V -'
-1

109

Chapter 1.5.
Furthermore

U:::::X - YGZ (modMa),
V:::::T (modMd.
Heuce

t6,(U)= Fq21 + 0(t-1/2).

Proof. Obviously the set of matrices of the form 1 + K with K E MI fonns a group.
So it makes sense for UB to find an invertible matrix 9 so that 9 - 1 E M 1. To construct
such a matrix gwhich puts :F ioto a diagonal form, we use the induction on dirn M - i.
Assurne that there exists gi such thatwith Ti ::::: D"J (modMd. In particular

Hut
0
( GZi

so

(GZi1 -YiG)
1

-YiG)
0
E Mi,

-1 _

(1
Yi G)
1
-GZi

E M2i.

Hence, ifwe define Xi, etc., by(XiZi 11Yi) (1GZi YiG)
(GZi1 -YiG)
1
1
then

-I _.

-.

(~i Tot,'.),Z,

Xi EX, - 2(YiG)Zi + (YiG)Ti(GZi)
EXi (mod M"Ji);Yj :::::Yi(1 - GTd + (Xi - (}iG)Zi)(}iG) E Mi+1;
Zi :=(1- nC)Zi + (GZ,)X, - (GZ,)Yi(GZ j ) E Mi+1;
TI =Ti + (GZi)Xi(liG) + Zi(YiG) + (GZdYi::Tj (modM 1 ).
Thus by looking at tbe sub-index, we may continue the induction.
Now BUPPose that we bave a matrix 9 of tbe required form which diagonalizes :F. Then

(

I+K
N

M) (XZ Y)(U 0) (1+K
M)
T 0 V
N
1+L'

I+L

110

Loeal Family Index Theorem

for same
I<
( N

M)
L

EM 1•

So

X+KX+MZ( NX+Z+-rZ

Y+KY+MT) _ (U(l+K)
NY+T+LT VN

UM)
V(l+L)'

Sinee X E M1 and K, L, M, N, Y,Z E Ml, we have(a) V = (T + LT + NY)(l + L)-l =: T (modMd, henee GV == 1 (modMd.
(b) U = (X + KX + MZ)(l + K)-l == X + MZ (modM a ).
(e) Y + lwT
UM - KY E .1\.1 1 , Henee multiplying on the right by G, we have M =:-YG (modM,). But this ia wbat we want for this lemma.

=

AB for the theorem, we may write

Now by tbe fact that V[O]series, we bave the following= 7[0] = D ia positive definite on &1, using the Yolterra
1

Lemma 4. There exist eonstants c, C > 0, auch that !e- t6t (V)1 ~ Ce- ct .
Henee, we havelt follows that

Similarly, we ean deal with the situation for derivatives with respect to the base. Henee, wehave the assertion.
CoroUary. The limit limt_+ooch( Ad exists, aod equals the ehern character cf theconnection V' 0 on the superbundle E = Ker( D).

Cbapter 1.5.

IU

I.5.5.b. The Infinite Dimensional Situation.
We use tbe same notation aB in §I.5.1. Wbat we diseuse in this section ie a fundamental
theorem of Berline-Vergne, which generalizes the result of last subseetion to the ease of afamily of Dirae operators. Assurne now that D is a family of Dirac operators such that
KerD l has a eonstant dimension, 80 that KerD ia a superbundle over M, If P: is theorthogonal projection from f. &z to Ker D Z , then Po E r( M, K: (E)) ia a smooth family ofsmoothing operators. Also we have the following easy

Lemma 1. The operator V o defined by the formula V o := PoA[1]Po ia a eonneetionon the auperbundle KerD.
For t > 0, let 6t be the automorphism of A(M,1I".&) which multiplies Ai(M, 11".&) by
t- i / 2 • Then At := t 1/ 2 6t A6;-1 is a aupereonneetion for the family of Dirae operators t 1/ 2 D.

Theorem. For t > 0, let
At := t l / 2 6,A6-; 1 = t 1 / 2 D + A(1] + t- 1/ 2 A[2] + ...
be the rescaled supereonnection with eurvature F, = t6 t (F). Then for t large enough ,uniformlyon eompact subseta of M X1l' M,

Proof. The atructure of the proof for this theorem ia the same aB the one for the finite
dimensional ease. First we filter the algebra

M := A(M, Endp(E» = r(M, 11". 1\ 1'" M 0 Endp(&»
by the aubspaees

Mi := LAi(M,Endp(&» .
. i~iIn the same way, we also get a filtration for tbe algebra N := A(B, K:(E».
Let G := (Gz : z E M) be the family of Green's operators GZ of (Dz)2, Thus loeally

1

00

G=

e-t(D'+PO)dt - Po-

Tben G preserves N. In fact , for every K E N, we may decompose aO((G + PolK) into
terms proportional to{'" (a;'e-'(D'+Po»)(a;' K)dt
with 0"1 + Q2
the form=

0". Thus by Duhamel's formula, we see that, in general, this integration has

112

Loeal Family Index Theorem=

where Ic
latl + 1 and Di E r(M, End." ( 1I'.E)). So the fact that < z!e-t(D:I+Po»)lz > deeaies
exponentially implies that the above integratal ia bounded. Now by the formula

we see that G K E }/. Similarly, we know that K GEN.
Let Pt= 1 - Po be the projection onto Im D. Ir K E M, we let

Lemma 2. (a) Let Rt2) be the curvature of the eonnection Va 00 the bundle Ker D,
and letF=(;n

be tbe eurvature of the superconnection A. Then X, Y, Z E}/ and(b) There exists 9 E M with 9 - 1 E Nt, such that

Furtherrnareu ==X - YGZ (mod}/3)
V::T (modNt}.

Proof. Ir G is tbe Green operator of D'J, aod Y, Z E Ni I then the operators Y G aod
GZ are alBO in M, but Y(1 - GT) aod (1 - TG)Z are in N i +1 . So we may construet 9 aain last section as a product of matrieea of the formG) E1+ (0N Nt0 ) .
(GZ1 -Y
1
1

Now the proof of this lemma is obvious from the one in the last section.
Since U :: Rt'J] (modN3 ) aod V == D'J (modNd. So for eaeh t > 0, the family of
operators Ot(V) is the sum of a family of generalized Laplacians D'J aod an element of
Pt MI Pt. Heuce, e- t6 t (V) is a section in N for each t > O. ThuB, by the uniqueness of theheat kerneis, we know that

Chapter 1.5.

113

With this, if U is a relatively compaet open subset of M, and A is the infinitum over Uof the lowest non-zero eigenvalue of the operators D 2 , then, by the Volterra series, we easily
see that over U,
pte-t6,(V) PI= O(e- t .\/2).

(Here, we use the foHowing convention: Ir A(t) : R>o -r(B, A:(E»), we write A(t)=O(f(t» if for aH e > 0, I € N aod each function tP E C~(M) of compact support, there ia
a constant C(I, e, tP) such that111r·(tP)(x) < xIA(t)IY > 11, :5 C(/, e, tP)f(t)for aH t > !.) Therefore, we haveeC ()-Ie-R[:JI- Uf 90(-16,(F) _But 6f (g) - 1= O(t- 1/ 2 ),1 200) 6t (g) + 6t (9)-1 (O(t- / )0c ( )O( e-0t >'/2) ) Ut
9 .

80 we have

From here, tbe theorem follows for I = 1. For derivations, we may proceed in the same way.
The details are left to the reader.
Corollary. The limit

holds with respect to each C'-norm on compact subsets of M.

51.5.6. horn The Real Situation To The Complex Situation
In this section, we present tbe results discussed in the last few sections in tbe sense of
Kähler geometry, by comparing them with those for Riemannian geometry.

1.5.6.3. Absolute Situation: Dirac Operators
First we present the complex theory of dirac operators. This may be treated by thefollowing
Theorem.. (1) Over a Riemannian manifold, the de Rham complex ia given by

114

Local Family Index Theorem
Furtbermore, we have(a) The bundle Ar M is a Clifford module defined byc(cr)ß = t(cr)ß - L(cr)ß

r

for a: E r(M,M), ß E A(M).
(b) The Levi-Civita connection on the bundle AT- M ia a Clifford connection.
(c) .The Dirac operator aB80ciated with the above data is the operator d + er, withthe adjoint of tbe exterior differential d.
(d) (Weizenböck's formula):(d + d-)2 = ß,"T-M -L Rtjj:/tJ:L1t Li .
iijld(2) Over a Kähler manifold M, the Dolbeault complex ia defined bywith tJ := L:~=1 €(dzi)~ locally. Furthermore, we have(a) The vector bundle of anti-holomorphic differential forms A(TJ,l Mt ia a Clifford
module defined by=f O,l + f1,0 aud frJ,b E (Tb,rJ Mt.
where f(b) The Levi-Civita eonneetion ia a Clifford connection.
(c) The associated Dirae operator ia V2( fJ + Er).

(d) (Bochner-Kodaira's formula):(8 + 8-)2=ß0,'+ L t(dzi)L(dzi )F K - (8z i, 8~i)'i,jHere K = A n (T 1,0 Mt ia the canonical bundle of M.

Proof. (1) By a local calculation in this ca.se, since 'V is torsion-free, we know thatd = t: 0 'V 1 d- = - L 0 V. So d + dc 0 V. Now by the Lichnerowicz formula, we have=(d + tr)2= /l"T- M + ~ L RtjJ:l(t:1: - LI:)(e ' - L1)e i Lj .
ijJ:l

But R;jJ:1 vanishes over the antisymmetrization of three indices, soChapter 1.5.

115

Hence we have the assertion.
(2) We know that without the Kähler condition, the vedor bundle I\(TJ,l)'" is stilla Clifford module and the Clifford action is aelf-adjoint. In general, the corresponding
canonical connection is not a Clifford connection. But once we have the Kähler condition,
the situation changes dramatically. In fact, by the Kähler condition, we know that the LeviCivita connection of the underlying Riemannian structure preserves the bundles T 1 ,o M andra,l M (which may be thought of aB the definition of the Kähler condition.) Hence we havethe assertion that the corresponding canonical eonneetion ia a Clifford connection. Next,
we have to prove that tbe associated Dirae operator is V2(tJ + 8·).

Let Zi be a local orthonormal frame of Tl,O M with dual frame Zi E (T 1 ,O M)'". Thend= L)E(Zi)V Z; + E(Zi)V Z;), [j = L: E(Zi)VZi.
ii

Hellce it is enough to show thattJ- =- L L(Zi)V Zi.
i

Let a be tbe one-form on M such tbat for ß, E AP,q(M) and ß,+l E AP,q+l(M),= (Bq, L(XO,l )ßq+d.
Since V preserves the splitting TM @R C =T ,o M EB ']"0,1 M,
Tr(Vo) = L(ZiO(Zi) - o(~ Z;Zi))'
a(X)

1

i

Thus

((L!(Zi)V Z,)ßq, ßq+ds:= -(ßq, (2: L(Zi)V zißq+ds: + Tr(Vo)s:.
ii

Now the assertion comes from tbe fact that the integration of the laBt term over M vanishes.
Finally, for (ä +a-)' do tbe same thing as for Lichnerowicz's formula. We have
(8 + 8-)2 = Va,' + L E(dzi)L(dzi)R+ (8z B,;)
1

j,

ij

witb Jt+ tbe curvature of 1\(']"0,1 M)'". Thus, it is enough to show that

L E( dii)L(dzj)W (8~j 8,;) = L €( dZi)L(dzi )FK • (8~j, 8,;).
I

ij

ij

By definition, we know that tbe left hand side ia equal to'L,g(Z;)L(Zj)E(Zi)L(Z;)(R(Zj t Zi)Zk, Z,)ijkl=L g(zt )L(Z;)(R(Zj Zi)ZI:, Zj)'
I

ij I:

Loeal Family Index Theorem

116

Now by the facts that
R(Zj,ZdZk + R(Zk,Zj)Zi + R(Zi,Zi:)Zj

and that R(Zk,Zj)=0=0, we have
R(Zj, Zi)Zi: = R(Zi:, ZdZj.

Hence we have the final assertion.

I.5.6.b. The Absolute Situation: Index Theorem
Theorem. (1) (Atiyah-Singer Index Theorem) The index of a Dirac operator ona Clifford module E over a compaet oriented even-dimensional manifold is given by thecohomological formula
Ind (D)=1M A(M)eh(E/S).

(2) (Hirzebruch-Riemann.. Roch Theorem) Tbe Euler number of the holomorphic
vector bundle E over B Kähler manifold M ia given by the cohomological formula:

X(M,E) ~1M td(M)ch(E).

Proof. (1) This ia a consequence of the result of Patodi aod Gilkey stated in subsectioo
3.7.a.

(2) We may deduce this formula from (1). In fact, if we consider the Riemanniao
curvature R to be the matrix with two-form eoefficients, then the curvature operator(V'A(r<',1 Mt)1 is L i j (RZi, Zj )e( zj )1.( Zi:), But the End( I\(ro,l M)· )-valued two-form RII(TO, 1Mf
equalsSo(V'II(TO'l M)·)1 = RA(To,1 M)·+ ~ E R( Zj, Zj).
i

We know that, by definition,1
pll(To. Mf0 E/ S = ~TrTI'OM(R+) + FE.
2

Here R+ denotes the eurvature of the bundle T1,o M and FE denotes the eurvature of E.
Now by the splitting TM 0R C
T1,o M EEl ro, 1 M, we see that=-A( A1)

R+= det( e R +/2 _ e- R + /2)'

I

Chapter 1.5.

117

Hence

Bince tbe Todd genus• R+R+
td(M)=det( R+)=det( R+/2
_R+/2)exp(-Tr(R+/2»).
e -1e-e
Therefore, we have our assertion.

I.5.6.c. The Relative Situation: Smooth Fibrations
Let 11" : M - B be a smooth family 8B in the previoUB chapter. We BBSume that M andBare complex manifolds of dimension n and m respectively. Then we have the following
exa.ct sequence of holomorphic tangent bundles over M:

0- T1,oZ _ T1,o M _ 1I"-T 1,oB - O.

Ti/M

B.

Also, as COO-bundles, we know that~ 'lf-Tl,O
However, in general, we do not havethiB isomorphism as holomorphic bundles. Hence TJio M ia not a holomorphic subbundle ofTl,O M. Now we state the complex situation for the srnooth family as a tripie ('1f, gz, TH M)
with a srnooth 2-forrn w on M of complex type (1,1), which has the following properties:(1) w ia closed;(2) THM and TZ are orthogonal with respect to w;(3) If X, Y E TZ, then w(X, Y) =< X, JY > .
Usually, we call euch a family a Kähler fibration witb associated (1,1) form w. In
tbis C88e, we know that (M,w) and (Z,gZ) are Kähler and B iB locally Kähler; i.e. there isan open covering U of B, such tbat there is a closed (1,1) formon U, which induces a
Käbler metric on TB. We also know tbat on 'lf-1(U), one may replace w by W + ).,'lf·rF forany )., > O. Since tbe fiber Z ie compact, if )., ia large enough, W + ).,1f-'-P iB a Kähler formon 1r- 1(U), which induces the metric gZ on Z aod ia such that THM = (TZ)l.. Denote by
WH,WZ the restrictions ofw to THM,TZ, respectively. Thus, on TM, we have the relation
W = wH + W z. We know that the pair (g Z , TH M) ia entirely determi ned hy w. In fact, weeasily have the following

rF

Proposition 1. Let W be a smooth 2-forrn on M of complex type (1, 1), ~hich has thefollowing properties:(a) w iB closed;(h) If X, Y E TZ, (X, Y) ~ w(J X, Y) defines a hermitian product gz on TZ.
For any x E M, let118

Local Family Index Theorem
Then, TH M is a smootb subbundle of TAt such that (1l", gZ, TH M) is a Kähler fibration
with associated (1, 1)-form w.

The bundle rO,1 Z is identified to TI,o Z by the metric gz. Therefore rO,1 Z inherits
the holomorphic structure of TI,o Z. 'lz induces tbe corresponding canonical connections
on rO,1 Z. Hence we know that ArO,1 Z is also a holomorphic hermitian vector bundle on
M. If is a holomorphic ~rmitian vector bundle on M of camplex rank Ic, for 0 :5 p :5 I, we
let EP denote the set of Coo sec tions over M of APro, I Z 0 E. We also regard EP as the setof Coo sections over B of an infinite dimensional bundle: For any y E B, the corresponding
fiber Ee ia the set of COO sectiom~ over Zt/ of APro,1 Z 0 E. SeteLet dz be the Riemannian volume element for tbe fiber Z. Then for any y E B, we have an
L 2 metricf < e, e' > (x)dx
Jz.,

=

=

on E y . Let (ZlXl + iyl, ... ,zl
zl + iy') be"a complex system of coordinates in one
given fiber Z, and let TZ be oriented by the hase (8/8x l ,8/ayl,. ",8/8x',8/8 y l). (So wehave 8/ayi = J8/8ri .) Let

8
8 zi

1

8

. 8

8

1

8

. 8

= 2(8 z i - 8yi ), 8ii = 2'({)zi + 8yi)'
1

1

and
Locally, let

-z

1

~

a ., := ~ dzl-' A 8{)ii .
}=I

Then for every y E B J the operator [}Z., acts naturallyon Ey • Let tJz.,. be the formal adjointof [)Z., with respect to tbe bermitian metrics on Ey . Motivated by the result of su bsection
6.a, we letäy h[)z",v'2fJ z"., D y fJy +

=

8; =

=

8;.

Then the Dirac. operator D y interchanges E: and E;. Let D±,,, be the restrietion of D" toAlso, by a local trivialization of the fibration 11", we know that äy ä;,
are first orderdifferential operators wbose coefficients depend smoothly on x E M, and Dy is formallyself-adjoint on E y •

Ei.

I

D"

Next we define a Clifford module structure on AT·o,1 Z (9 E: If X E T1,o Z, denote
X· E T·o,1 Z the I-form Y E TcZ ........ < X, Y >, we define c(X) E End(AiO,1 Z (9 E) byc(X) := V2X· A . Ir X' E TO,I Z, we let c(X') := -V2Lx'.
Wi th this, we can gi ve another description of Dy . In fact, if we let e 1, ... ,e n be an
orthonormal basis of TZ, W I , •.. Wl an orthonormal basis of TI ,0 Z wi th t he correspondingJChapter 1.5.
basis tuj, wi ,wi on ']'0, I Z, r
we have

1,0 Z, T-O,l Z

119

respecti vely. Then by the fact that Zy is Kähler,

Therefore we have
Proposition 2. For any y E B, DJI = L~=l c(ek)\7 e...
JUBt aa in the real situation, we introduce a connection \7lf.e on E as folIows: for any~ section of E, if Y E TB, then

\7~"h := VYHh,
where Y H ia the lifting of Y in Tu M .
Theorem. (1) The connection \7lf0' does not depend on tbe metric on B, and preserves
the herrnitian metric on E.
(2) AB a 2-forrn (V"0')2 ia of complex type (1,1). Furthermore, for any U E TI,o B, V E
ro,IB,

Proof. (1) may be proved aB in the real situation.
(2) We know, by a loeal calculation, tbat tbe curvature of \7"'0' is given by(\711'0')2(y, Y') = R Z (yH, y'H) @ 1 + 1 ~ R!(yH, y'H) - VT(YH,ytH) ,for Y, y' E TB witb R Z , ~ tbe curvature of TZ and f; respectively, and T the torsion of
V$. Thus by the eondition for a Käbler fibration, we have tbat V Z on TZ preserves thecomplex structure of TZ and induces on TI,o Z its canonical connectioll, and T is of type(1,1). So we know that (\7"0,)2 is of complex type (1,1). On the other hand, if (yl, ... ,1/)
ia a complex coordinate system of B witb (8/8 yO) the corresponding basis of TI,O B, etc,
then by a Iocal calculation, Bince R Z , JtE, T are of type (1,1), we also have that"'t;roE: D =V"oE: a+ v..·e Er=dyOc(wj)[R Z «8/8 yO),wj)01+ 1 (9 Hf «8/8 y U), tuj) - V T «(8/8J/G),üJj)]+ dyOc(wj)[R z «8/8 yO),Wj) 01+ 1 @R!«8/8 yO),wj) - V'T«8/8gCl),wj)]'
Since \?oE preserves tbe grading in E, and so V"oE:tJ (resp. \7r o etJ-) increases (resp.
decreases) tbe degree in E by 1. But R Z , R', \7T da not change tbe grading in E, hencea\7r o E: =dyOc(Wj )[R z «8/8 yU), Wj) @ 1+ 1 @ If « 8/8 y O), Wj) - V'T«(8/8r ),tDj)];Q

v'1: t Er =dyOc(Wj )[R z «8/8 yU), Wj) ~ 1o+ 10 [f«8/8fJU), Wj) - VT«(8/8VG),wj)]'

120

Loeal Family Index Theorem

Therefore, by eounting the degreea of both aides, we have the assertion.

Remark. Ir E ia a finite dimensional eomplex hermitian vector bundle on B, endowedwith a hermitian eonneetion 'V whose eurvature ia of eomplex type (1,1), then by
Newlander-Nirenberg theorem we know tbat there is a unique holornorphie strueture on
E such that 'V is the "eorresponding eanonieal eonnection. For tbe infinite dimensionalsituation, as stated above, we ean still attacb a unitary eonneetion whose eurvature isof eomplex type (1,1). Also Vr.. ttJ = 0, so formaIly, we have a kind of 'holomorphie'on E. In this sense we usually eall such a eonneetion a holomorphic connection.
We do not use the notation of eanonieal connection here as by the result in tbe realsituation, tbis eonnection is not the right one for us to study the problem at tbe levelof differential forms; it does not give us the niee cancellation. For tbis reason, we haveto use tbe Bismut supereonnection.
We end this seetion by tbe following explicit formula for the Bismut supereonnection
in tbe eompiex situation.
Let X, Y be two vector Reids on B. Then we have the horizontal lifting XH, YH to THM.
Let [X, Y] be the commutator of XH, YH and T(X, Y) E TMIB be the projeetion of -[X, Y]
along THM. Tbe map T deftnes a tensor in COO(M,TM1B 01\'l.THM). Let T:= Tl,O +J"'Ü,1
be ·tbe deeomposition of T aeeording to its type in TM1B and c(T)c(T1,O) + c(ro,l)tbe corresponding decomposition of Clifford action c(T). Tben, we know that tbe Bismut
auperconnection in our case is just given as follows:=t,Moreover, if we seale tbe metric for tbe fiber by a factor
we know that tbe associatedrescaied Bismut Buperconnection B.B8Ociated with this new metric is given byChapter 1.6.

121

Chapter 1.6
Relative Bott-Chern Secondary Characteristic Forms
With Respect To Smooth Morphisms 11: Existence
From the construction for the classical Bott·Chern secondary characteristic forms at
the end of Chapter 4 by the Mellin trasform, we know that such a theory depends on theexistence of ceratin trace classes which have the right asymptotic behaviors when the timegees to zero or gees to infinity. In this chapter, we will give a construction for relative
Bott-Chern secondary characteristic forms with respect to smooth morphisms. Similarly,
the basic idea ia to use the Mellin transform. In general, this process is rather complicated.
So we first deal with a special situation for the infinite dimensional case, in order to get aneasy statement for the exponential decay of our objects when the parameter goes to infinity.
Then, we study the most general situation, by using the key observation which comes fromboth the finite dimensional situation and the special infinite dimensional situation mentioned
above. The references for this chapter are [BGS 88] and [Fa 92].

§I.6.1. A Special Case In The Infinite Dimensional Situation
In this sedion, by imitating the process in 4.5 for the finite dimensional situation, wegive a construction of relative Bott-Chern secondary characteristic forms for certain. special
CBSe8 following [BGS 88], from which we may get a good feeling for the construction in
general.

1.6.1.a. Bismut's Buperconnection
We use the same notation as in the previous chapter: (11'Kähler fibration with the aasociated (1,1) form w, etc. Let

M - 4 B, gZ, TH M) 15 a

be aholomorphic chain complex of finite dimensional holomorphic vector bundles on M
with hermitian metries Pi on €j. Set

122Conatruction w.r.t. Smooth MorphisffiS

Let V j be the canonical connection of (ej I pj) for j = 0, ... , m , V{ =: $'V j I L{ := ('V{)2
aod R{ := _~(V{)2. Let v· be the formal adjoint of v. Set V := v + v·.
For 0 :5 j

:5 m, we make the various conatructions as in 5.6.c for ej.

Denote by

Er Er E the corresponding infinite dimensional hermitian vector bundles on B wb ich weIIjendow with the (unlabelled) 'holomorphic' hermitian connection 'V..... Also we have theunlabelled families of opemtors ä,Er, D on Ej as weil as the vertieal Clifford multiplication
operators. c( e,).

=Let T be the involution defining the grading on Ej I i.e. T±1 on Er. We also maketbe eonvention that v, v·, V set on E j lilre 1(1 ~ V)i therefore they sntieommute with c(e,).
'Hence we have a 'bolamorphie' double ehain eomplex of infinite dimensional vector bundles
on B:

0

0

-

0

0

1
E[

1

v
---.

'f~

-

113

!8

1ä
E'0

1
0

v

E'1
!

0

v

-

v
---.

-

-

v

v

0

1
El1JO
18

-

0

113

-

0

E'm
!
0

Taking the grading naturallYl we also set

E:= $j,pEj, E+ '.- €Xl'J+P even EPj

I

E- .'- €Xl'J+P odd EPj 'The operators ä,ä- I D , v, v·, V are odd in End E. Hence for u ~ 0 , we have tbe superconneetion'11(. + JU(D + V)
on E. Hut as we bave already seen in the real situation, this superconnection is not theright one for our purpose. More precisely, by a complex realization, we have the following
Bismut supereonnection: For u ~ 0,
Au := '\11(. + VU(D + V) _ c(~
4v u

wbere

[.6.1. b. Local Family Index Theorem
With the same notation as above, from the local family index theorem in the realsituation whicb· was proved in Chapter 5, by taking the correspondence at the end of thelast chapter l we have the following

Chapter 1.6.

123

Theorem.. 1. Let' P be the vector space of amooth fonns on B with complex type(p, p), 0 :5 p :5 dimE. For any u > 0, the smooth differential forms on B
Tr,[exp(-(\7 lr • + v'ü(D + V))2)], Tr,[exp( -A~)]are in Pj and they are c1osed.

(2) (Local Family Ihdex Theorem.) Uniformlyon compaet subsets in B,[2'1ri]liITlu_oTr,[exp(-A~)]=lz

td(-Rz)Tr,[exp(-R!)].

Remark. Even tbrough Tr,[exp(-(V lr • + .jü(D + V))2], Tr,[exp(-A~)] are in tbe
same cohomology dass, in general, Tr, [exp( - (V lr • + ..;ti(D + V))2)] does not converge asu -- O. It is at thia part that we must UBe tbe Bismut superconnection.

1.6.1.e. Number Operators
Tbe double eomplex E bas a horizontal and a vertieal grading. Let NH, N v be tbe
number operators corresponding to these two gradings: NB and Nv act onby themultiplication with j and .t respectively. Thus N = N H + Nv is the total grading number
operator. We know that thia number operator ia the right choke of tbe number operator
if we use the superconnection V r • + JU(D + V), when we do everytbing aB in the finite
dimensional C88e. For example, we may have tbe double transgression formula associatedwith Tr, [exp( _(\7"'. + .jü(D + V))2)]. On the other hand, since finally we use the Bismut
auperconnection, 80 we bave to change number operators, in order to make our theory
go through. Now aupp06e Nu ia the right nurnber operator witb respeet to the Bismut
superconnection. For getting the right cancellation, similarly to the finite dimensional CaBe t
we need the following basic relations ,EI[\7'1'., Nu] = 0, [8, Nu] = -lJ, [8·, Nu] = Er,[v, Nu) = -v, [v·, Nu] = v·, [c(T1,o), Nu) = _c(T 1 •O), [c(T'·l), Nu] = -c(T'·l).
Therefore, by a direct calculation, we have a natural choice for Nu. Ta expiain it, we makethe following observation.
First note that we da not change the horizontal data, so it is enough to modify Nv. For
this purpo&e, we first evaluate Nv in a more geometrie way, via the vertical Kähler form wz.
In fact, by a local calculation, we know that, aB an element of the Clifford algebra C(TZ),
Wz is given by

l.e.

124

COIlBtruction w.r. t. Smooth Morphisms

Here, I ia tbe relative dimension. So from tbe fact that L: '=1 wJ 1\ L~j on E~ is given by the11multiplication by 1:, we know that
1

N

v

.
= -twz+ 2'I
CFrom here, witb the abov'C basic relations, we let

. eil
N Vu
, := -twz + -2
u WH A +-,2

I.6.l.d. Double Transgression FonnulaWith above notation, we have tbe following
Theorem.. (1) For any u > 0, tbe amooth differential form Tr,{Nuexp(-A~)] is in P.
(2) (Double Transgression Formula):uTr,[exp{ -A~)]=- 2~(ÖB + äB)Tr.(.,IiJ(D + V)+ ~jJ)exP(-A~));
Tr.(.,IiJ(D + V) + ~jJ)eXP(-A~)]2= (8 - 8-8 )Tr,{Nuexp(-AJ].

In particular,a Tr, [exp(-A u2 )] = --8
1 -B aB Tr, [ Nuexp(-A 2 )]'
-a
u
u
u

Proof. Hy the construction, (1) is trivial. For (2), by the choices of the Bismut
superconnection and tbe properties of tbe number operator listed in the last subsection,
one may exactly imitate the proof of tbe corresponding assertion for the finite dimensionalsituation, to give a complete proof for tbis part. The details of this translation are left to
tbe reader.

I.6.l.e. Asymptotic Behaviors Of Certain Forms
Hy the result in the last subsection, in order to imitate the definition of the classical Bott-Chern secondary characteristic forms for finite dimensional case, via the Mellio
transform, we now need to consider the asymptotic behavioIS of

Tr, [exp( -A~)],Chapter 1.6.

125

Tr,[( y'U(D + V) + ~~)exp(-A~)Land

Tr, [Nuexp( -A~)].
when u - 0+ and u -+00 respectively. In this respect, we have the following

Theorem. (1) There exist ~ even differential forms Ao,A I , ... in P such that for
any Je E N,k

Tr,[exp(-A~)]= L A u-1 + o(ul:)
j

j=O

where

Ao

= (2:i)1htd(-Rz)Tr, [exp(-Ld].

(2) There exist ~ odd differential forrns Ba, B II ... such that for any Je E N,=

Moreover, B o O.
(3) There exiat smooth differential fonns C_ II Co, ... in P such that as u - 0,

(4) The various 0( ul:) are uniform on compact subsets in B.

Proof. By the local family index theorem 1.5.4, we have the asymptotic expansion in(1), since the super-trsce vaniahes on all elements ofClifford degree strictly less than 21, andthe fact that the corresponding 0 ia uniform on the compsct subsets in B. Furthermore, by
the diacUB8ion for the local family index theorem. in section 5.6 for the compeIx geometry,
we hsve the expression for the term A o•

Now 1 we consider (2). By Duhamel's formula,'Il,[exp( -A~ + (VU(D + V) + ~~) du)].

c(T)='Il,[exp(-A~)] + 'Il,[(y'U(D + V) + 4JU)exP(-A~)]du.
Therefore, it suffices to give the 88ymptotic expansion Corc(T)'Il,[exp( -A~ + (JU(D + V) + 4JU) du)].

126

COllstructioD w.r.t. Smooth Morphisms

For doing so, we may UBe exactly the same proeedure for tbe proof of the Ioeal family index
theorem to deduee tbe aaymptotic expansion, since, aa an essential generalized Laplacian
the strueture of tbe super-Liehnerowicz formula is just the same. With this, BQ = 0 ia a
direct eonsequenee of (1) and (3), provided that we UBe the double transgression formula in
the previous subsection.
So, to eomplete the proof of the theorem, we need to prove (3). From the above proofof tbe aaymptotic expansions in (1) and (2), we see that if we could give a formula, inwhich ooly the terms Tr,[exp( -A~ + Bu ) occur, where, aB essential generalized Laplacians,-A~ + B u has the same structure Ba what is for the super-Liehnerowiez formula, we thencan use the same procedure as what we did for tbe proof of tbe loeal family index theorem
to give the aaymptotic expansion. In fact, this is the general method for tbe proof of theexistence of asymptotic expansion in tbe sequel. Here we demonstrate it by the example
with

:u

(uTr, [Nuexp( -A~)]).

We are supposed to show tbat its values at u = 0 ia a amooth form C_ 1 on B.
We first recall tbe situation for the finite dimensional CMe. At that plaee, we got theassertion

Therefore, the assertion (3) ia not really surpriaing. But now we get a eertain trouble,
aB the bigher Grassmannian degree terms in A~, Nu Beale with negative powers of u1/"l.
Fortunately, finally when we count the Clifford degreea as we did before, we will find thatthe terms witb a low u-power also have low Clifford degree by the properties of tbe Bismut
superconnection and the number operator listed at section 5.6 and subsection 6.1.c. Thusby the fact that the super-trace vanishes on elements with low Clifford degrees, we find thecorrect cancellation.
In practice, we do as follows, whieh ia tbe same as for the finite dimension situation:
First, we have a generalization of tbe double transgression formula: For u > 0, b ;;::: 0

Hence, hy the fact tbat
Tr,[Nuexp( -A~ + (y'ü(D + V) + c(~) du)}du
4y u
8c(T):l= ab Tr.[(y'ü(D + V) + 4v'U)ex p (-A u + buNu)h=o,Chapter 1.6.
we have, if we replace127

ib by t Grb,
Tr,[Nuexp(-A~ + (JU(D+ V) + c(7))du)]dU
4v u

1 8'l
{
'l
]
= ( {JB -8- B )"28b2Tr,exp(-Au+bNu)b=O,
On the other hand, by the fact thata (uNu) = -iw z+ N H + ~,2{J u

we get{}'l

8u Tr,[exp(-A u + buNu ))=- -21u (fIJ + bB)Tr,[(y'ü(D + V) + 4c(1))exp(-A~+ buNu )]uvc(T<l,O»)+b(Tr,[exp(-A~-(v'U(8+v)+..;u )da4 u
C(T<°,l»)- (.jU(8· + v·) + .;u) da - iwz da da + buNu)]dlJ d4
4 u
1
p(-Au'l + buNu )]).
+ Tr,[(NH + "2)ex

Differentiating with reapect to band evaluating at b = 0, we then get:u (uTr,[Nuexp(-A~)])=Tr,[exp(-A~-(v'U(ä+v)+

c(T'l,O») .
..;ti )da4 u

_ (v'U(ä· + v·) + C(r<O.I))) da _ iw c da da)]dtJ diJ
4jüZ1+ Tr,[(NH + 2)exp ( -A~)])- (/JB + ilB)(~Tr.[N"exp(-A~ + (y'ü(D + V) + ~~) du)]"".
Therefore, put all this together, we get:u (uTr. [N" e.xp(-A~)])=Tr,[exp( -A~ - (.jU(ä + v) +c(T<l,O))
4y'U ) da_ (v'U( ä· + v·) + c(T<°' 1))) da _ iw zda da)]dtJ diJ

4Jü

'1 ]
+ Tr, [(N H + 21)exp( -Au)
)
,

'l

.

BI a
[
'l
]
- a {} "2 8b'l Tr, exp( -Au + bNu ) 6=0·aB

Construction w.r.t. Smooth Morphisms

128

Hence, we expressed the term

as a combination of the farms of

Tr, [exp( -A~ + B u )],as stated above. Thus, by the method used in the proof ofthe local family index theorem, i.e.,
first, to use tbe normal coordinate to localize the problem, then, aB an essential generalizecl
Laplacian, to use the structure for -A~ + B u in the sense of the super-Lichnerowicz formula
to give the discussion over tbe Euclidean spaees via the generalized 05cillators to give tbe
asymptotic expansion. More precisely, we haveE

I
Tr,[(NH + 2)exP(-A~)])= • Fi'J + o(ui:)
j=O

with
1
f
I
Fo = (21ri)1 Jz td(-Rz)Tr,[(NH + 2)ex p (-L,.,p,)),

and

a1

1
ab1 Tr,[exp(-A u+ bNu)h=ok,,'= ,Li Dj'; + 0(1.' )k)=-1with D_ 1 , D_ 1 closed forms. Thus, we eomplete the proof of the theorem.
From the proof above, if we -write aB terms down in apreeise form from the loeal
discussion over Euclidean spaees aB what we did for the proof of the loeal family index
theorem, we also ean have the foBowing

Chapter 1.6.

129

Corollary. With the same notation as above,

C- 1=(2~i)1

l i~td(1-Rz )Tr 8 [exp( -L(.,p.)},r8

Co ~(21ri)' Jz 8b(td(-Rz - bl)h=oTr,[exp(-L,.,p.)}+ I (2~i)ll td( - R z )Tr [exp(- L, .,P.)]
8+ (2~i)fl

td(-Rz)Tr,[NHexp(-L(.,p.)]

1 B

- -d
2 E o.

Indeed,So, by the fact thatuNu = -iuwz + ~wH + U(NH +~)contains DO nagative powere of u, we may get above expression of C- 1 by imitating theprocess in the proof of the local family index theorem. For Co, we see thatliIllu_o :u (uTr , [Nuexp( -A~)])= Co.

Thus, by the last part of the proof of the theorem above, we see that only the part forTr,[exp( - A~ - (y'U(ä + v) +C(T<I,O»
4JU ) da- (,Jü(O· + v·) + c(~;))) dä - iw zda dii)J"" da=mattem. But then, we may first assume that v
0 to deduce the result. In general, we
need to kDOW that fact that the 0 order operator [8·, v] + [8, v·] ha.s the weight u, so it does
not contribute to t~e limit. ]n this way, we get.
_
C(T<I,O»)
liIllu=oTr,[exp( -A~ - (..jU(8 + v) + 4JU ) da

_ (v'U(ä· + v·) + C(T<°,I»)) da _ iw~ da dä)]da dll

4Jü=~ { a8b[A(R z - ibJz)h=oexP(--21Tr[Rz])'I'r, [exp(-Lc.,p.)L(2?n) }z130

Construction w.r.t. Smooth MorphisIDS

where Adenote tbe Hirzebruch A genus. Hence, we get

SO, by using the relation between complex Kähler geometry and Riemannian geometry liatedin the final section of the last chapter, we get the corollary.

I.6.l.f. The Construction In A Special Case
We bave already described the asymptotic behavior of
Tr 11 [ Nu exp (- A~ )]aB tJ 0+. Therefore, if we caD also prove that it decays exponentially when u - +00,
then we may use the Mellin transform to construct a good objeet. In general, however,this is not tbe CBBe , say, we do not always have the condition that the double complex(E I tJ + v) is ä.cyclic. So in order to go furt her I we malte the basic additional aBsumptionthat (E , 8 + v) is acyclic. Theo, as in the finite dimensional case, tbe eigenvalues of thecorrespondiog Laplacian are strietly positive. So by uaing Volterra's series, it not difficult
to show tbat wheo tJ - +00,
Tr,[exp( -A~)L

Tr,[(VU(D + V) + c(~)exP(_A~)]4v u
andTr, [Nuexp( -A~)]aB decay exponentially aod uniformlyon compact subsets in B. Therefore, we can use theMellio tranaform to give the following

Theorem. Witb tbe same notation as ahove, asaume that (E, ä + v) is acyclic. Theo ,(1) For sEC, Re(,,) > 1, let
1

r+ u'Tr, [Nuexp( -A~)]-;;-.
duoo«(.,P"lf(S) := fes) Ja

As a notation, u8ually, if there ia no confusion , we mayaiso denote «(.,P.,. as (E. Then
(E ia well-defined, and ia an element in P.

(2) There exists a meromorphic continuation of «(.,p.,.(s) to the whole complex planesuch that this extension is holomorphic at s = O. In particular, it makes sense for us totalk about q.,p"Jr(O).

Chapter 1.6.

131

1.6.1.g. Certain Properties oe Tbe Construction
Here, we prove that the object constructed above satisfies the corresponding modification of the axioms for relative Bott-Chern secondary characteristic forms with respect tosmooth morphisffiS, i.e. the axioms su btraeting the term associated wi th 11". {, since we here
assurne that (E, 8 + v) is ~yclic.
Theorem. Ir (€, v) is acyclic, then(1) We'have the modified axiom 1, i.e. the double transgression formula:

1

I
8B
8(E(O)= - .1- 1 td(-Rz)Tr,[exp(-L Cp.)]·(21n) z. (2) We have the modified axiom 3, i.e.
(E I (0)1= -21r11(')' Z td( -Rz) ({.,p. (0).
I

Proof. (1) is nothing but the integrated form of the double transgression formula:
Since we have the correct decay at infinity and the right asymptotic expansion at zero, theintegrating process works weil.
The proof of (2) ia based on adeformation prOCe5s. More precisely, it comes from thefollowing two statements:(a) For t> 0, let (E,t(S) be the zeta function clssociated with the chain complex(E, 08 + v), then as an element in PI P,(E t(O) = (E(O) + A Logltl·
I(b) When t - 0+, we have[211"i](E t(O) + A' _,t

Jfz td( -Rz) ChBC({., p.).

Here A and A' are smooth forms of B.
Suppose we have (a) and (b), formally, after we consider the constant terms, we complete the proof of the theorem. But, in practice, it is not so simple: The difficult is that ingeneral p' is not elO8ed in P. Hence, in the convergenee arguments, we have to be moreearerul.
We next give a proof of (a). The b88ic idea for proving this ia to use adeformation
process. That is, we prove the result by studying the relations between the zeta functions
associated with the following complexes:

132

Conetruetion w.r.t. Smooth Morphisms(i) The Bismut supereonneetions:First, for (E, 8 + v), from the previoue diseussion, we get the associa.ted Bismut super·eonnection
Au := '1-:. + y'U(D + V) -

c(1).
4
y u

Thus, if we consider the eha.in complex (E, [) + av) for (y, a) E B X C, whieh ie acyclic for
a '# 0, theo the corresponding Bismut supereonneetion is._ "'lI'. + daT+Öd- Ö
aa- +yUC(D + VII) - c(T)1::'

Acu·- v
·

4y u

a

va

Here we set VII := av + äv·. Similarly, for t 2:: 0, we seale B, Er by tbe factor .../i, then we
may get the associated Bismut Bupereonneetions aB follows

A~ :='\7 + jü(VtD + V) - ~~;
a
a

t

- {)

C( r;

Acu:='1+da- +daa_+yuytD+V
,
a
a

(1)

c(T)
- 4.C·yuWith above, if we look at the dependenee of those elements on v, i.e., we let
Au =: Au(v), Ac,u =: Ac,u(v),then we have the foUowiog relations among the above supereonnectioDB.

(ii) The number operators.
With the Bismut supereonneetions as above, we easily know that the corresponding
number operators are Nu, Nu! Nut and Nut.
(iii) Tbe zeta functioos.
Henee, we also know tbat the associated zeta functions are respectively as folIows:(E(S)=- rts) EX> u'Tr, [Nu exp( -(Au)'))du,(E,C(S) = - q1s)

EX> u'Tr, [Nuexp( -(Ac,u)')]du,(1 (s) =- rts) EX> u'Tr, [Nutexp(-( A~)'))du,Chapter 1.6.
andckc<.) =-r;.) [0133u'Tr, [N.,exp( -( Ab,.)' )]du.

Also, if we consider them 8B a function of v, then, we get

As a corollary, we see that

and<k,c,tI'(O) = (E,C, ~/(O) - (E,C, "*(0) logt.
(iv) Tbe local family index theorem.
On the other hand, by the local famiJy index theorem for 'Ir x Id c , we see thatliIIlu_o[2'11"i]Tr,[exp( -A~,u)]=ltd( -Rz) Tr,{exp( -R(.,p.)].

So, we see that there is Da da or da term in the right hand side. Tbus, later OD, we may use
tbe trick of counting tbe Grassmannian degree in C to deduce the result.
(v) The expansion of (c'(O) with respect to C.
First, we consider tbe acyclic chain camplex (E, aä + v) on B x C· for a # O. Far(E,C, tbe corresponding zeta funetion, there exist differential forms 0o, BI, 0_ 11 and O2 on
B depending smoothly on (y, a) E B x C·, such tbat

(E,C'(O) = 00 + 81 da + 0_ 1 da + O2 dada.
Similarly, by considering the GrBBBmannian degree with respect to C, for a E C·, there
exist smooth forms 8} on B, depending smoothly on a E C· such that<h ,E' (0) = 8~ + 8~ da + O~I da + 8~ dada.
Thus, ie i a : B - B x C· is tbe embedding y 1-+ (y! a)! then

wbile
and

O~(y, 1) = i~(b,E' (0),(J~(y, 1) = i~(h,E' (0) = (~' (0).

Construction w.r. t. Smooth Morphisrns

134

Therefore, to study the relation of (E' and (Ete', we need to study 0 and ot.
(vi) Bo and 0&.
First, we study 80 . Since, by the double transgression rormula, we haveh=h

r4I d"B [2rij(E I (0) =dBxecfBxc [2ri](E,e'(0)td(- Rz )Tr, [exp (- ~ ,p)];td( -R z )Tr" [exp( -Re.,p.)]·In particular, dBxcdBXC(E,C'(O) does not contain da or dä terms. Hence, by the relationof 8BxC,äBXC with those for B aod C, we bave2

8 80 _ 8B 80 1 _ [)B 80_ 1 _ [)B8 B 0
2
8a8ä
8ä
8a

= O.

So, by the facts that 00 is a radical function of laI, that 8~;a' aeting on the radical functiooof lai r, coinddes with=

we havej"+

80 0 (r) = ![880 (1) 4[8 B (80_1 )
8r
r ßr
'1
8a

+ äB (80_ 1 ) + [)B 8B (8 2 )](b)b db].
8a.

Thus, by integration, if I ja the linear operator COO(R:+) inta itaelf such that

j

,.

r

/ ~ 1(/) : r f-to 1(/)(r) := 4 1 /(b)b Logt; db,
we have

Lemma. With the same notation as above, for a E C·,

00 = (8(0) + ~rO (l)Loglal + 8 B 1( ~;) + aB 1( 8:~1) + [)B 8 B 1(0 2 ),On the other hand, if we let rt : C - C be the map defined by a ~(E,e,~,(y,a)'(O)= r;(E,C,tI,(lIt a /J't)'(O).
In this manner, we get
1,

1 80 0OO(y, Vi) =(E (0) - 2a;:-(1)Logt

8 8 1(80 1 )( _1) 88 1(80_ 1 )( _1)
8ä Y, Vi +8a Y, Vi

+.

-8

B

1

+ a {} 1(02)(Y, Ii)'

:7t' ThenChapter 1.6.

135

Now we are ready to consider oö. For this, we use that fact that
(1,c'(1I,a/ (0)= r; (E,e,u'(lI'~)'(O)- (E,e, ~(O) Logt.

In particular 1 we getot(y, a) + O~ (y, a) da + o~ 1(y, a) dä + O;(y, a) dada

= r; (E,e,u'(lI'~)'(O)- (E,C,7i (0) Logt.

But, by definition, we get·r;(E,e,u,(v,~)'(O)=ot(y, .Jt)
1 t(

1

a

a_

t

+ ..Ji 0l V, ..Ji) da + ,;t0 -I (y, ..Ji) da
1 t

a

t

vi

_

+ -02(V, r;) dada,
so we get9~(y, a) +ot (V, a) da + O~ (y, a) da + O;(y, a) dadaI'( a)
1 t( a)1 t (a)_=90 Y'..Ji + ,;t0l y,,;t da + ViO-1 V, Vi da./t) dada - (E,C'7i (0) Log t.

+ fO~(v,

To go futher, we need an expression for (E,C,~(O).

(vii) (E,C,~(O)Ta give the value of (E,C,~(O), we start with (E(O). With the same notation as in tbeßBymptotic expansion theorem and its corollary in subsection e, by definition, we see that

(E(O)
and

'1
=-

(E (0)

o

= -Co

1

:2] - -CI - Co)du(Tr,[Nuexp(-Auluu_]O:J Tr, [Nuexp( -A;)] duu + C_ + r'(l)Co.
I

1

We could do the same thing for (E,C(S). For that, we may introduce the follawing familyof cloeed differential forrns on B:'

lz i~td((2~i)1 lz+
lz+ (2~i)1 lz

C_! (u) := (2~i)1

Co(u) =-Hz )Tr, [exp( -(\7(..p + JUV)')],:b (td( -Hz - bI)h=oTr. [exp( -(\7 (,p + JUV)')](2:i)1td( -Hz )Tr.[exp( -(\7(.,P + JUV)')]td(-Rz)Tr,[NHexp(-(V{.,p. + vU'V)2)]

1 B C
:2
- '2d EoTr,[exp(-(V{.,p. + v uV ) )].

Construction w.r .t. Smooth Morphiams

136

Here Eo ia E o calculated witb € = C endowed with its canonical bermitian metric.
Directly from the definition, note that-1

Eo =EoTr, [exp( - V"{.,p,)]+ (2:i)I.h ~i td( - Rz )(Ju

2Tr, [V exp( - (V (,p. + v'UV) )].=0,

and heuce2d B E o =d B EoTr,[exp(-V"{.,p.)]-(2~i)1l

we imrnediately get
C- 1

iwtd(- Rz )( :u Tr, [exp( -(V(.,p. + v'UV)')]u=O,= C- (O),1Co= Co(O) + C_ '(O).
1

Similarly, with the definition from above, replacing V"CI'. bya_aV"(. + da 8a + da 8aand V by VII, we mayaiso define Cc,o(u) and CC,_I(U). In this manner, using the same
prOCeBS as above, we have the following
Lemm.a. With tbe same notation as above ,( .E,C,~ (0) ='-C0 (0) _ CC,-l'(O).
t
Indeed , here oniy v ia changed to 7; I but v dOeB not appear in Co(O), 80 we have theassertion.
Now we return back to (vi). Write out the closed form CC,-I'(O) with respect to C,
we find that there exist amooth fonns K 11 K - l I K 1 on B, such that with respect to theGrassmannian degree in C

CC,_I/(O) = lal 1 C-t'(O) + aKt da + aK_ I da + K'l dada.
Thus, by the d-closed property of CC,-t'(O), we haveB,

8 K 1 = C_ I (0),a-8 K_ = C- 1, (0), a-8 K'l = K il 8 B K = -K_
1

1

1•

So, by comparing the Grassmannian degree with respect to C, from the last part of (vi)ahove, we get,a~ C_1'(O)90 (v , a) =9o(V, 17) + (Co(O) + [al)Logt;

vt

t

,la
aKt
91(v,a) = t;9t{y, j;) + -Logt;
yt
vt
t
.
t (
) _ 1
( a) aK_ t
9_ 1 Y, a -. j;9_ 1 Y, 17 + --Logt,
vt
vt
t
t
1
a
K'l
92 (y, a) =-9 2 (y, 17) + -Logt.
t

vt

t

Chapter 1.6,

137

Thus,

:0. 9dY, a) =t :0. 9~ (y, av'i) - K Log t;
1:a 9- dy ,a) =t :a(J~l(y,av'i) - K_1Logt;
93 (y, a) =t9~(Yl av'l) - I<2Log t.
So{}aI({j_9 1 (y,1fi)) =4
vt

1 a
1/

1

t

1
)bdb
vtb
b db

{}_9 1 (y,b)Log(

a

1

1
=1
=

";;

{j
1
{jo. 91(y,
)Log

Vb

fi

tb

2

8
b db
c 80. 9~(y,1)Logtb - K 1
1

1
1

t

b db

Log(b)Log tb2 ,Thus, if J ia the operator acting on COO(R+) such that

f 1-+ J(f) : t 1-+

1
1

bdb
-t b

!(b)Log-- '

t

then

{}

8

1

Lo~

2

I(8ä 91 (Y, Vi») = J(8ä(JiI G =d - K1(-t- + Logt + 't - 2).
In the same way, we have expansions fara l lI(8ä(J-l(Y, Vi)), I(9 2 (y, y'i»'
Putting them together, by the fact that(Jb{y, 1) == Ck' (0),we havet "1 a9 0'( ) Logt
CE (0) =(E(O) + (Co(O) - 2a;:-(I))Logt + C- 1 0 -t-:0.

+ {jB(J( Bi IG=d)(t) + [jB (J(+ SB{jB(J(9 2IG=dHt)
B

-B

-B B

:a

9:.. 1 IG=d)(t)

logt

2

-(8 K 1 +8 [(-1+8 8 K2)(-t-+Logt+'t-2) ..
Tbus, by tbe relation that

we have tbe the following

138

Construction w.r. t. Smooth Morphisms
Proposition. With the same notation as above, we haveo'(Et '0)(=(E(0) + (Co(O) - 1 BB ( 1) - C- 1, (O))Logt

2a;+ 20_ 1 /(0)(1-

f)

-8
B
+ (7 8( J (B8ä Bi la=d)(t) + ö (J( 8a B2Ia=d)(t)+ aB 8 B (J(B 3Ia=1 ))(t).

In particular, we get (a).
(b) We riow give tbe asymptotic expression for (~t(O) as t - 0+. For this, by thedefinition, or, better, by tbe expressiort of ('(0) used in 'tbe proof of (a), we need to know
the behavior of the corresponding integrant Tr, [Nucexp( -(A~rl)L when tbe parameter ugoes to zero and infinity respectively.
(i) We start with tbe situation for u - 0+. By definition, we know tbat this is equivalent
to studying the asymototic expansion ofwhen u' - 0+. Here, by definition, Nu' = N H + Nv,u"

We separate the above into two parts according these for Nu" For the part with N H,
as u' - 0+ we get
I

Tr,[NHexp(-(V + ND + V;v _ C(~)2)]
4vu'-(2~i)1lTd( -Rz )Tr, [N Hexp( -('Y + v'üV)')].

So it is sufficient to study the behavior ofTr, [N v,u,exp( -('Y + ND + v'üv _:j,?,)2)),

as u' - 0+. Now we use the same method as what we did in the proof of the theorem insubsection e: That is, to express.

r:

Tr,[Nv,u,exp(-(V + vu' D +c(T)vu V - 4y';l))]2Chapter 1.6.

139

aB a eombination of the super-trace for the heat kernel of certain essential generalized Laplacians. For doing so, by a loeal diseuBSion, we see thata~' (u'Tr, [Nv,u1exp( -(A~' /U)2)]),_ c(T 1,O)
=Tr,[exp(-(A~ /U)2 - {N8 +Ci )da4vu'- (Ha- + c{ro;» da _ iwz da da»]da da
4vu'

I
1/
+ 2Tr,[exp{-{A~U)2)]- dBTr,[Nv,ulexp(-{A~'/U)1 + (ND +:jJ,) du,)]du'.

In particular, after following wha.t we did for the eorollary of subsection e, we find thatThat is, we have the following
Lemma 1. For u> 0, aB t - 0+,

Tr, [ NUfexp ( - ( Auf )2)] = C- {u) + Co(u) + Ou{ut).
ut1(ii) Now we consider tbe uniform eatimates aB u - +00. For thia, we may use themethod of Berline aud Vergne in section 5.5. Thus, by the fact that the complex E isacyclic, we have
Lemma 2. For any eompact subset K of B,' there exist CK > 0, d K > 0 such that foru ~ 1, t > 0, Y E K, we have(iii) Now we are ready to obtain the asymptotic expansion of (E f with respect to t, aB

t - 0+. Ta state tbe result, we make the following definition: For ~(s) > 1, Set
1

(CO

1

[CO,

..\0(8) = r{s) Jo,..\1(8) =r(s) Ja

duu'Co{u)-;;duu C- 1{u)-;.

140

Construction w.r.t. Smooth Morphisms

Then Ao and Al may be extended to meromorphic functions on the whole complex plane,
which are holomorphic at s O.

=

Proposition. There existB an element ß E P such that as t - 0+
(E,t(O) =+ +'\' (0),\ti(O) + ßt + o(t),and 9(t) is uniform over compact subsets in B.

Proo{. First, by definition, we know that(~' (0) = r (Tr, [Nurexp( _(A~)2)] _ C- 1(0) + C-1'(O)U _ Co(O)) du1

Jo+ ]100utt2]du C- 1(0)Tr, Nurexp -(Au) ) -;- + - t -

[

u

(

_ r'(I)(Co(O) + C- 1'(0);.

tThus, by Lemma 1, for u :5 I, we have IOu (ut) I :5 Cut. Thus if t - 0+, we have
1

1

1

o

du
Ou(ut)-j :5 Ct.
ur (Tr [Nu,exp( -(A~ )2)}_ C- 1(0) +u7- 1'(0)u _ Co(O)) duu
Ja

So

1

8

r C_ (u) - C_1(0) - C_ '(0)u du

=~

1

1

1

t Ja

+

u
o

u

du
(Co(u) - Co(O))- + o(t).

11uOn the other hand, by Lemma 2, we have]+00 Tr. [Nu1exp( -(A~ )'))duC- (u) du ]+00 C ( ) du
( )=-1 ]+00 --+u -+g t.
1

0

t

1

U

U

1

U

Here €(t) ia such that lilIlt_o€(t) = O. Hence, if we let t(t) = E(t) + O(tL then(t '(0) =~[E

r C- (u) - C-1(0) - C_1'(O)u du
1

1

t Jo

u

U

f+co C_ 1 (u) du + C_1(O) + r'(l)C_l'(O)]

J1

U

1
1

o

U

du

(Co(u) - Co(O»- u]+00 Co(u)du+ r'(l)Co(O) + t(t).
1UChapter 1.6.

141

Therefore,(1' (0) = '\'-1 (0) + ,\' 0(0) + t(t).
tThis completes the prüof of the proposition, and hence (b).
Now we may finish the proof of the theorem. First, we know that (1; c' (0) has an
asymptotic expansion simtlar to that in the above lemmas, simply replacing B by B x C·.
Moreover, we easily see that~gt
I
Bö 1 Ia=I, !!...g'{ja -1 a=I,

g~la:;;1

have similar expansions. Thus, by the fact that(J( b»(t)=/.1 Logtb dbb = Logt - 1 - "t;1(J(l»)(t)

=

1

t

J

b db

1
t

/.

Log

1

tb = 2

LogJt ,

we know that aB t - 0+,:s :sHere 0{ (1i 4) are Coo forms on B snd K(t) - 0 aa t - 0+. Furthermore, one may
verify that such expansions may be differentiated and we' can apply the operator 8 B on botb
sides.of the tbe above equation and 8 B K(t) - 0 a.s t - 0+. Similarly, we may do aB of thesefor8J( 8a 8:.. 1 Ia =d(t), J(8ila=d(t).
Hence, using (a) snd (b), if we identify the .constant term in the expansion of (1'(0) aB

t - 0+, we find'\0'(0) - (E'(O) - 2q-l'(0) E P.

But we know that C_ 1'(0) E P', so'\0'(0) - (E'(O) E P.
On tbe other band, by tbe fact thatCo (u) ;; (2~i) IlTd( - Rz )Tc. [NHexp( -(V + "t'UV)')),

aB in the finite dimensional case, we have that

Ao'(O) -(2~i)1lTd( -Rz )({,P'(O).

Construction w.r.t. Smooth Morphisms

142

Here, <€.,p.(s) is the zeta funetion associated with the exact sequence (~.,p.) on M, which
is a finite dimensional version. Therefore, hy the degree counting,<E'(O) - _1.-1 f td( -R z ) (€-'P.'(O) E pi.
(211'1) }z
This completes the proof öf the theorem.

Remark. We mayaiso discuss the dependence of (E' (0) on the choiee of the metrie 9Z
on the fibre. For doing that, we also should use adeformation process as in the above proofof (2) of the theorem. More precisely, we may first consider the variation of the metric bya rational family tg z + (1 - t)gl z. In this ease, we then deformation the double copmlex(E, [j + v) aB stated in subseetioD a vertically. Thus the similar discussion aB above could
offers us a corresponding result for tbis change: We should first express the associated terms
as a combination of the supre-trace of certain trace elasses, say Tr,exp( -A~ + B u ). Then,
one may use a local diacussion for th"e essential generalized Laplacians -A~ + B u I whichbave similar structures aa what are for the super·Lechnerowicz formula, so that we could
wirte the associated asymptotic expansions down.. For the gereust situation, one may use apartition of unity to deduce the final answer. We will not give the fuH details here, aB tbe
principle behind this now becomes quite elear, and the discussion ia ratber duB and tedious.

§I.6.2. The Construction In General
From above, we know tbat if we put the acyclic condition on (E, [j + v) then thecorresponding

decays exponentially wben u - 00: Sirrce in this case, the correaponding Laplacian ia
positive and self-adjoint, hence the eigenvalues are strictly positive. Hut in general, this ia
not always the C88e. In this seetion, following Faltings, we give a method which removes
this technical assumption.
[n fact, it is not very difficult. Recall that from sbeaf theory, for aoy two complexes E.
and F. of coherent !heaves with a quasi-isomorphism 4>. : E. - F., then there is the cone
construction, cone(4).), such that we have the following exact sequence:

0- F. - cone(,p.) - E.{1]- 0,
which is defined as folIows:

E.[l]P := EP+ 1 ,dE.[I]= -dEoandcone(,p.) := E.[l] EB F.,

143

Chapter 1.6.
where the differential operator is given by(

dEO'[lJ«P.[1])dF .

.

In this sense, at the level of cohomology classes, cone( t/J) may be thought of formally as thesummation of F. and E'[ll' Le. the difference of F. and E .. In this section, we prove that
this can go througb even when we study everything at the level of differential fonns. In
order to da BO, we bave ta find out the key rule behind the very complicated constructionsin tbe previaus section.

1.6.2.a. Key Observations
We start with tbe finite dimensional situation. When necessary, we suggest tbat tbe
reader refers to eection 4.5.
Suppose M is a complex manifold aod (E., v.) is a finite complex of bolomorpbic vector
sheaves on M. Let Pi he the bermitian metrics on Ei' Tben E := (!JE; has a hermitian
metric P and a canonical connection 'l. Let v· he the formal adjoint of v with respect tothis metric. Tben we have the superconoection 'lv := 'l + v + v· on E, and ('l + v + V·)1
is of type (1,1). We know that[21ri]Tr, [exp( -('l + v + v· )2)]
represents the Chern charactercb(E.)=:E(-1)ich(E;);

in cohomology. Also, the cl8BSical Bott-Chern secondary characteriatic fonns me88ure thiscooatructioo under tbe change of metries at the level of differential forms.
Hy tbe construction in sectiona 4.5 and 6.1, we know that tbe number operator plays avery important role. Previously, we gave a precise description of the number operator. But
bere we look at this in an essential way, Le. we consider the number operator as the one
being the me8Bure of the change of metrics. There are two ways to change the metrics: Oneis for the vecto~ aheaf E, Mother ia for the Kähler metric 9M. We discUBB them separately.
First we discuss tbe change of the metric P on E, hut instead of discussing it in general,
we only deal with tbe infinitesimal variations: Suppose we have a one-parameter family of
metrics on E. defined by< eI, f2 >~ :=< eI, e2 > +e < fI, N(e2) > +O(€,2),
where N ja a self-adjoint endomorphism of E.. (For example, if we 8BBume that (E.,II.) is
acyclic, and we multiply tbe metric on E; by a factor ti with t > 0, tben we know that the144

Conatruction w.r.t. Smooth Morphisms

ordinary number operator N, wbicb is defined by the multiplication by j on Ej, is given by
the above definition.) With respect to this change of metrics, we know that 'V~ = V" + v
remains unchanged, while 'V~ = V' + v- changes by -€ [N, 'V v]. Since the conjugation does
not change the trace, 50 we may conjugate with 1 + ~ N to get a more symmetrie situation,
Le. tbe variation in connections ia given byThus, we have

8 Tr, [exp( - 'V~ ,)],=0
8€
'

=

:!

Tr" [exp( - 'V: -~(V v [N, 'V~ - 'V~]])]'=OI= ~ Tr I [[V V I (V'~ - V':, N])exp( - V'~)]=~dTrI[[V'~ - V~, N]exp(-V~)].
On the other hand, we know that

dTr, [Nexp( - V'~)]=Tr" [[V' v, N]exp( ~V';)]= Tr, [[V~, N]exp( -V'~)] + Tr,[[V'~, N]exp( - V;)].
Hy comparing the holomorphic aod antiholomorphic parts, we see that the two terms are
just 8 and 8, respectively, applied to Tr,[Nexp( - V'e)]. So finally, we have

So locally, we may let
chBc(E" N) := [211'i]Tr, [Nexp( - V'~)],
we will baveßdcJCehBc(&., N) = ßg ch(E., P,,),

In particular, if E. is acyc1ic, we may have the integral version, w hich is just our constructionof c1assical Bott·Cbern secondary characteristic forms in section 4.5, since, in this eBBe, at
infinity, we bave the exponential decay.

Now we consider tbe change of gM. In thia case, the Kähler form W M = - ~ L dZj I\dzj
ia changed by !C(WM), Here dZj denotes a loeal orthonormal basis. (Tbe re8BOn for us toeonsider this change comes from the definition of the Bismut superconnection.) This change
effects the metric on &0 An~l in two ways:
First the volume on M chan'ges, and then so does the hermitian metrie on n~/. For
the first eBSe, if we let Q E End(~~o) denote the hermitian operator, which infinitesimallyChapter 1.6.

145

generates the change of metrics, then this first effect ia measured by the scalar number
operator Tr(Q). For the Becond one, the number operator is given byThus the total number op.erator is given byThus, far in tbe relative situation, we may have the follows:
Let,... : M -+ B be a Kähler fibration. Tben we study tbe metrics UM, := UM +on M. By the definition of the Bismut superconneetion, we know that 'the Bismut
superconnection may be tbougbt of 88 the limit of the corresponding ordinary Clifford
superconnection 88 8 -+ 0+. (For this, see tbe definition in section 5.1.) Therefore, we mayneed to consider the corresponding limit for the total number operator N, 88 introduced
above. We change tbe notation and denote tbis limit as N. The formulas at the beginningof this subsedion for the finite dimensional case now turn to be the one which describes thedependence of the Bismut superconnection B on ux. For example, we then could have
S-lUB

B

2· -

2

BE Tr,[exp( -B ))':0 = 8BTr, [Nexp( -8 )],or, formally, by the local family index theorem,

To demoDBtrate how the above relation works, we go as folIows: Introduce a new parametert on repladng the original metric gM by t-1gM, which could introduce a family of Bismut
auperconnectioDB B,. Then, for t -+ 0+, the asymptotic expansions of the above relation go
88 follows: (For this, cODsult 6.1.d.)On the len hand aide, we obtain the derivative of f z td(M/ B)ch(E), which is dBdB ofhtdec(M/B,Q)ch(E).

On the right hand aide, we know that the t- 1 term in Tr,(N texp( -Bn) is equal to aclosed form f z ~td(M / B)ch(E) , 88 what we gave in the proof of Theorem 1.g. Furthermore I tbe corresponding tO term is tbe sum of.

!2 Jz( Tr(QIT.)td(M/B)cb(t') and }zf Aec(Tr,Q)cb(l').
So, we may get the coinddence. In fact, for tO, tbe first term is the contributioD of the faetor
exp( ~Cl (Tr )) to tbe secondary cbaracteristic dass tdec(Tr , Q) associated to the Todd genus

Construction w.r.t. Smooth Morphisms

146

=

iCl

td(T", ) A( T" )exp((T., »), so the sum of these two terms is j ust f z td ac (M/ B, Q)ch( E).
This proves the claim above.
AB we said above, the Bismut superconnection may be thought of as the limit of theassociated Clifford superconnection with the blow·up of the metries 9M" = 9M + S-1 tr- 9B
when s -. 0+. We introduce a parameter t by rescaling the metric 9M as t- 19M, whichatrects the fibre part, amt then we have the super-Lichnerowicz formula for them, and soon. In th~ next subsection, we use all these observations to give the corresponding concepts
for tbe cone construction.

1.6.2.b. The Situation For The Cone Construction
Let 11' : M - B be a Kähler fibration aud (E, p) be a 'll"-acyclic vector sheaf on M.
There is a natural morphism
11: 71:.[ -. 11"_([ @ An~,1).

Here 1(.[ is with the natural push-out hermitian metric 'll"_p. Hence we have the associatedmapping cone, cone(v). On this mapping cone, we consider the Buper-Dirac operator
A cone := 'V "'.(& ,p) + B + 11 +

v·

v;.

v·

with
the limit of the formal adjoints
when 8 - . 0+. This contains terms of positive
Graasmannian degree because tbe puB-back metrie on ",·OB may not coincide with its
subspace metrie induced from nM. Such a difference in metrics, in the limit, is given by
the number operator N, anddiffers from the naive v· by a factor exp(N). Furthermore,
we rescale the metrie gM aB t- l gM, by introdueing the parameter t. At the same time, themetrie on 7r.[ @ An~ 1 is scaled by an addi tianal factar t l with I the relative dimension of 11",
to aecount for the volume forms. (This makes sections of 7r.E uniformly square integrable
on tbe fibers of 11".) In this way, we have Acone,t, aod we denote tbe associated number
operator by Ne. Tben by section 5.2, we know tbat there exist heat kerneis exp(-A~one t)
for the essential generalized Laplacian A~one t. As one may imagine, in order to give therelative Bott-Chern secondary cbaracteristic forms associated with tbe ir-acyclic hermitian
vecter sheaf (E, p), we use the Mellio transform ferv·

Tr,[N t exp(-A~one,t)]·Therefore, we need to censider the behavior of this super·trace when t -- 0+ and t -- +00
respectively.

1.6.2.c. Conditions For The Mellin TransformLet 1( : M - B be a Kähler fibration, ([, p) a 1l"-acyclic hermitian vector sheaf on
M. For the metric g"t := t- l (gM + 8- 11l"·9B), consider the Laplacian ß"t = t ß,,1 on the
mapping cone ofChapter 1.6.

147

On COO(Y,1r.. E @ I\n~l) the metric is scaled by an additional factor t , to account for the'volume fonns.
By the res u I t in su bsection 6.1.d, we know that it is enough to consider the effect of theterms associated with 11. As an illustration, we treat first tbe absolute CMe, Le. wben Y is apoint. In this C88e, 11 and aB its derivatives are uniformly trace dass. Hy Duhamel's formuls,
the corresponding super-traces for Bismut superconnections have an asymptotic expansion
when t - 0+. Furthermore, by tbe fact that 11 only influences terms with positive t-power,
we have the same formula aB in subseetion 6.1.e. On the other hand, the exponential decay
is a direct consequence of the fact that the complex ia acydic and thus aH eigenvalues are
positive.
The b88ic idea to treat tbe relative situation in general ia that we first consider theobject for the ordinary Laplacians, then consider the limits for t, aod finaUy let 8 - 0+ to
get tbe result. If we proceed in this way, one may ask that why the final resulta are juat tbe
same as when we first let 3 - 0+ and tben take the limit for t. The answer is simple: theconstructions do not depend on the paths we choose. As an example, we eonsider the heat
kerneIs exp( -~"t). We want to prove that when t .... +00, exp( -~"t} decays expooentiallytogether with aU its s-derivatives: By a loeal discussion aB in the proof of the loeal family
index theorem, it ie enough to sbow that the associated Laplacian have strictly positive
eigenvalues. Hence, we need to sbow the following
Proposition. With the same notation aB above, tbe eigenvalues of ß"l are uniformly
bounded below by a fixed , > 0, whicb is independent of 3.

=l
Proof. Let y E B and Z
1r- (y). Near Z, we identify M with Z x TB,rJ' Scaling tbe
coordinates on tbe second factor by Jä, we know that ßO,l is a direct sum of tbe Laplacianof TB,l' aod tbe relative Laplacian on the fiber Z. On the other hand, by our construction,
we know tbat tbe cone is acyclic &long the fiber, aB we assurne tbat E is 1r acyclic, 80 the
relative Laplacian has positive eigenvalues. So by the perturbation pro CesB , it follows thatfor each y, tbere exista a smaIl neighborhood Uy in B and an tl' > 0 such that for any
COO-section witb support in Uv , and for 8 small enougb, we bave

< ß"t(/),/ > ~ 2ty < /,/ > .
Moreover, by tbe compactness of B, we may choose finitely many UrJ and hence a t > 0,
which is independent of y. Therefore we may UBe the Sobolev estimation uniformly in s withthe metric 9,,1 by the finiteness stated above. In particular, we have a uniform Garding.
inequality for ß"l- From this, tbe assertion may be dedueed easily: In fact, we may choosea small open cover Ui of B sucb that if / has support in Ui, then< ß"t(!),! > ~ 2, < !,I > .
Now chooee Coo functioDB 4Ji with support in Ui such that Ei t/>~ = 1 is a partition of unity.
Tben tbe Sobolev norm of [d"ll t/>il is O( y'i). So, for a Coo section ! on M 1 we have<ß"l(!),! >=E < tPiß"df),tPif >i=

L < ß"l(tPi/),tPiJ > - L: < [ß"l,tPil(!),tPif > .
iiConstruction w.r.t. Smooth Morphisms

148

Since 4Jil has support in Ui, the first term is ~ 2t; < 1,1 >, while the second one ia bounded
by a fixed multi pIe of y'8( < 6.,,1 (I), 1 > + < I, 1 >). Thus for s small enough, 'we have
<6."l(/),/>~f: </,/>.

This completea the proof.
With this lemma, we. know that aB t - +00, sdimB exp( -t6."l) decays exponentiaUy
together wi th aU its s-derivati ves. In other words, the asymptotic s-expansion of exp( - tLl" 1)
aB weil aB a1l remainder terms, decay exponentially at infinity.. From the fact that the 8expansion starts with s-dimB, and the sp-dimB_terms have Clifford filtration degrees in B
at most 2p, we know that when t - +00, the super-objects for the cone construction decay
expansionaHy, since tbe super·traee vanishes on the low Clifford degree terms.
Finally, we consider the asymptotic expansion for t - 0+. As we stated above, we firsteonsider the 8Bymptotic expansions 88 t -+ 0+ for a fixed s. Hence, we UBe a cut-off in B
and seale the B-coordinatea by
ThuB 6."t beeomes aperturbation t6.z + 6.TB,.' We
know that the coefficient of t dimB exp( -tLl"d is G oo in Vi, Furthermore, the terms with
low t-power also have low Clifford degree in B. On the other hand, the terms involving
11 have at least a Vi in front of them, and have Clifford degree zero. Hence, they do notcontribute to the leading terms. So, taking the super-trace and letting s - 0+, we get tbe
formula similar to that in subsection 6.l.e. That ia, when restrieted to the diagonal, there
is no negative t-power, and the eonstant term is independent of 11, Similarly, we mayaiso
do this for tbe objects by twisting the number operator, as it eould be represented as aeombination of eertain trace classes. In summary, we bave the following

Vi,

Theorem. (1) There exist Goo even differential forms Ao,A t , ... ,G-l,Ga, ... in P
such that for any k E N,
C

1,'
Tr, [exp( -Acone,t)]= ,L...t, "Aj1;+ o( U c),okTr, [N t exp( - Acone,t)] = L...t Gj u1 + o( U c ),
1

,,",'

j=-l

where 0 is uniform on compaet subsets of B. Furthermore Ao, G- 1 aod Co are independent of 11.
(2) Wben t -+ +00, Tr, [exp( -A;one,t)L Tr, [Nt exp( -A;one,t)] decay exponentially.

I.6.2.d. The CODstrnction In General
With the results in the previous subsections, we may make the following

Proposition and Definition. (1) For sEC, Re( 8) > 1, let

«(.

.t

,p.,'II"r+ u"Tr, [N u exp( - A~one,u)] ~.
du(s) := r( s) Ja

ooChapter 1.6.

149

Then ({"P., .. (s) ia well-defined and is an element in P.
(2) There exists a meromorphic continuation of ((.,p., .. (s) to the whole complex planesuch that this extension is holomorphic at s
O. Hence it makes sense for us to talk
about
(O).

=

q.,P....

Now we have tbe relative Bott-Chern secondary characteristic forrns by the following..

Main Theorem. ChBC(E,p; 1r,P.. ) = (21ril({.,p.,.. (0).
The proof of this theorem is given in the next seetion.

§I.6.3. Checldng The Axioms
To prove the theorem at the end of last sedion, we have to check the axioms for relative
Bott-Chern secondary characteristic forms with respect to smooth morphisms.

1.6.3.a. DOWDStairS Rule
We start with the proof of the foUowing equality:

Tbis is a direct consequence of the double transgression formula. In fact, by the doubletransgression formula, it is enough to show that the limits of the corresponding heat kerneis
associated with tbe Bismut superconnection for the cone construction behave aB follows:

AB t - +00, tbey decay exponentially; and aB t - 0+, the limit isf. (ch(E, p)td(T.. , P.. )) - ch( 1I'.E, 1t'.p).
For t - +00, the assertion is obvious by Theorem c. On tbe other hand, as t - 0+,
the lirnit hBB two parte: one comes from the local family index theorem associated with
1I'.(E ® An~·l), whicb gives tbe term11".( ch(E p)td(T.. , P.. )).
I

The otber part comes from 1r.(E,p), which gives the term

einte tbe cone construttion now will just give U8 the differente between them. So we haveaxiom 1.

Construction w.r.t. Smooth Morphisms

150

I.6.3.b. Functorial Rule
This axiom simply states that the relative Bott-Chern secondary characteristic forms
are compatible with Hat hase changes. But this is easily checked hy the fact that everythingin our definition for relative Bott-Chern secondary characteristic farms ia compatihle with
the Hat hase change. The~efore, we have axiom 2.

1.6.3.c. Triangle Rule For Vector Sheaves
There are two different ways to check axiom 3: One is obtained by using Theorem l.g
and Theorem 5.5. The other is obtained by using the pI-deformation.
We first give a proof using Theorem l.g. With this theorem, the concluaion is easy, as
we know that the cone construction ffiay be thought of as the difference of terms associatedwith 7f'. (E r&l "n~,l) and 1r.(E, p). Thua, write each term down, in which only the push-out
matters, we have axiom 3. Next, we use the pI-deformation to check the axiom.
The pI-deformation methad reHes on axioms 1 and 2. In fact, by the canstruction ofcl~ical Bott-Chern secondary characteristic fOrIns in section 1.2, we know thatchBc(E.; PI, P1., P3)
.

=

1.

pi

QoglzI2]ch(DE2 , Dp2)'

Then by axiom 1, we have(;r x IdpI).(ch(DE2,Dp2)td(TMxP1IBxPI,9MxPI/BXPI»=ddcch BC ( DE2 , DP2, 7f' X Idpl, 9M x pt I Bx pI)+ch((1l" X Id p t).DE2,(1l" x Id p t).Dp2).
On the other hand, we have1r.(ChBC(E., PI, P2, P3) td(TMI B, gMI B»
=

1.

pi

[loglzI2]((1r X Idpl ).(ch(DE2, Dp2)td(TMxPI/BxPt, gMxPljBxPI »).

Therefore,
1r.(ChBC(E.,plt P2,p3)td(TM IB, 9MIB))

=

1.
+

1.

[loglzl ] d~chBC(D[21 DP2, 7f' X Idpt, 9MxP t /Bxp,)
2

pi

1.

pt

[loglzI2]ch((7f' X Idpl ).DE2 , (7f' X Idp, ).Dp2)= pt dd [loglzI 2]ch BC ( DE2 , DP2,
+

C

1.

pi

1r X Idpt, 9Mx pli

BXP')

loglzI2 ch(( 1f' x Idpl). DE21 (7l'" X Idpl). Dp2)'

Chapter 1.6.

151

Then by the following equation of currents,

we have
7T.

(ChBC(&~' PI, P2, Ps) td(TM / B, gM/B»=i~chBC(D&21Dhi 7T X Idpl 9MxPl/BxPl)
I- i~chBC(DE~, DP2; 11" X Idpl, 9MxPt / BXpl)+

1. [loglzl~]ch((pi

11" X Id p l). Dt:'~, (:Ir X Idpl). Dj12).

Now, by axiom 2 or better by tbe constructions, we know thati öchBC(DE2 DP2; 1T X Idpl, 9MxPl/BXPl)= ChBC( iöDE'l iöDp'lj 11", 9M/ B)= CbBC(E2 ,p'l;11", UM/B),
I

I

and
i~chBC(DE'l, DfJ2, 1r X Idpl, 9MxPl/BxPl)= ch BC ( i;" DE'l' i:;' Dp'l j 1r, UM/ B)= ChBC(EI EBE3,Pl EB/l3j1r,UM/B).
Obviously, our construction is compatible with the direct SUffi, so we havei:;'cb Bc (DE2, Dp'li 11" X Idpl, UMxPI/BxPI)= ChBC(E 1 ,Plj 1I",9M/B) + chBC(Es , P3; 1r, UM/B).
Then, by tbe fact tbat
CbBC(1I".E., 1r.Pll 11"./l2, 1r.P3)
=

1.

pt

log]zl'lch((1r X Id p l).DE'l,(1r X Id p l).Dp2),

we bave

CbBC(E'l' P2i '11", 9~) - ch BC (E1 , PI i 11", U.. ) - ch BC (E3, P3i 1r, Ur)chBC(E., p.) td(T.. , g..
ch BC (7T.E., 1r.P.).

="".(

»-

Tbis is just axiom 3.
Remark. From tbe proof above, we know that we may use tbe degenerate triangles inaxiom 3. That is,152Construction w.r .t. Smooth Morphisms

Axiom 3'. (Degenerate Triangle Rule For Hermitian Vector Sheaves) For any I-acyclic
vector sheaf E with hermitian metrics p and pI. We have

ChBC(E, p; 1,9X/Y) - chec(E, pI; 1,9X/Y)= 1.(chBC(E; P, p')td(Tx / y , gx/y)) - chec(/.E; I.p, I.p').
In addition, let (E , p) ~nd (:F, T) be two I-acyclic hermitian vector sheaves on X. ThenchBC(E e:F, p e Tj I, 9x/y)= chBC(E, p; I, 9X/y) + chBC(:F, T; I, 9x/y).

I.6.3.d. Triangle Rule For MorphislDB
After the proof in the last subsection for hermitian vettor sheaves, one may naturally
use pI-deformation to deduce axiom 4 for smooth morphisIIlB from the following
Axiom 4'. (Degenerate Triangle Rule For Morphisms) Let I : X - Y be a smoothmorphism of complex compact manifolds. Let (E,p) be an I-acyclic hermitian vector
sheaf. Let gx/y, g'x/y be two bermitian metrics on Tx / y . ThenchBC(E, Pi I,gx/y) - chec(E, P;I,9'x/y)=1.(ch(E, p)tdec(Tx / y , gx/y, g'x/y)).
Here td BC (Tx/ y , 9X / y, 9'x/ y) is tbe classical Bott-Chern secondary characteristic form
8880ciated with the Todd dass with respect to the metrics gx/y, g'x/y and the relative
hermitian tangent sbeaf. In addition, if 9 : Y - Z is another smooth morphismwith I.E being g-acyclic. Suppose that the associated short exact sequence of relative
hermitian tangent sheaves is split, then

ChBC(E, p;g 0 I, 9X/Z)=ChBC(/.E, I.p; g, gy/z) + g.( ChBC(E P; /, 9x/y )td(Ty/ z , h y / z )).
l

In practice, we proceed 88 folIows. Recall the data: We have three Kähler fibrationBf : X - Y, 9 : Y - Z, goi: X - Z and an I-acyclic vector sheaf E on X such that g.E
is g-acyclic. Thus we have the following natural morphisms:

f.E L f.(E 0 An~,l);g.(/.E) L g.(/.E ® An~,l);(g 0 f).E g.:j (g 0 f). (E ~ An~~~),
PrOfi them, we have associated cone constructions, and the relative Bott-Chern secondary
charaeteristic forms with respect to f, 9 and gof, respectively. So, in order to discuss theChapter 1.6.

153

relations of tbe relative Bott-Chern secondary characteristic fonns for these morphisrns, weconsider the following diagram:g.(f.E)

L g.(f.E0tS2~,1) Lg.(f.(E0An~,1)0An~,1)

11

11

11

g.(f.E)

L. g.(f.E 0 An~,l) Lg. (f. (E 0 An~,l 0 f· A n~,l))

I1

11

11

g.(f.E)

1. g.(f.&0An~,1).L

g.(f.(&0An~~1/))

I1(g 0 f).&11

Lg.(f.E '9 An~,l)1I

L(g 0 f).(& 0 An~~l/)11

11(g 0 f).E(g 0 f). (E 0 An~~l/)'9.ti

Thus, we do as before to formulate the cone constructions with respect to the morphism 9, j
aod j. We also introduee the assoeiated superconnections, number operators and then tbeir
beat kerneIs respectively. Finally, one haa similar elements by using the Mellin transforms.
Denote those final elements byebBC(eone(g», eh Bc ( eone(j), eh Bc (eone(j).
Then, tbe above diagram and the fact that ch Be corresponds to tbe difference of the complex
glve
ChBc(cone(g) + chBc(cone(j» = ehBc(cone(g 0 I»).

Also the difference of chBc(cone(j» and chec(eone(j» is given by

1,ch(E,p)tdBc(X, Y, Z).

(Tbis is a consequence of the discussion of the remark at the end of subsection l.g.) So, inorder to check axiom 4, it is enough to show that
ChBc(cone(j»=g.(ehBc(eone(/» td(~,g,».

But tbe proof of this equality is not difficult. We have

I.E .!... f.(& 0 ~n~·l);g.(/.& 0I\n~,I)

L 9.(/.(& 0I\n~·1) 0 An~,l).

Thus, at leaat, formally by the results in the previous sectioos, we know that tbe additionalterm I\n~,l will give the td(1j,g,). Hence we have the assertion. Even in practice, thisstatement werks weIl, since when we take the integration with respect to the parameters,
our eonstruction does not depend 00 the path we eho08e. Therefore by chaoging the orderof taking the limits with respect to the parameters, we have the ahove relation.as we did inseetion 6.2.
Up to now, we have already proved the existence of the relative Bott-Chern secondary
ebaracteristic forms with respect to smooth morphismB. There is another statement for theuniqueness of our object. We will p08tpone the proof of this uniqueness after we have thearithmetie Riemann-Roeb theorem in part 11.

154

Axioms w.r.t. Closed Immersions

Chapter 1.7
Relative Bott-ehern Secondary Characteristic Currents
With Respect To Closed Immersions I: Axioms
In the previoUB chapters, we proved the existence of classical Bott-ehern secondary
characteristic forrns and the existence of relative Bott-ehern secondary characteriatic forms
with respect to smooth morphisms. In this chapter, we obtain the axioms for the relative
Bott-Chern secondary characteristic currents witb respect to closed immersions.
We first consider tbe classical Riemann-Roch· theorem for closed immersions. Let i :M' ~ M be a closed immersion of complex manifolds. There exists an exact sequence:

On the other hand, for any vector aheaf Tl on M', tbe direct image i. Tl is usually not a vector
sheaf on M: It is ooly a coherent sheaf on M. Hy classical sheaf theory, there exista a vector
aheaf resolution of i.7J on M:

o- En -•.• -EI - Eo - i.7J - 0,or

E.-i.7J-O.
Then the c1assical Riemann- Roch theorem for c10sed immersion says that we have the following equality at the level of cohomology classes, Le. in CH(M)Q,ch(i.7]) = i.(td(N)-l ch(7J)).
Thus by the fact that i. TI= Li (-1)i Ej, we havech(E.) = i.(td(N)-l ch(7])).

(For all of tbis, see 11.1.)
Now we consider the problem at the level of differential forms. Then we may put metries
on the exact sequence of normal sheaves. Also, even through i.7J is only a coherent sheaf,
we may still put tbe metrics on Ej. Just aB for smooth morphisffiS, a natural question ia
how we caD meaaure the change of (7], 91/), after the action of the c10sed immersion i, at theChapter I. 7.

155level of differential forms. It is for this reason that we introduce the relative Bott-ehern
secondary characteristic currents with respect to c10sed immersions, ehBc (11,9,,; i, 9d. (Herewe have to UBe the language of currents, M, at least formally, the i.-image of a form may
be written as the product of this form with the Dirac symbol 0M of M' in M.) In practice,
we have the downstairs rule as follows:.

dd C ch BC (11, 9,,; i, 9i)= td(N, 9.v)-1 ch(11, 917) 6~ - ch(t:., p.).

Nevertheless, the situation ia not sa simple. We know that the metrics on E. are notunique and, in general, we cannot control them very weil. In order to introduce the relative
Bott-ehern secondary characteristic currents with respect to closed immersions, we need atechnical assumption on the metrics, which is nothing but the so-called Bismut condition(A), which gives certain compatibility condition for the ßBSOciated metrics. The references
here are [B 90] and [We 91].

§I.7.1 Basic Facts Associated With Closed Immersions and Resolutions
1.7.1.a. AssumptioDS and Notations.
In this subsection, we recall same basic facts 8880ciated with closed immersions andresolutions of tbe direct image of a vector sheaf. Even througb we only deal with tbe closed
immersions, since there is 00 further difficulty or complexity, we work with the following
data:(1) A closed immersion i : M' c........ M with M' U? Mj. Each Mj is a compact connectedcomplex submanifolds of dimension Ij + I' of a dimension I + I' complex manifold M,
such that if j #; i', Mj n Mj, ==(2) A submersion 1r : M - B which restricts to a submersion 1r : M' - B, wbere B is acompact connected complex manifold of dimension 1'. Tbe fibers Z of 7r are compact
connected complex submanifolds of dimension I, and tbe fibers Yj for Mj are compact
connected submanifolds of dimension Ij. We let Y
Yj , and deootei: Y c........ Z tbe
induced closed immersion.
3. A hermitian vector sbeaf ('1,9'1) on MI witb a vector sbeaf resolution of the coherent
sheaf i.TI by achain complex of vector sheaves on M:

=

e.

=Ur

We also UBe tbe following notation: TR is the real tangent bundle snd T the (1,0) partof the bundle Ta ®R C. For 1 ::; j ::; ", NRJ ia the real normal bundle of Mj in M j aod Nj
is the (1,0) part of NR,j ~R C. We often write NR, N instead of NR,j, Nj, etc .. For any
vector bundle, tbe dual is denoted by a symbol .; for example, Ti is the dual of TR.

156

Axioms w.r.t. Closed Immersions
We have the following exact sequences

o-TZ -TM -1r·TB -0,
o-TY -+ TM' - 1r·TB - 0,
o -TJ\1' -+ TMIM' - N - 0,
o -TY -+ TZIM' - N - O.
Also there exists an exact sequence

Moreover, the complex (f, v.) is acyclic on M - M'.

1.7.l.b. The Homology of (E., V.)IM'.
For 1 ::s j ::s n, let Trj : Nj - MJ be the natural projection. Ir E is a vector bundle on
Mi, we will also denote by E tbe vector bundle 1fj (E) on tbe total space of Nj. Consider
Mi as tbe zero section of Nj, via kj. Tben on N j , we have the following exact sequence:

tbe K08Zul complex, with i y tbe interior multiplication for y E Nj.
For any z E Mi, there exiat holomorphic coordinates (z 1 , ... , zl) on an open neighborhood U of z in M such that Mj n U ia represented by zl
0, ... , zl-I;
O. On
Mj n U, Nj is spanned by tbe farIns dz 1 , ... , dz f - ;, whicb extend to tbe whole open set

=

=

'

U. Hence Nl on Mj n U extends into a holomorphic vector bundle on U, aay Nj. In thisway, Y = Li-I; z.l: b is a holomorphic section of Nj (tbe dual of Nj on U), which exactly
vanishes on Mj n U.
Nowextenda to a holomorpbic vector bundle. we choose U small eoough, so that 1]j IM/nu;

ijj on U. Hence, we get a Koazul complex (A( Nj ) 0 ijj, i y ) on U. By the local uniqueness ofresolutions, tbere exists a bolomorphic acyclic chain complex (A, a) on U such that on U,

Next, we use the above isomorphism to st udy the homology of the complex (€, v)l M' :For z E M', denote by FO,r, ... , Fm,r the homology grau ps of tbe chain complex (e, v)r.
Let FrffiW Fk,r' For aoy holomorphic trivialization U of (e, v), and any X E (TRM)zl
we may define the derivative 8x v(z) of the map z l-+ v(z) in the direction X. Hy the factthat v 2 = 0, we know that 8x v acts· naturallyon tbe vector space Fr. This action does notdepend on the local trivialization. (In fact, if 8~v(:z:) is the derivative of v in the direction=

Chapter 1.7.

157

X with respeet to another holomorphic triviaüzation of~, there exists Az:(X) aeting linearlyon €o,%, ... ,€m ,r such that
8'x v (z)=8x v(x) + [At(X), v(z)}.)Obviously, 8xv(z) decreues by 1 tbe grading in Fr. Since V is a holomorphic section ofEnd(~), if XE ti°,l)M, then 8xv(x) O.

=

With the above notation, we have the following

Proposition. (1) Tbe vector spaces FO,r, ... , Fm,r are the fibers of smooth vector
bundles Fo, ... , Fm on Mj, which inherit a canonical holomorphic atructure from thebolomorphic vector bundle €o, ... ,€m.
(2) For an~ x E Mi I X E (TRMj)rl 8 x v(x)=O. Hence the linear map

ia weil defined, depends smoothly on x, y, and (8,v(x»2(3) On the normal bUDdle Nj , the complex=O.

is a bolomorphic Z-graded ehain eomplex, whieh ia canonically isomorphie to the holomorphic Z-graded K08zul complex (I\(Nj) rs '1j, i y ).

Proof. (1) Since for any z E Mi nu, there is an isomorphism

we know that Fit,1/: h88 constant dimension on Mi. But the FkB are the homokJgy groups ofthe holomorphic ChaiD complex (€, V)IM 1., so they are the fibers of the emooth holomorphic
vettor bundles on Mj.

J(2) By the fact that 8x v corresponds to LN(X), where N(X) ie the compoDent of X in

Nj I we e88ily have 2.
(3) From above, for aoy y E Nj, there exists an isomorphismon Mj nU. Therefore, it ie enough to prove that this i80morphism is canonical. This laststatement is a direct consequenee of the fact that the restrietion map lS canonical.

158

Axioms w.r. t. Closed Immersions
I.7.1.c. Bismut Condition (A)We see from the above discussion that there ia a natural algebraic isomorphism on N,

On the other hand, if we put metrics on the vector bundles in qUestiOD, there are induced
metries on each side of the above algebraic isomorphism. Now a natural question is: whenis this algebraic isomorphism an i80metry? It is to achieve this that Bismut condition (A)is needed.
Put herrnitian metrics gj on €j for j = 0, ... ,m. Hy the Hodge theorem in Chapter 3,
we know that if v· is the adjoint of v with respect to the metrics, we haveHence there is an induced metric on FJ: from €J:. We denote this metric by gF... We saythat the herrnitian metric g" satiBfies Bismut condition (A) with respect to the hermitian
metrics gN Bnd g" if the identifications of holomorphic chain complexes on N byalso identifies- the metrics above.
Proposition. Given herrriitian metrics gN snd g" on N and 7], there exist hermitian
metrics gl; on €J: whicb satisfy the Bismut condition (A) with respect to gN aud g".

Proof. We start wlth aoy hermitian met ries g' J: IMI on €J: IMI. Let v· denote the adjointof v on M' with respect to these metrics. Therefore, for 0 :$ k :$ m, by the Hodge theorem,
there exists a decomposition

and this splitting iB orthogonal with respect to g' J: IMI. Now we may modify these metrics:
Obviously, there is a hermitian metric gJ: IM' on €J: IM' so that
(1) The above splitting for €J: ia still orthogonalj(2) The restrictioDS of gJ: IM' and g' J: IMI to v(€J:+d EB v· (€J:-d coincidej(3) When F,i; is equipped with the metric induced by g'.tIM', the canonical identification
F,i; ~ I\J:(N·) 0 '7 also identifies the metries.
Now by a partition of the unity, we may extend the metrics on ~b which satisfy Bismut
condition (A). This completes the proof.
Next, we give same consequenees of Bismut assumption (A). This kind of result ia
uBually proved by the so--called Mathai-Quillen 's method, developed in the paper {M Q 86].
First we need some more notation. There exist hermitian metries on the bomology
bundles Fo, ... , Fm. For any YEN, we have ii E N and Y =: Y + ii E NR. We know that

Chapter I. 7.

15989v- ia the adjoint of 8,,11. Hente 8y V =: 8$111 + 8g ll- is a self-adjoint operator aeting on
F. Let '\JF =: EBgt'\J F.. be the eanonieal eonnection on F, then F lifts naturally to a vector
bundle on N, which we also denote by F. Alsoia a superconneetion on the superveetor bundle F on N. Let N H be the number operator
in End(~) whieb maps fE €,t to kf E €,t.

Theorem. With the notation as above, if the metries go, ... , gm on the veetor bundles€o, ... ,€m satisfy the Bismut 8BSumption (A) wi th respect to tbe metries gN and g", on
N and '7, then we have the following equalities of differential forms on M':
iN Tr,[exp( _B 2)] =(21ri)dimN td- 1(_('\J N )2) Tr, [exp( _(V"')2)],[Tr,[N H ex P(-B 2 )] = - (2,,"i)dimN(td- 1)'(_(V N)2) Tr,[exp(-('\J'1)2)].
Proof. Under Bismut assumption (A), we have the identity of holomorphic hermitian
chain complexes on N, (F, 8v11) :::::: (AN- l8l '7, i v )' In particular 8v11- = fiA andI[ Tr, [exp( _B 2 )]=Tr[exp( _(V N )2)] [Tr, [exp( _(VII~- + i v + fiA)2)].

We identify N with N- by the metric of N. The algebra A( N-) is an Na-Clifford
module. Namely, if U E N, V E N, setc(U) = -iv2tu, c(V) = -iv2V A.
So if Y = y + ii, thenThe connection V N splits the tangent bundle TaN into a horizontal part, and into avertical part which may be identified with Na. If Y E Na, X E TNRI let Dx Y be thevertical component of X. If (fo) is a basis of TRN with dual basis (fO) in TiN, we letoThen we have160

Axioms w.r.t. Closed Immersions

Under Bismut assumption (A), (V'F)2 is the action on I\(N-) of the curvature tensor(V'N)2 of the connection 'Y N on N. Furthermore, if el, ... ,edimN ia a complex orthonormal
basis of N, let eI, ... , edim N be the eonjugate base of N, thenAs in [MQ 86L we may replace (V'N)2 by a skew-adjoint endomorphism A of N with A
invertible. (Later, by the continuity, we ean drop the eondition that A ia invertible.) The
co-action A' induced by A on I\(N-) is given by the analogue of above relation:

Observe that if X E NR,

Then, by the above expreBBion for 8'J1I + 8g v-, we haveaad soexp(

ic{A-IDY)-12) c(X) exp( -ic(A- 1 DY)

v'2

2i

) = c(X) + v'2 < A- 1 DY, X > .

Therefore

Since the supertrace vanishes on supercommutatofS, we haveTr,(exp(-A' +ic(D Y)

V2 )J = Tr,[exp(-A )]ex p (2"1 < A- 1 DY,DY ».
I

But c1assicallyTr, [exp( -A')] = det(I - exp(A)).
Hence, we getL

Tr,[exp(-A' + iC(~Y) _ 1~12)]IYI~
1=det(I-exp(A)) ], exp(-- - - < A- 1 DY,DY ».
N
22

Chapter I. 7.

161

Tbe Pfaffian of _A-l, witb tbe canonical orientation of NR, ia given by a~:(':.:), 80 thatwbich completes the proo~ of the first relation.
After the above discuBBion, we easily have the Becond relation. A local calculation gives

On the other hand, we haveTr, [NH exp ( -B

2)]

a [

F

2

= ab Tr, exp(-«V ) - bNH -.

ic(DY)

.,ß.

2

IYI
+ -2-»lb=o.

By integration over N, witb a similar discusaion as above, we getiTr.[exp( _((V'F)2 - bNH _ iC(:;r) + 1~12 ))}=(21ri)dimN t d- 1 (_RN - bIN)Tr[exp(-(V'l)2)].

So, we have tbe eecond assertion.

I.7.1.d. Wave Front Sets
Let r be a current on M and denote by W F("'() the wa.ve front set of /. For the defini tionand tbe basic properties of wave front sets, see [Hö 83]. We know that WF("'() is a closed
conic subset of Tß.M - {O}. If p : TiM - M is the natural projection, p(WF(..,.» is exactly
tbe singular support of /, wbose complement in M is tbe set of points z such that / is ~on a neigbborbood of z. We letthen the elements in 'D}/,-R. are smootb on M - M'. Furtbermore, there exists a natural
topology on 1)}/,-R given as follows:
Let U be a sffiall open set in M, which we identify with an open ball in R 21. OverU, we identify TiM with U x R2/. Let r be a closed conic set in R 21 such that if
z E U, r n Nä~
0. Let '() be a smooth current on R 21 with a compact support
included in U and let m be an integer. Ir / is a current, denote by
tbe Fouriertransform of lP1 (w hicb is considered as a current on R 21). For auy "'( E V JoIÄ. 1 let

=

vrr

162

Axioms w.r.t. Closed Immersions

We say that a sequence of currents {-rn} E V}/,.~ converges to '1 E V}/..R if
1. lirI1n_oo'1n = i in the sense of distributions; and2. liIIln-oopu,r,lp,rn(-rn - '1) =o.

As in the case of differential forrns with P aod P', we also let

P,tf, := {w E V}lÄ. : w is a sum of currents of type (p, p H.
And

p~:o := {w E Pt/, : w = 80: + äß with 0, ß E 'D,,~).
Ir M' = 0, we write pM, pM,O instead of pU" pff,'O respeetively.

§I.7.2 Axioms For Relative Bott-Chern Secondary Characteristic Currents
With Respect Ta Closed Immersions
Suppose we have tbe following data: A closed immersion i : M' ~ J\f with hermitian
metrics on N, TM and TMI. Let ("1,9,,) be a hermitian vector sheaf on M' with a vector
sheaf resolution of i.fJ on M:
11

11

11

r.

o-- Ern -- ... -- EI -- Eu -- J. "1 -- O.
Suppose that Pt are herrnitian metrics on E. which satisfy Bismut condition (A) with respeet
to gN, g". In the same spirit 88 in Chapter 2, we introduce axioms for the relative BottChern secondary characteristic current with respect to closed immersions, ch BC ("1, g,,; i, pd,
on M, as folIows:

Axiom 1. (Downstairs Rule) In there is an element ChBC("1,9"ii,Pi) E p!:f,/P:!,.o,such that

Axiom 2. (Base Change Rule) Let f : M -- M he a holomorphic morphism. Assurne
that fistransversal to J\1', i.e. for any xE f-l(M'),

Tben we haveAxiom 3. (Triangle Rule For Hermitian Vector Sheaves) Leto-- fJl -

"'2 -

TJ3 -

0

Chapter I. 7.

163

be a short exact sequences of vector sheaves on M'. Then we may find the resolution
Ce. for i. '1e in the above sense, wi th the condition that, for all j,

is a short exact sequence. Put metrics satisfying Bismut condition (A) on them. Then
we have3I) -1).l: chBC('1.b 9'1.; i,pd
.1:=1m=i. (td -1 (N, Pli )ChBC('1·,9'1.)) -L( -1 YchBC(E. J, P,J)

;=0
.

lD

pM/~,OM'

r M, .

Axiom 4. (Triangle Rule Far C108ed Immersions) Let i' : M" ~ M be anather closed
immersion such that M' and M" intersect transversely, i.e. if z E M' n M" I thenLet i" : M' n M" ~ M be the induced closed immersion. For any vector sheaves 7](resp. 7]/) on M' (resp. M"), let '7]" =: 7] / IM' 0 '1/ IMII. Then, in Pt/'UMII/ Ptf-'uMI', wehavech BC ( 1]" ,9'1" .1 J'11 1 Pi" )=ch(c ' " p' .)ch Bc(1], 9'1; i, Pi) + i.(td- (N, 911)ch(1], 9'1) 1" chBC(fJ/, 9'1'; i', Pi'))1

andchBC ( 7] 11 ,9'1"'••"
J
Pi" )
1
=ch( E" p.) chBc( '1',9'1 i ' , pi / ) + i~ (td- (N', 911' ) ch( fI', 91/' ) i- chBc( 7],91/; i, Pi)1

1 ;,Axioms w.r.t. Closed Immersions

164§I.7.3 Existence Theorem
Far Relative Bott-ehern Secondary Characteristic Currents
With Respect Ta Closed Immersions
We have the following
Existence Theorem Of Relative Bott-ehern Secondary Characteristic Currents With Respect Ta Closed Immersions.
With the same notation as above, let i : M' --+ M be a c10sed immersion of compact
Kähler manifolds. Let (1], g'J) be a hermi tl an vector sheaf on X. Then there exista aunique current in pU" eh Be ( 71, g'J; i, Pi), Buch that the axioms stated in the previous
Bubsection hold.
We will prove this theorem in the next chapter, whieh are taking from [DGS 91]. As in
the case of smooth morphisrns, this theorem LS proved by the following steps:(1) The introduction of a family of superconnections with a parameter t > O.
(2) Investigation of the eonvergence of the associated heat kernels when t --+ 0+, andt --+ +00.

(3) The use of the Mellin transform to construct the relative Bott·Chern secondary characteriatic currenta.
(4) Proof of the axioms atep by step.
It ia clear that in this procesB, the number operator plays a very important role.

Chapter 1.8.

165

Chapter 1.8,Relative Bott-Chern Secondary Characteristic Currents
With Respect To Closed Immersions 11: Existence
We now give the proof of the existence theorem for relative Bott-Chern secondary
characteristic currente with respect to c10sed immersions as stated in the last chapter. The
basic ideal for doing so is as folIows: First, we need to find a family of superconnections with
one parameter t > 0; then investigate tbe convergence of tbe 88SOciated beat kerneis whent - 0+ and t - OOj thus witb the right convergence, we may construct the relative BottChern secondary cbara.cteristic currente by using the Mellin transform; finally, we check tbe
axioms case by case.
We study the above 88pects in different sections of tbis ebapter. The basic strategy
is tbat we first discuss the general formula witbout Bismut assumption (A), aod see how
far we ean go. Then we adopt Bismut 8BSumption (A) so BB to deduce the final axioms.
Essentially, this cbapter comes from [B 90], [BGS 90], [BGS 91] aod [We 91].

§I.8.1 Convergence oe Beat Kernels Associated To Certain Superc:onnec:tions
We use tbe same notation as in the previous chapter: i : M' e.....,. M denotes a c10sed
immersion of complex manifolds, etc.. Let V E =: EB(j'VE• be tbe canonical connection on=: EB(j'e.· Set V = v + v·. For u ~ 0, let Au = VE + ylUV be a Bupereooneetion on the
superveetor bundle
with A := Al. There are natural hermitian metrics on tbe homology
bundles Fo, ... , Fm' For any YEN, we have ii E N and Y =: y + Y E Na· Since 8g v· istbe adjoint of 8., v, hence 8y V =: 8"v + 8g v· is a self-adjoint operator aeting on F. Let
V F =: EBg'V F • be the e&Donical connection on F. F Iifts naturally to a vector bundle on
N, wbich we also denote by F. In particular,ee,ia a supereonnection on the superveetor bundle F on N.
Using the above superconnections and the result of Cbapter 3 and Chapter 5 about theexistence of heat kerneis for generalized Laplaci611s and their beyond, we know that there
exist heat kernels for A~ aod B'. Hence it makes sense to talk about Tr,[exp(-A~)] andTr, [exp( -B')]. Dur first result for convergence ia the following micro-local estimations.

166

Relative Bott-Chern w.r.t. Closed Immersions

Theorem. (1) AB U M

00,in ptf" we have the following convergence of currents on

Tr,[exp(-A~)] - [ [ Tr,[exp(-B 2)]]6MI;(2) For aoy natural number k, there exists a constant Cl: > 0 such that for aoy smoothdifferential form Jl on. M and u 2: 1,(3) Ir U, r,lP, mare taken 88 7.1.d, there exists a cOßstant C> 0 such tbat for u 2: 1,

pu,r,~,m('I'r.[exp( -A~)J - [ [ Tr.[exp( -B2)]OM') :s

5,..

Proof. By definition, (1) is a direct consequence of (2) and (3).
(2) We first consider the case with k
special CBBe is also valid in general.

= O. Later we will see that the proof for thisAssume first that the compact support of Jl is inclused in M - M'. Since the linear
self-adjoint map V E End(€) is invertible on M - M', by Duhamel's formula, we know thatlz

JJTr, [exp( -A~)] - 0

uniformly together with its derivatives.
Take now ZQ E M', let z = (z1, ... , z'+I') be holomorphic coordinates on an open
neighborhood U of ZQ in M, such that locally, M' is the vector subspace (z1, . .. ,ze)o.
There existB an open neighborhood V of ZQ in M' and € > 0 such that if D~ is the open
ball, center 0 and radius E in ce, then U
V X D r • We then need to prove the statement
for the case that Jl has compact support included in U.

==

Let Uu : V x Dt.,fü we haveV x D r be defined by uu(z, y) := (x, ,*). Sy a direct calculation,r JJTr,[exp(-A~)] = j(YnV)xD~",.,. (u~Jl)(u~Tr,[exp(-A~)]).

Jz

In this way, we reduce the problem to a local discusaion. Obviously, the vector bundle
11~€ on V x Dr-/U is equipped with the metric l1~g{l aod the connection u~V'{ =:Thus,by u~V V(z,*), we have

V't.

=

1

f JJTr, [exp( -A~)] =}z(YnV)xD.",.,.

(U~JJ) Tr, [exp( -(V'~ + y'UV(z, ~ »1].
VU

Chapter 1.8.

167

But for u ;?: e- 2 , we havef

JeynY)XD •..",.

(O":1J)Tr.{exp(-(\7~ + v'UV(x , ~)2)]vU=J('/"ny)xf(0":Jl)Tr8[exp(-(\7~ + ,;uV(Z, ~)2)]
D.
v
U

+ f

JeynYXeD.v---Dd(O":J-&)Tr.[exp(-(\7~ + .;tiV(Z, ~)2)].
yU

Now we let P be the projection from ~ to F defined locally by the orthogonal projection
operation ~~ to F~ for auy z E M'. Hence for any smooth section f of F on M', we have

The first result in the theorem is then a direct consequence of the following

Lemma. For U ~ +00, we have the following estimates:(1)

I{}YXD t(O":JJ)Tr,[exp(-(\7~ + y'ÜV(Z, ~)2»)- {(i- Jl)}Y

1.

1111$1yU

Tr, {exp( _(V F + PV~ V p)2)]1:5~IIJlllc.eM);

yU

(2)

(3)

Proof of the lemma. First, we give the proof of (3). Hy Duhamel'a formula, it ia
sufficient to prove the foUowing
Sublemma 1. There exists a constant c > 0 such that for aoy z E M', Y.E NR,~1 JE
F~,

18v V(z)fI 2 ;?: cIYI~Rlfl}·

In particular, on N, Tr,{exp(-B 2 )] decays faster than exp(-Clyl~) when lylN ~ 00.

=

Proof cf Sublemma 1. We only need to prove tbe assertion for IYI 1. In this CaBe,
the complex (F , 811 11) =::: (AN- ~ ,., I i'J) is acyclic. Thus by Bodge theory I 8v V ia a self-adjoint

168

Relative Bott-Chern w.r.t. Closed Immersions

invertible operator. Hence by the compactness of the sphere bundle SNI'l on NR, we deduces
that there exists c > 0 which is a lower bouad for the smallest eigenvalues of (By V)}es ......
So we have Sublemma 1.
I'l
Next we prove (1) aod (2) in the lemma. For doing so, we need some more notation.
By Prop. 7.l.b, since· M' is compact, we know that there exists a constant b > 0 suchthat if z E M', V 4l (z) has no eigenvalue in the interval ]0, 2b] . .T herefore, if! > 0 is smaH
enough, z E V aod lyl :5 !, we may choose b > 0 so that b is not an eigenvalue of theoperator V 4l (z, y). Hence if for 0 :5 k :5 m, we let ~t(resp. ~~( )) be the direct sum of01', Z,I/
01', :,1/( )the eigenspacea of the restriction of V 4l (z, y) to ~c,(:,y) corresponding to eigeovalues which
are strictly larger (resp. smaller) than b, thenare smooth vector bundles on V x D. with

Furthermore, we know that as Z-graded vector bundles, ~--Iv = F. Rence ~+Iv is exactly
the subbundle F.l of ~Iv orthogonal to F. Let p± be the orthogonal projection operator
from toObvioualy on V ~ V X {O}, P- ia just P deflned before the lemma. Thus if
we let h, IF be the identify maps on ~ and P, the orthogonal projection operation Q from~IMI to p.l is given by Q = Ie - P. And on V, Q P+.

e e±.

=

Let

e+, e-

which are connections 00
aod ~ respectively. As in the smooth morphism C8.5e, we
consider the difference of these two connections, aod denote it as S, i.e., S := 'Ve - 'V$. If
we ideotify the fiber e(:I:,ll) with the fiber e(:I:,O) = e:l: by the parallel transport along the lines E [0,1] ....... x + sy with respect to the unitary connection V$, the linear map V(z, y) acts
as a self-adjoint operator on the fiber
aod preserves the splitting e~ =€EIIn thisway, we introduce the operators V± as the restrictions of V on

e:

e±.

et e;.

-

Since a I-form on V x Da is the sum of a I-form on V and a I-form on Da, we may
denote by H the set of one forms of the first kind, i.e. on V, and by H.l the 1- forms oftbe second kind, Le. on Dt . Let (z 1, .•. , zl+1') be holomorphic coordinates on V. Then we
deflne'V~ V(x, y) = L(dza'V~ V(z, y) + diav~ V(r, y)).

vt

Using the identification e(r,v) ~ er, we find that
V(z, y) lies in A 1(TRM'0Ende)r. Hence
if we identify Y E R4lej with the vector field (x, Y) - (0, Y) on V X R 4le j , where we let
ej := I-lj, then [Y,
0, [Y,~] = 0 for any Q. In particular, V~(Pvt V(z)P) is awell-defined I-form on V which takes values in End(P). Now we need the foHowing technicalb] =Chapter 1.8.

169

Sublemma 2. (a) If z E M', X E (TRM')r, then V'k V(z) maps F into. FJ.. With
respeet to the splitting €IM'
F ffi Fl., if z E V, U E (TRM)r, we have=05:J;(U) = ( _(V+)-lQ(V'tV)P.

In partieular,(V'FflP(V'{ V)Q(V+)-l)U0.

= p(V'{)2 P - P(V'{V(V+)-2V'{V)p.

(b) For any z E V, Y E R 2ej ,PV'~(V't V)(z)P = V'~(PV'~ V(z)P).
Suppose that Sublemma 2 is proved, we also need the following

Sublemma 3. (a) If V and E > 0 are small enough, there exists a constant C> 0 Buchthat if (x, y) E V X D«, thenV(z, y)2 ~ Clyl 2 •In particular, for ..\ E f, x E V, Y E ce with y "I O,IYI :5 E.jü, ..\Je - .jüV(x,Endir is invertible.
(b) If x E V, y E N r , Y = y + y, we have*)E

IPV'~ V(z)PI :2: Co lyl·
In partieular, ror ..\ E r, x E V, ..\IF. - PV'~ V(x)P E End F:J; is invertible.
(e) For A E r, z E V, y E ce with y #; O,IYI :5 E.,jü, let Y ::::: y + ii and defineA(u,z,y,.\) E End(er) by(.\le - y'UV(z, ft))-l :::::P(.\lF - pvt V(X)p)-l p+ ~{ ~P(.\lF - p.V'~ V(z)p)-l PV'~V'~ V(z)P(AlF - PV't V(z)p)-l p - Q(V+)-l(z)Q}

Then ror g >°+A(u,x,y,..\).

amaIl enough, there exists a constant C >°such that

Suppose we have tbe above two sublemmas, ror tbe first estimation in the lemma, wetben need the foUowing

Proposition. For auy ..\ E C with IIm.\1= 1, let I( 1.&,.\) be{(I~ - (A 2I~ - uV 2 (z, ft))rl((V'~)2+ v'üV'~V(z, ft))}-I(AI~ - v'üV(z, ftW 1 •170

Relative Bott-ehern w.r.t. Closed Immersions
Then for u - +00, /(u,..\) - /(00, ..\), where

1(00,..\) :=P{/F - (..\2IF - (PVtVP)2)-I((V F )2 + V F pvtVP))}-1((..\IF - PVtVp)-l - (..\2IF - (PV~VP)2)(PV~V(V+)-IQ).

Proof of the propo~ition. Let

J(..\)=:P{IF - (..\2 Ir - (PV~ V p)2)-I((VF)2 + V F pV~ V P))}-I(..\IF - pvt V p)-I,

then it easily follows tbat=

where r is the oriented coutour in C defined by IIm..\1
1 and taken clockwise, since thesecond term in 1(00,..\) ia an even function of ..\. On the other hand, we may also eaaily have

Apriori, the right hand side of thia equality is a differential operator of degree I, while theleft hand side is of degree 0: There ia same cancellation here, which may be explained aB
follows: Since [V~, V] belangs to 1\ 1(TR.M)@End~, (Ale - V~ - ..;uV)-1 can be expressedas the surn of tWQ differential operators of degrees 0 and 1 respective:(Ale - V e - VUV)-I={I~ - (..\ 'l /~ - uV 2 )-1((V e)'l

+ v'ü(V e, V])} -I ((..\I~ - jUV)-1 + (..\2 Ie - u VA)-IV d.

Hy the fact that the ..\·function

ia even, we know that its integral on r vanishes. From this,exp( -(V e + y'üV)2)=~ [ exp(-..\2){le - (..\2le - uV2)-1((V~)2 + VU[V~, V])}-I(..\le - VUV)-ldA.
27ft

Jr

Hence, we getChapter 1.8.

171

Similarly, we have

The conneetion 'Vt on € converges to the conneetion 'V~, which is the pull- back of theconnection V( Iv on the ve-ctor bundle €lv by U oo ' This means that in a given trivializationof €, the connection-forrns of vt converge uniformly to the connection-form of V~ tagether
with their derivatives. Hence if we define E( u, x, y) by

then

C

IIE( u, x, y)11 5 y'U(1 + Iyl)·
Moreover, we know that if F(u,x,y) ia defined by

JUv~ V(x, :!u) = JUV~ V(z) + 'V~v~ V(z) + vtJ. V(x) + F(u, x, y),thenClIF( u, x, y)lI 5 JU(!yl + lyI2).

Note that since V~ V (x) mapa Fr into F;- by sublemrna 2 (a), we know that if >. Er, aB
u-oo,(>'h+v'üV(x, :!u))-lJüV~V(Z)(>'I(- y'ilV(z, :!u))-l=(V+)-l(Z)(vt V(z))P(>'IF - pv~ V(z)p)-l P- P()'IF - pv~ V(z)p)-l P(V~ V(x))(V+)-l + O()u).

But by Sublemma 2 (b), we know that=

PV~(V~V)(x)P V~(PV~V(x)P),BO, togetber with Sublemma 3 (c)(..\h+v'UV(z, :!u))-lV~(V~V)(z)(>'h - jUV(x, :!u))-l=P().]F + pvt V(X)p)-l PV:;(pV~V(x)P)(>'IF - PV~ V(z)pr- 1 P
1+ O(JU)'
On tbe otber band, by definition,

P(vtJ. V)(x)P = PV~J.(Pvt V(z)P),Relative Bott~Chern w.r.t. Closed Immersions

172

so similarly, by using Sublemma 3 (c), we have(..\h+v'üV(z, ~))-lV'~..I.V(z)(..\l( - y'üV(z, ~))-l=P(..\lp + PV'~ V(Z)p)-lV'~..I.(PV'~ V(x)P)(..\lp - P~ V(Z)p)-l P

1+ O( yIü)' 'Therefore, for ..\ Er, as u -- +00, the form I ( u, ..\) converges pointwise to tbe form I (00, ..\).
Also, for ..\ Er, we have

Therefore, in the above expression , the norms of the various 0("*) ean be dominated by*(1 + 1..\13 ). This praves tbe proposition.
With tbiB proposition, tbe proof of tbe first estimation in the lemma becomes quiteeasy.
. Indeed, if G(u, z, y) ie tbe form defined by(O'~Jl)(Z, y) := (O'~i· p)(z, y)+ G(z, y, u),thenFinally, for any ..\ Er,lexp( -..\:;1)1= exp( -I Re..\fl + 1)

and 80 far any p E N, I..\IP exp( -..\:;1)1 is integrable on r. Thue, by (*), let u -- 00, we get thefirst estimate in the lemma. Tbe second estimate is proved similarly: In this esse, insteadof using r, we have to use the contour

Colyl .
Colylr y =: {z E C : Re(z) 2: -2wlth IIml(z) = 1 or Re(z) = -2-}'
The details are left to the reader.
Next we give the proof of Sublemma 3 and Sublemma 2 that were stated above.

Proof of Sublemma 3. (a) By the uniqueness of the loeal resolution, we know that
if V and € are small enough, thenFurther , there is a metric h on the right hand such that

Chapter 1.8.

173(1) The (A.l:)(J' are mutually orthogonal.
(2) The splitting is orthogonal.
(3) The metric on the complex (AN- 0 ry, i v ) comes from metrics gN and gif on Fr and ry
respectively.
Let Z; be the adjoint of LV with respect to the metric gN. Ir ii E
element of Fr by the metric gN, then Z; = fi A . Therefore,

N ia identified with an

Let v- be the adjoint of v with respect to h. Set ii =: v + v-. From (ty + t;)2 = IY1;N'
we know that there exists a constant C > 0 such that if (x, y) E V x D r , thenWe use this estimate to deduce the assertion.
Now fix (x, y) e V x (Dr - {O}). AB all the estimates will be done at the point (x, y)in the sequel, we omit (x, y). Alos, both y2 and y2 preserve Ker(v), by Bodge theory, thelowest eigenvalues CRD be calculated by considering these on Ker(v), 80 let f e Ker(v) n €.l:aod set 9 := v-( V2)-1 f, we have f = vg. Hut Hadge theory tells us that 9 = va + v- ß far
certain a, ß. Hence f
vv- ß. SO v- ß 1I-(y2)-1 f.

==

Taking the metrics, we haveaod
But for V and E > 0 small enough, there exist constants C > 0, C' > 0 such that on V x D r ,C'llll1 $llI1~ $ Cllll~·
Hence,< (y2)-lf,f >h ~ C < (ii 2)-lf,f >ia'

Tberefore, if '\, ~ are the lowest eigenvalues of y2, ii 2, there is a c> 0, which is uniform on
V x D r1 such tbat ,\ ~ ci Hence, we have the estimate (a) by using(b) Note that if x e V, Y e N~, Y(}y Y(x) ==y + Y e NR,r, thenpvt Y(x)P = pvt V(x)P.

174

Relative Batt·Chern w.r. t. Closed Immersions

Thus, by Sublemma, we find that there is a constant Co such that

pvt V(x)PI ?: Colyl·(c) Let 1(% be the identity map of ff:: and define A+(u,x,y,..\) by.

(..\/(+ - y'UV+(x,

Jun-

1

=: - )u(V+)-l(X, 0) + A+(u, x, y,..\).

Similarly, note that Bince V - (;r) = 0, we may define A - ( u, x, y, ..\) by(..\1(.- -y'UV-(x,

Jun-

1

=(..\1(- - V~V-(x))-l+

V~V·-(xn-lV~V~V-(x)(..\I(- - V~V-(xn-l11;;(..\/(_ 2y u

+ A-(u,x, y, ..\).
Obviously, A= A + + A-. We next deduce the estimate for A from those for A ±.

We begin with an estimate for A +. First, we havecV+(Y ))-1( ...,'I(+ - vI'x, Vü~u(V+)-l( , y'"~u))-l .
= (V+l-j;' -ru {/t+. - y'"lThus by finite increments, we have

Hut for any ..\ :F 0, we have.
I
1
IInu\j2
mfdER 1-dA 2 =~.

Hence, far ..\ E r, Le., IImAl = 1, we haveAlso by finite increments, we getChapter 1.8.

175

Therefore, put the above aU together, we know that if x E V, lyl:5 !..fii, IImAj= I,

Now we deal wi th the. estimates for A -. We define B( u, x, y) by

Obvioualy, if x E V, 1111:5 c:..fii, we haveelyla

IIB(u,z,y)ll< - ,uand hence

IIy'üV-(z, ~) - V~V-(z)11 :5 Gelyl·

On the other hand, deflne D(u, x, y,A) by(Afe- - y'UV-(x, ~))-l =:(A/e- - V~V-(x))-l+(Afe- - V~V-(x))-l(JüV-(z, ~) - V~V-(x))(A1e- - V~V-(x))-l'+ D(u,x,y,A).
By finite increments again, we find thatlID(u, x, y, A)II :5 sUPcE[o,l]{II(Ale- - cV~V-(z) - (1- c)y'UV-(z, ~))-11l3}
IlJüV-(x, ~).- V~V-(z)1I2.

Th~refore by the fact that in Ende;, V~V-(x) and v-ex, ju) are self-adjoint, together
with the estimate for B(u, x, y), we havee

IID(u,z,Y,'\)II< lyl ".
- u
In particular, if xE V,lyl :5 t...jü, A E ~, we have

So we bave the estimate (h). Thia completes tbe proof of sublemma 3.

176

Relative Bott-Chern w.r.t. Closed Immersions

Proof of Sublemma 2. (a) Ir f is a smooth section of F, V f = O. Rence if X ETaM',(v~ V)f + Vyr{x f = O.
In particular, we get (v~ V)f E FJ.. Hut on V x D~, if U E TaM,
• vtv = V3v + S(U)V - VS(U).

Rence by· the fact that on V, v~ V preserves the splltting
pvtVQ= PS(U)VQ,

eI = F EB FJ., we haveV

Qyr(uVP = -QVS(U)P.

So we have the structure formula for SJ; ( U). Others may be obtained from a direct computation.
(b) Hy definition, we haveV~(vtH V)(z) = vt(V~V)(z) - [S(Y), V~V](z)and

V~(V~ V)(z) = vt V~ V(z) + [(V~)2(y, N), V](z),since Y commutes with the vector fields
Fr aod F;-. So, from above, we get

P~(VkV)(z)b, da. Hut (a) teils us that v~ V(z) ioterchanges= P{Vk V~ V(z) - S(Y)V~ V(z) + vtH V(z)S(Y)}P.

Hence, by the fact pvt V(x)P = Y~V(x)P, we have

PV~ V(z)P = ('v~ V(z) + V(z)S(Y))P.
Thus for aoy smooth section f of F on V, since V F f = PV Ef
we haveaB stated berore the lemma,(V:;'(Pv~ V P))f = P{Vk(V~ V + VS(Y))f - (V~ V)(yr{Hf - S(H)f)}.
Thereforeyr(H(Pvfv P) = P(V~

Using again the fact thatvt V + (V~ V)S(Y) + (v~ V)S(H))P.

Vh V(z) m~ps Fr into F;-, we finally have

P(S(Y)V~ V(z))P = P(V~ V)(x)(V+)-l(x)(V~ V(x))P
P(vt V(x)S(H))P = -P(V~ V(x))(V+)-l(x)(V~ V(z))PThis completes the proof of Sublemma 2 and hence part (2) of the theorem for J: = O.

Chapter 1.8.

177

Now we show how one ean get similar estimates for a general k. Using a partition ofuni ty, we need ooly diseuBB the ease in whieh 11 has its su pport as in ease (b). Let z = (z I, X2)
be a holomorphic system of coordinates on V such that 1I"(z)XI. Then by our choice ofthe coordinate y, 1T(ZI,x2,Y)
XI' Therefore we may lift any smooth real vector field Xl
on B to the vector field (X ll 0, 0) in the coordinate system (XI, :1:2, y). For ahort, we also
denote it as X I. Now, we need to study the behavior of==

Obviously, if IIm..\l = 1,

Now V aod V$ preserve tbe splitting ~ = ~+ $ €-; while on M', V vanishes on ~- = F
aod maps ~+ = Fl. iota itself. Therefore, V~l V(x), ... , (V~)kV(X), ... aU vanish 00 F
aod map Fl. into itself. With tbe same method in the proof of Sublemma 3.(b), aB U --+ 00,
we get a similar aBymptotic expansion for the left hand side of the above relation. Thus we
see that the form (L X1 )1 f z JlTr,[exp(-A~)] haB a limit as u --+ 00, and that the norm ofthc difference in CO(B) with the limit ean be dominated by ~1IJ1l1c.+dM). So we have theresult for a general k.

The Proof of Part 3. From thc proof above, this' estimate is not diffieult to obtain.
The point is to replaee tbe eorresponding concept at thc right place.
Now we briefly explain how tbis to be done. With the notation aB above, recaU that
dimM/ = Ij + I'. Let z E Ct;+l' aod fj E Cl!; be variables conjugate to z aud y and setz = (z,y), € = (i:,y). Denote by <> the real aealar product of z and Let t/> be aaIDootb current with support in U. Take Q > aod let a be tbe cone

z,e

°

r

e.

Then obvioUBly, our 8B8ertion comes from tbe foUowing

Lemma. For any natural number m, there existe C > 0 such tbat if eE r

Q
,

U

2: 1,

we bave

Proof of the lemma. Let Pm be the differential operator witb constant coefficientsin the variable z sucb tbat

178

Relative Bott-ehern w.r. t. Closed Immersions

Integrating by parts, we get

jxl 2m l f ei<z'(>IjJ(z)Tr,[exp(-A~)] _f ei<z'~>i-ljJ(z) f Tr,[exp(-B 2 )]1
1M
1M
1N=1 f ei<z,(> Pm(tPTr,(exp(-A~)]) - f
ei<z,i> Pm(i-1jJ f Tr,[exp(-B 2 )])I.
1M
1M I><c·j
1N
•
I

,I ,

,
On the other hand, if A is a partial differential operator with eoeffieientB on C + A
may be seen as aeting on the variable x E V. Henee we ean apply the results in the proofof 2 to the fibration V X D~ - V with fiber D t . That is, if V is small enough, there exist
C > 0, C' > 0 such that if z E V, U 2: 1, y E ce, lyl:5 e-/Ü, thenNow by the fact that on compact subsets of M - M', as U - 00, tbe form Tr,[exp( -A~)]
converges uniformly to 0 faster than exp( -Cu) for certain C > 0 and similarly for tbe
derivatives, we may assume that U ia a small open neighborhood of %0 e Mj of tbe form

V x Dt chosen aB befare. Let JJ be a differential form on U. Denote by IIJJII, II~II the supof the norms of JJ aod of tbe partial derivative on U. Hence if IYl :5 e."fü,Therefore, by tbe fact tbat the form Tr,,(exp( _B 2 )] snd its derivatives deeay as lYI faster than exp( _C"!yI 2 ) with c" > 0, we get

I f JJAq:Tr,[exp(-A~)] -

1u

00

f
i-JJ ATr, [exp(-B 2 )]1
1v ><c·j~ 5u(lIJJII + 11 ~~ 11)·=

In particular, put A
Pm, then by the fact that the first derivative of the function
e'<z,{> in tbe variable y ia bounded by 1Yl, we getlil 2m l f e,<z'{>fjJ(z)Tr,[exp(-A~)] _ [1M

1M

ei<r,f>i-ljJ(z) [ Tr,[exp(-B 2 )]1
I

1N::; 5u<1 + IliD·
Hut IYI :5 O'lil, so we have the a.ssertion, which eompletes the proof of the lemma aod hence
the theorem.

Chapter 1.8.

179

§I.8.2 Number Operators and the Double Transgression For~ula
1.8.2.a General Double Transgression Formula
In this section, we will give a general double transgression formula. With the same
notation Ba above, for 0 < k < m , let MI: be the set of smooth hermitian metrics onthe vector bundle {I:. Set M~: n~ MI:. lf 9k E Mi;, x E M, let B~· be the set of
endomorphisms of {k,r which are self-adjoint with respect to the rnetric 9k. We also identify
the tangent space T,.Mk with the set of smooth sections of Bg· on M: In fact, if 9~ E
End({J:,is an infinitesimal deformation of hl; in Mk, then 9;1 g~ is the corresponding
element in Bg·. Let dM be tbe exterior differentiation operator on M. For ge E M, let w
be tbe connection-form associated witb the corresponding canonical connection V~. ThendMw is a 2-form whicb ia the equivalent representation of a 2-forrn r on Mx M. ObviouslYI
we know tbat
1-[V (ge)-ld Mg e]

€on

=

e,

with values in End {. Recall that A = Al = V~ + V. We letA'= V" + 11·, A" = V" + v.
e'eil

Then, we bave the following
Proposition. 1. The following identities holddM Tr, [exp( -A 2)] =8'Il, [; - [v· , (ge)-1 dM g~]) exp( -A 2)];
'Ir, [(1 - [V·, (g~)-1 dMg e]) exp( -A 2) = - 8 Tr, [((ge)-ld M ge) exp( -A 2)].
Tberefore,

dMTr, [exp( -A 2)]= -88 Tr,[((ge)-ld M ge)exp( -A 2)].

2. Similarly, for aoy odd Gr8B8rnannian variable z, we havedM 'Ir, [exp( _A:2 + z(ge)- ~dMge)]=!aTr, [(A", (ge )-1 dM gel exp( -A 2 + z(ge) -1 dM g~)]2,
- ~8'fr, [[A', (ge)-ld Mg e exp( -A 2 + z(ge)-ldMg e )].
Proof'. Consider { as a vector bundle on M x M. There is a natural canonicalconnection Ve on which restricts to the canonical connection V~ wi tb ge E M on Mx {g~} ,and is trivial on {O} x TM c T( M x M). As a superconnection on { over M x M, for
Ve + V, we havee,

By Dubamel's formula,Tr,{exp( -(Ve + V)2}]=Tr, [exp( -A 2)] - Tr,[(r - [11·, (ge)-ldMg e])exp( -A 2)] + C

180

Relative Bott-Chern w.r.t. Closed Immersions

with C being of degree ;::: 2 in the Grassmannian variables in 'r M. So

Therefore, by eounting the degree, we havedM Tr,[exp"(-A 2 )] = 8Tr,[(I- {v· , (g{)-ld M g{])exp(-A 2 )]8Tr, [(r - {v·, (g{)-l dMg{])exp( -A 2)] = O.
But by definition, we know tbat

so we have the aecond equality.
The proof for 2 is very similar and is left to the reader.

I.8.2.b. Number Operators And The Double Transgreesion Formulae- e

Let NH :be the operator in End(~) which map8 f E 6: into kf E ~.c. We eall NH
the number operator for €. AB an application of the result above, we have the following

Theorem.. For any u;::: 0, the smooth fonns Tr,[exp(-A~)] and Tr,{NHexp(-A~)]are in PM. Moreover for aoy u > 0, we have
I)
:2 ]
v·
:2 ]
I)u Tr,[exp(-A u ) = - &1'r'[y'Uex p (-A u )=- 8Tr.. (~exP(-A~)];
v·

N

yU

u

Tr,[ J::'exp( -A~)] = - &Tr,[ -!!exp( -A~)]
11
2
NH
2
Tr,{ r.:exp(-Au)] =8Tr,[-exp(-A u )]'yUUTherefore, for u > 0 I
f)

:2

l-

2

-I) Tr, [exp( -Au)] = -88Tr,{N Hexp( -Au)]'uu

P roof. For u > 0, replace the r.netrics h~ = (h{ 0, . . . , h{ m) byh{u= (h{o ,U -lh{l ' ' ' ' l U -mh{"') .

Chapter 1.8.

181

Then the adjoint of v with respect to h~ is uv·, where v· is the one with respect to h~. AlsoH[
• (h~ )-1 ~h~J ( h~u )-1 ~h~
8 U u -- _ NU ,uv,u
8U u - V • •If we let Tu E End € be defined by Tu! = u 1/ 2 f for ! E 6:, we haveHence, by result in the last subsectioß, we get the first and the third relations. Similarly,
by interchanging holomorphic part and anti-holomorphic part, we get the others.
I.8.2.c Convergence Properties
Since, later, we will use the Mellin transform to define the relative Bott-ehern secondary
characteristic currents, so we need to asymptotic expansions. In this subsectioD, we give
convergence results forTr,[v'UVexp( -A~)), Tr,[NHexp( -A~)].
Tbere are tbe following

Theorem. (1) There exists a constant C > 0 such that for any k E N, any smoothdifferential form JJ 00 M aod u ~ I,and(2) AB u -00, we bave the following convergeoce for currents on M:

Tr,[JUVexp(-A~)] - 0,

andin Viv. (M);R
(3) If U,f,ip,m are taken aB in 7.l.d, there exist canstants C, C' > 0 such that faru;::: 1,

182

Relative Bott-ehern w.r.t. Closed Immersions
and[L

pu,r,~,m('rr.[NHexp(-A~)J-

Tr.[NHexp(-B')]6M')::; ~.

Proof. Hy the fact that if K is any amooth section of EDd(~), then the analogue ofTheorem 8.1 still holde forTr, [K exp( - Au )2)],IN

whose limit aB U - in/ty will then be tbe current
Tr,[PKPexp(-B 2 )]6M we see thatthe statements about the relations concerning N H may be similarly proved by following theproof for Theorem 8.1, we will not do it here. Instead, we use Theorem 8.1 to prove theassertions for .;üV.
I.

For (z,a) E M x C\ set i(z,a) = (1I'z,a) E B x C·. Then i has essentially the same
properties aB 11". Let j be the embedding M' x C· ~ M xC·. The vector bundles ~k (resp.
'1) extend naturally to Mx C (resp. M' xC). Then on Mx C·, we have the exact sequenceof sheaves

0- OMxC·(€m) ~ OMxc·(€m-d ~ ... ~ OMxC·(€O) ~ j.OMXC·('1) - O.
The natural canonical connection on ~ over M x C· ia given by Ve +a E C· with Ja - 11< by Theorem 8.1, we know that!'lilllu_+ooLda-lä + da-A. Thus if

IJTr, [exp(-(V{ + da :a + dä :a + v'U{ av + v' ))')]= [ i'lJ

L

Tr, [exp( _(V F + da:a + da :a + a8_11 + ao,v')')]'

Also, with the correct parameters, we know that the difference between the expressions
appearing in both eides above is dominated by 5t11~lIcll+t(M)' Hut by Duhamel's formula,
we know that there exist fOrIns "Yu and "Y on B, such that over B x {I},hIJTr.(exp{-{V{ +da: +dä: + v'ü(av+ v'»')Jaäh= j.JTr,[exp( -A~)] -h

JJ Tr, [y'Uvexp(-A~)] da. - hIJTr,[v'üIl'eXP{-A~)ldä+ 7ud~da;
[

i~ IJ

L

F

Tr. [exp{ _(V + da :a + da :a + a8,11 + ao,v' )')1=[i'lJ

LL- [lJdäTr.[exp(-B')]- [i'lJdaL

Tr.[o_vexp{-B')]

Tr.[8gv·exp(-B')] + 7 dadä .

Thus, assertion (1) ia a direet consequence of the following

Chapter 1.8.

183

Lemma. With the same notation BB above,

Proof. Note that if y- E N is considered as a vector field on the total space of N, thenclearly LV('~F)2 O. Also=and thenLS/(B 2 ) = 8S/v.
But LV is a derivation of the Z2-graded algebra Ar NR@EndFj by Duhamel's formula, andthe fact that Tr, vanishes on supercommutators, we have

Therefore tbe form Tr, [811 vexp( _B 2 )] h88 no component of maximal degree in the directionof the vector fibers N, and so

Similarly, we could have[

Tr.[ögv'exp(-B'l]= O.

In this mannel, by a similar process aB in the proof of Theorem 8.1, we may aBlo get
the rest.

§I.8.3 The Construction oe
Relative BottOoChern Seeondary Characteristic Currents
With Respect To Closed Immersions.
I.8.3.a. A Construction
Hy Theorem 7.1 and Theorem 7.2, we know that for u -00, the differenceie bounded by u- 1!2. Also, we know that when u - 0+, the ßBymptotic expansion exists
from tbe general discussion for the heat kernels. Therefore its associated Mellin transformmakes sense. With this, we eßBily have the following

184

Relative Bott-Chern w.r.t. Closed Immersions
Proposition-Definition. 1. For SEC,°<

Re(s) <!' let

Then (ds) is a well-defined eurreot 00 M.
2. For aoy smooth .corm iJ on M, there exista a unique meromorphie extension of
Jo+ JJ(e( 8) to tbe whole eomplex plane such that this extension is holomorpbie at
S
0.
3. Let <((O) denote tbe eurrent on M such that for aoy smooth differential form Jl on

co
=

M,

Then
d

1 .

1

<HO) = o {Tr,[NH(exp(-A~) - exp(-A5»]}~u+ j+co {Tr, [N Hexp( -A~)] _ ( f Tr,[NHexp( -B2)])6M/} du .

iN

1

u

-f'(l){Tr. [N Hexp(-A5)] - ( [ Tr, [~Hexp(-B 2)])6MI}.

Ie aB before, we let [2ri] be the natural operation on the graded algebra D(M) suchthat
[2rilj(J:]= (21ri)k f[k] ,then we define the relative Bott-ehern secondary characteristic current, denotedby eh Be (1], g'J; i, Pi), to be [2ril<{(0), but with g{ defined by Bismut eondition (A). Usually,
if we do not have Bismut eondition (A) on the metrics g{, we denote the corresponding
eurrent by

From seetion 8.1, we know that for a given vector sheaf resolution of i.1], there are
metrics on ~k, such that the Bismut condition (A) is satisfied. Hence the above definition
for relative Bott-Chern seeondary charaeteristic current makes sense. Next, we have tocheck the axiol118, from which it follows that ehBc(1], g'J i i, Pi) does not depend on sorne of
the special data used in tbe definition.

§I.8.4. Checking The Axioms
In this section, we cheek the axioms listed in section 7.2. As stated at the beginningof this chapter, we first work without Bismut assumption (A), but once this asaumption ia
made, the translation ean be easily done by using Theorem 7.1.c.

Chapter 1.8.

185

1.8.4.3. Axiom 1.
Let-y(g<) =: (

\'/ 2

2~t

[2"'i](Tr.[y'üVeXP(-A~)])du
Jr'
2u
o

Here, we fix anee for a11 a,pe square root for 21Ti. Then the transgression axiom is verified
by the following

Theorem. (1) Tbe current ChBC(17, gl1; i, Pi; ~,g{) lies in Pt/,.
(2) Tbe following identities of currents holddCchBC( '1, gl1; i, Pi; ~,g{) = - ")'(g{);

d ")'(ge) =cb(~, gel -(L[2ri]Tr, [exp( -B 2 )])6M

I •Therefore,

If g{ satisfies Bismut eondition (A), we have

Proof. (1) Hy Theorem 2. band Proposition 3.a.3, we see that ebBe (7J, 9'1 ; i, Pi; ~ , g{)
is a combination of currente of type (p,p). So, for tbe first assertion, we only need to provethat WF«({(O)) ia included in Ni. To do so, by Proposition 3.a.3, it ie enough to prove thattbe term forfl+ oo has its kerne I in Ni. For short, we also denote this term by ft oo , i.e.,/+00 := /+CO (Tr,[NHexp(-A~)]-J j, [NHexp(-B )])6M du
2

1

N

1

I )-.

U

fl+ ooBy Duhamel's formula, we know that
ia amooth on M - M'. On the other hand, if wechoose U, r, 4>, m 8B in subsection 2.c, we know that/

Pu,r ",m (1+00 :5 C /+00 du/2 <)Henee, we have W F(chBC('1, h'1; i,Pi)) C Ni.
(2) We use tbe result in section 2. We let

1

u3

+00.

186

Relative Bott-Chern w.r. t. Closed Immersions

then for any smooth form IJ,

8ut

1.M

dl''Y( =lim/l_ +00 [/I {1. dl'Tr, [VU V exp (- A~ )]} duJa M
2u~ -li~_+oo Ja[/I {1.M JldTrJ[vuVexp(-A~)]}du.
2uThus by Theorem 2.b, aod Theorem 2.c, we have

1.M

p.d'Y(=1. IJ'I'r,[exP(-A5)]-1. i·Jl [Tr,[exp(-B 1 )].
M

M'

JN

This proves the second equality. Similarly, by definition, we know thatNow by the closed properties of associated currents, we know that

1M ~IJTr,[NHexp(-A~)] =0
1.M' i·(dCIJH1.N Tr,[NHexp(-B =0.
1)])

Thus, similarly, aB we did above for the second eqality, we, mayaIso have the first one.

I.8.4.b. Axiom 2.
8y Proposition 2.a.3, we know that if we letthen in V N- (M), by Theorem 2.c and Theorem 4.a,R

Therefore, what we need to check 'is the funetorial property of each term in { }. But thisis quite easy. Since f is transversal to 1\1', we know that f- ~ provides a resolution for f·,.,Chapter 1.8.

187

and N = f. N. By the fu~ctorial properties of eharacteristie forms, we only need to show
thatf·OMI 0M"=

But, such a result ean be proved by using a local approximation of 6M , with a sequence ofsmooth eurrents, whieh eonverge 'transversely' as in example 8.2.8 of [Hö 86]. Henee wehave the assertion, again ~y the fact that f is transversal LO M'.
I.8.4.e. Axiom 3.
In the same way, as for smooth morphismB, we use the pI-deformation teehnique todeduee tbe most general triangle relation from the degenerate triangle relation, Le. we let
7J3 = O. As an illustration, we give the following

Theorem. (1) Let 8(gF) be tbe smooth form on M' defined by

8(gF) =:iN [21ri](Tr, [exp( _B

2

)]).

Then 8(gF) E pM ' is dosed.
(2) 5uppose t ER 1-+
E MF ia a smooth map. Let t ER 1-+ B t be tbe eorresponding
family of superconnections on tbe graded vector bundle F. Define

gr

[I

[

X(g;) := Jo dt JN [2 1r1l (Tr, [(gi)-l(8gF)-Iin exp( -B;)]).

Tben X(g[) E pM ' and its dass in pM' / pM' ,0 only depends on ga, gr. So we may~efine this dass as X(h[, hf). MoreoverdcfX(g[) = 8(g[) - 9(g[).
(3) We have the following relatio~eh BC (11, Pi

gr ji, Pi) - ehBC(11, P; g&; i, Pi) =X(Ur, U[)6MI - ChBC(ej g&, ur)·(4) Ir Bismut assumption (A) holds for the· corresponding metrics for all situations,thenx( I Ur) = tdäb(N ; u~ , )eh(11, 90) + td -1 (N , )ehBC (.,,; gO , Ul )

Ur

gf

gf'

in pM' /pM',O.

Proof. The first two statements may be proved by the method similar ta these for
Theorem 4.5.& and Theorem 6.l.d. We leave the details ta tbe reader. For (3), we have to
use pI-deformation.
.
On M x pi, we equip (e.) with hermitian metrics g( whieh restrict to tbe metrics
On M x pI, tbe wave front set
WF(ebBc(11,91J;i,Piie,9()) is induded in Ni, aodu& aod Ur on M x {O} aod M x {oe} respectively.

WF(loglzI2) n WF(ehBc( 11, gfl; i, Pi j ~,g())

= 0.

188

Relative Bott-Chern w.r.t. Closed Immersions

Henee, by Theorem 8.2.10 of (Bö 86L we know that tbe produet of eurrents
Logjzl2ehBc( 7J, gT/; i, Pi; (, gelis well-defined. Also the uaual rules of differential ealeulus ean be used. Thus,

8fJ .(800glzI2] ehSC(7J, gr,; i, Pi;(, gel + 2 .(Loglzj 28chBc(1],g11; i, pi;e,ge)),=288.(Loglz 12 )ehBc(7J, gT/; i, Pi; ~,ge) - Loglz 12 2tJ8i'rZ.ChBC( 7J, gT/; i, Pi; e, g()

2~z

~z

~1

=ch Bc (J7, g"; i, Pij~, g{)6 MX {O} - ehBC(J7, g1l; i, Pi;e, ge)6

MX {oo}- LoglzI2(9(gF)6MlxPl - ch(g{)).

Ir we integrate the above equality along the fiber of 1\1 x pt - M, we haveeh BC ( 7J, 9 11 ; i, Pi ;

-( {

}pt

e, g{) - chBC (7J, 9 i Pi; e, g{ )11 ;ILoglzI 2 9(gF)]6MI + ( LoglzI 2 eh(g{) E p.~,IO.
}plHy the similar proeess for X(ga, gr), instead of eh BC (7], g'1; i, Pi j ~, g{), we get

X(gO, gl) +J.

LoglzI 2 0(gF) E pM/,O.

pt

Henee, we have (3), since(4) If gr aatisfies Bismut assumption (A) with respect to (gr', gi) for i = 0, 1, we eanfind a smooth family of metrics t 1-+ (gf' ,gd on N and J7 whieh interpolate between the two
initial points above. Ir gr is the metric on F associated with the metrie (gf 1 U,), the family

95 and gr. Also, the operator (g{")-l ~ aets naturallyon
tbe exterior algebra AN-. One verifies easily that Bince F = AN- 01],

t 1-+ gr interpolates between(hi)-la;[ =(g{')-I a:;: ® 1 + 1 ® (g.)-l ~'.

Let (V'f)2, (\77)2 be tbe curvatures of the corresponding canonieal connections for(N,gN),(1],g1l). AB in subseetion 7.1.c, we find that-L

Tr.[(gi)- 1=(2~i)dimN

8;[ exp( -B~)].

N:b [td- (_(\7~)2 - b(gt')-1 8~ ) Tr[exp( _(Vi)2)]
1+td- 1( _(\7~)2) Tr[exp( _('\77)2 _ b(gt )-1 a~c )lh=o.

Chapter 1.8.

189

By the results in subsction 4.5.b for the classical Bott-Chern secondary cbaracteristic forms
with respect to Todd characteristic forms, we have the assertion. So we bave the proof ofthis theorem. In particular, (3) and (4) verify axiom 3.
1.8.4.d. Axiom 4.
Let M' be another complex submanifold of M with i : M' <.....+ M the closed immersion.
We set M" =: M' n M'. As for the immersion i, we could give the same result8: We now
ooly have to put : and ." on the notation for M' and M" respectively. Since M' and M'
are transversal, we know that
Nil
NIMI/ EI} NIMI/.

=

Also N IMI/ and NIMII are exactly the normal bundles to j : M" <.....+ M' and ; : M" <.....+ M'
respectively. Thus if we let

and

Tl' := 11IMII®ijIMII,
with r" = r@r, we have the following exact sequences of sheaves

0-4 0!ti,((€,v)I!ti,) ~ ].OMII("IMII) -+ 0,o-+ OM,((i, ii)IM ' ) ~ j.OMII( 11/ IM/I) -+ O.
Moreover, for the homology groups, we have the identification of holomorphie vector bundles

Thus, similarly, we have the relative Bott-Chern secondary cbaracteristic currents withrespeet to i, i, i" respectively. Note that i and i are transversal and we know tbat the pullback currents i·chBC(ij, gl'J; i, 1'1) and i·ch BC (l1, g'l; i, pd on M' and M' are well-defined, by
Theorem 8.2.4 of [Bö 86],88, on wave fronts, statements are affected. Tbus by axiom 2, we
have'
eh BC ( J'. 11, J.. P'I ; J,• P' ) .• chBC (1], 9' I; J,
Pi ) ,•• - •• ~ •.•(- rJ '":' )chBC(' 11,' 9 ;',Pi)=' ehBc 1],9 ;',Pi·
-

-

-

"'I:

=.
_

In particular, the corresponding wave front sets are included in Ni and Ni respectively.
Now by theorem 8.2.13 of [Bö 86], the currents of both eides in the axiom 4 are actually
elements in P:f'uMI' and axiom 4 makes sense. Next, we check the relation.
Let
Qu =: Tr,[exp(-A~)L ßu =: Tr,[NHexp(-A~)LRelative Bott~Chern w.r.t. Closed Immersions

190
etc. Sinceby definition, we have

Furthermore, by Theorem 1 and Theorem 2.c, we know that

(1) As u -+ 00, the curren ts 0u, ßu, Ou , ßu , cr~ ,ß~ have limits cr oo ,Poo ,iftoo , ßoo ,0';.., , ß:;', by
Theorem 1, Theorem 2.c.

(2) Po, ßoo, ßo, ßoo, ß~, ß~ are closed ..
{} 1 - (3) &;O'u = "t/J8ßu, by Theorem 2.b.
(4) Setetc. Tben, by (2), (3),Ou =00 + a8iiu,
cr u =cro + 88TJu.

Therefore, we have

Chapter l.B.

191

And, similarly, for A E [1, +00[,

J1

A. (ß"u - ß")- JA. (ßu - ß) du + O'A.'1A- - 01'11-00 -du =0'0
00

1

U

U

A.

J

+
•

1

-

- du

(ou - Ooo)ßoou

- du - 8 JA. (ßu 8i1u - ß008iloo)du- 8-JA. 8(ßu - ßoo)TJuu

1

1

U

- JA. (ßu - ß)
-00 -du + OA.'1- A - 01'11

=Qo

1

U

- JA. (71u - '100 )ßoo
- -du
+ 88
u

1

_JA 8(ßu - ßoo)iJu -du - 8 JA. (ßuoiJu- - ßoo 8ijoo)- -du.

- 8

u

1

1

U

Now we take the limit of the above expresaions for each case M A - +00. Obviously,
by Theorem I, and Theorem 2.c, the only problem will come from the last two terms. We
check them separately.
For any smooth differential form JJ on M, we know that if u ~ 1, from definition andTheorem 2.c,Hence in 'D'(M),

J

A

1

du
(71u - 7100)- U

We also know that

1.+00 (TJu - 7100)-'
du
1

U

.

C

pu,r,cp,m(TJu - 7]00):5 Vii'
so in 'D N- (M), we have
R

J
A

1

du
(71u - '1(0)- U

1.+00 (fJu - 71(0)-'du
1UHy the fact that i and i are transversal, we know that, in V:V. +N. (M),.

R

Thus, we get- -du - {JaBa A. ('1u - TJu )ßoo

J
1

U

1.

00

R

-

du

(1]u - 1]u )ßoo -

1.

U

192

in

Relative Bott-Chern w.r.t. Closed Immersions

V:v- +N- (M). On the other hand, since ß(X) is closed, we haveflR.

Similarly, in

V:v- +1V- (M),ftfl

In particular, we may also have-) -du 8_0 JA ß(X) (-TJu - 1]00
I

in

U

8-0 j+(X) ß(X) (-1Ju - TJ(X)- ) -du ,I

U

V:v- +N- (M). Now we need the following
R

flLemma. With tbe same notation as above,(1) fl (ßu - ß(X) )8iiu :u - fI(X) ({3u - ß(X) )äiju :u , in V~ _+N- (M). In partieular,
A

R

8

J

A

I

-

du

(ßu - ß(X) )8iju -

- 8

U

j(X)
I

R

- du
(ßu - ß(X) )8fJu - ,-Uin V~_ +N- (M).
ft

R

Suppose we have this lemm?a then we may take the limit for A at any piaee above. In
. I
. pM
/
,0
partleu ar, we get In M'UM' M'UM"

1+ (ß~ ß~)

du

du

j+OCI

du

U

I

u

1

u

fi (ß~ _ ß'd) du +JoUI

= iio{ foo (ßu - ßo)-+
1

+a

1

- du
(ßu - ßo)-+

1 -

OCl {

a

u

00

_

U

(ßu - ßoo)-}j+oo (ßu- - ß(X)-}.
- duFrom here, we easily have the first assertion by certain trivial substitution.

Proof of the lemma. We only prove (1), aB the rest is very simiiar. Obviously, thediffieulties in this eonvergenee only oecur near MN. For any x E Al/1, let U be a small openChapter 1.8.

193neighborhood of x, and let r, tp, m be as usual but with respect to M'. We know that, byTheorem 2.c,-c

Itp(ßu - ßoo)(~)l :5 v'ü(l + 1~I),cpU,r,lp,m(ßu - ßoo) :5 Vü'

1<'ö~(€)1 :5 C(l + 1~12),
PU.r,,,P,m(;7u) :5 C.
Therefore , by definition and the condition NitIM ' n NRIMI[Bö 83], we know that

= {O}, from Theorem 8.7.4 of(1) There exists a natural number k such that for u ~ I,

(2) For any smooth current (J with compact support in U I if .!l ia a c10sed cone in R:u suchtbat
ß n (Ni + NaJ{O}=on (M' U M') n u, tben for u 2: I,

Therefore, in

V:V. +N. (M),
R

R

J

A.

1

du
joo
- du
(ßu - ßoo)8ijU-'u .
1U(Pu - ßoo)8iju -which completes tbe proof of the lemma and hence the theorem.
We bave now finished our account of the theory of Bott-Chern secondary characteristic
objects in various contexts, except for the uniquness, wbich will be proved after the proof ofarithmetic Riemann·Roch theorem for l.c.i. morphisms. Next, we discu88 the aingularities of
the relative Bott.Chern secondary characteristic currents with respect to closed immersions I
wbich will be used in the ternary theory.

§I.8.5. The Singularities ofRelative Bott-Chern Secondary Characteristic Currents
In this section , we prove that, in general, the current chBc(71, g'7; i , gi; €,~) is smoothon M - MI and not locally integrable on M. In the process, we determine the singularities

194

Relative Bott-Chern w.r.t. Closed Immersionsof chsc(f}, g'1; i, gi je, ge) near M'. In particular, if Y ia anormal coordinate to M', thennear M', chsc( FJ, g'1; i, gi je, g~) -- IY 1- 2di mN, hence the integral of chsc( FJ, gl'J; i, gi; e, ge)on the complement of a 6 neighborhood of M' in M is equivalent to c Log6 as 6 - O.
Furthermore, we can calculate chsc(f},gl'J;i,giie,9() as a finite part by Bubtracting off thelogarithmic divergence. All the results in thia section will be used in next chapter to discUBB
the deformation theory of relative Bott-Chern secoodary characteriatic currents .

.

1.8.5.a. The Singularities
We UBe the same notation aB above. Identifying N with the orthogonal to T M' in TM
aa c;oo bundle, we aBBume that gN is exactly the metric induced by gTM. For Xo E M',
let U he an open neighborhood of Xo in M, and z = (zl, ... , zl) be a bolomorphic systemof coordinates on U such that V
M' n U ia represented by ZA:+1= z'
O. Setx = (zl, . .. ,zA:), Y(zA:+l, ... , zl). Then x is a coordinate system on V. For E > 0, let
D~ he tbe open ball with the center 0 aod radius E in ce. Then for e small enough, we bave
V X D~ C U. Usually, we UBe the convention tha~ x E V, Y E Dt . If (x, y) E V X D" weconsider y as an element of N ~.

=

=

= ...

=

On V x D~, we haveThus, for aoy 0', we have the decomposition 0' = L;~:N O'P according to the partial degreein the Grassmannian variables of Ni. We deoote the maximal degree element in tbis
expression by O'm&X. In particular, on M', i.e. at (x,O), a max does not depend on thecoordioate system (x, y).
For u > 0, we bave the superconnection

Then we have the differen tial form Tr [N H exp( - B ~)] on N, which in the sequel, we consider
aß a form on V X ce. Thusjaod as u - 0+,

Tr,[NHeXp(-B~)]mllX= G(u).

Hence on N - {O}, we have a smooth form
oo

ßF:=

r+ [21Ti]Tr,[NHexp(_B~)]maxdu.uh

In fact, as u - +00, the form Tr [N Hexp( - B~ )] decays exponentially, so ßF is well-definedand depends on the coordinate system (x, y).
jChapter I.S.

195

Theorem. (1) For any a > 0, yEN - {Ol,(2) Let w (€ , he) be the smooth form on M - M' defined by the restriction of the current

to M - M'. Then there exists C > 0 such that l if (x, y) E V X DeIHence the current w(€, ge) - ßF ia integrable on V x Dr:.
(3) Let 9(€, ge) be the smooth form on M - M' defined by the restriction of the current

to M - M'. Here., is defined in subsection 4.a. Then 9(€, gel ia locally integrable on
M, and coincides aB a current with(4) If the metric ge satisfies Bismut condition (A) witb respect to gN and gq thenIßF = -(dimN - I)!

IYI2~;mN(td-ll'(N,gN)Ch('1'9'),rt:N'

where ..\ is the volume form on N with respect to tbe metric gN.

Proof. (1) Let tPt be a group of diffeomorpbisms of N: y 1-+ e'y. Tben the group tPt isgenerated by the vector field Y = y + y. Also

So if Ta is the map (X,y) 1-+ (x,ay), then T:ßF= ßF. But naturally we bave.pF (X, Y) -- a 2dimNßF ( X, ay ) .

Ta

Hence we have (1).
(2) In tbe sequel, the COllstants C may vary from line to line. In the definition of1
ebBe in section 3, the first integral o { } ~ defines a smootb form on M, aod so does notcontribute to the singular part of the current
Dear M'. So we only need to caDsider
the other terms.

f

,gO)

Let CX u and 6u be the farms on M and N respectively given byRelative Bott-Chern w.r.t. Closed Immersions

196
Then

+00
],1

1+

00

du

au(z, y)- =U

1!l121r2

duar2u/I~12(z, y)-.
U=

On the other hand, if we denote U u
Tl/fil then by the proof of Lemma 8.1, we know that
if V and c are small enough, there exist C, C' > 0 such that for u ?: 1, x E V, lyl ::5 eVü, we
have•

lu~Q'u(x, y) - 01 (x, Y)l :5 ~exP( -C'lyIJ)·

Thus by the fact that

(O'~a)(x, y) =JdimN

Lo u-P/Ja{(x, yu~),

we know that for 0 < 1] ::5 1, IYI ::5 e,

So if lyl ::5 C,ßI then 7] :5 1, andSo we have proved

Lemma 1. With the same notation as above, if lyl ::5 c, 0 ::5 P :5 2dimN,/

Now we can deduce assertion 2 from the foUowing

Lemma 2. AB lyl - 0,co(a) lylJdim N I fl+ Q'~(x, y)~u 1:5 Clyl.
(b) For 1 ::5 P :5 2dimN - 1,

Chapter 1.8.

197

Suppose we bave this lemma, then note that since the function l/lyl 2di mN -1 il;! integrable
near 0, by (1), we have (2).
(3) First, we bavevuTr, [By Vexp( -B~)] = -iyTr, [exp( -B~ )].
Therefore
vuTr,[8vVexp(-B~)]max= O.

Similarly aB in Lemma 2, we see thatAs a consequence, 9{~, g{) is integrable near M'.

fl+ ooft { }{ } duu by integrals
Replacing in tbe analogue tbe integrals
duU , we find that
if 8G(~,g{) is tbe density of tbe smooth approximating current ..,.G(~,g{), then we have theuniform estimate for a 2: 1, % E V, lyl:5!,

lyI 2di m N IOG(€, g{)(%, y}1 ::; Clyt·
Thus as u - 00, OG(e,g{) so 8{€, g{) = .,,(€ I g{).

O(~,g{). But ."G(€,g{)= OG(€,g{) and ."G(€,g{) - ."G(€,g{),(4) This follows from the fact that
Tr.{NHexp( _B~)]max= _ (iu)dimN (td- 1)'( _(V F )2) Tr[exp( _(V'1)2)]exp( _u~YI2 )"'twhich may be proved as wbat we did for Theorem 7.l.c.
Finally we end this subsection by the followi~g

Proof of Lemma 2. Since
if Y 1:- 0, we have(!JUt 6f(z,!VU 1:1) = ~2u (x, 1:1)'(a) By Lemma 1, the assertion is a consequence of the facts thatand 2 dimN 2: 2.

198

Relative Bott-Chern w.r.t. Closed Immersions(b) By Lemma 1, note that for p ;:: 1, tbe integrall+COI"I~/I:~

y

du

y

U

(eJU)P6 i (x,eJ'ü-1)1

is bounded as lyl - O. Also, the expression on the right hand side of lemma 1 is bounded,so we have (b).
e. Sinee 2 dimN ~ 2, we haverlJlI~II:~

I Jo

d

(e.fij)2dimN

6idimN (x, €VU 1:1) Uu I :5 Cly!2.

Thus by Lemma I, we have

co
co
r+
[2ri]o:~dimN(z, y) du _ r+ 6;~~mN (x, -IYI) du :5 Clyj.
11
u
Jo
Y

Ilyl2dimNtl

So we have the assertion (e).

L8.5.b. The Principle Part.
The form w(~, g{) ia not integrable on M in general. However, by Theorem a.(2), we
know that it has a well-defined principal part, whieh defines a eurrent. We now compare
the eurrent ebBe (11, g" i i, gi ; €, g€) with this prinei pal part.

l'heorem. (1) For 11 > 0, let M" denote the set of points of M whose Riemannian
distanees to M' are greater than 11. Let JJ be a smooth even form on M, then as 11 > 0
converges to 0,

has a limit, which we denote by f M jjWC(€, g€)(2) We bave

iM JJch (11, g"; i, gi; {, g€)
BC

=].M

jjWC(e,g€) -].
M'

f'JJ

r (2LogIYI- r'(I))[27Ti](Tr,[NHexp(-B

iN

2

)]).

(3) If the metrics g€ satisfy Bismut condition (A) with respect to gN and g", thenL

(2LogIYI- r'(I»)[21fiJ(Tr, [N Hexp( -B')j)dimN-1 1=- (td-1)'(N,gN)ch(7],gl1)( L.1:=1

k + Log2) .

Chapter 1.8.

199

Proof. With the same notation aB above, we choose geodesic coordinates in the directions of TRM, which are normal to TRM' with respect to the given Euclidean scalar
product ofTRM. For E > 0, set B:--:= {Y E R 1 e;IYI ~ c}; U
V x B:-- is then a smaHneighborhood of x in M. We identify R 1e with the real normal bUDdle NR to M' in M. Set=

w := [21l"irlw(~,ge),.

ß:= [211"i]-lßF(~,ge).

(1) We may assurne that the support of JJ is contained in U. The form j"JJ on V lifts
naturally to a form on V x R 2e. Moreover, j" JJ haB partial vertieal degree 0 and coincides
with JJO(x, 0). Hence it makes sense to consider the following identity

By the condition that JJ is smooth, we haveThus, by Theorem 5.a, we have

MoreoverSo by the same theorem, we have

But by Theorem 5.a.(1), we have that if T]::; E, then

Also,
{

1M'!

lIYISt(i"JJ)ß(y) = (LogE - LogT]) { i"JJ { i n {3.

,

1M

I

1s

N

Now the assertion ia a direct consequence of the following local result:Lemma. Let SN := {Y E Na; IYI = 1} be the unit aphele in Na. SN is naturally
oriented with n the unit vector in Na normal to SN aod pointing outwards. Suppose

200

Relative Bott-Chern w.r.t. Closed Immersions

a is a smooth form 00 N such tbat there exist C, C > 0 for which lai :5 cexp( -Clyl~).
Then, on M', we have1+

f er = fiN

coineP: ads .

iSN-coIn fact, by this lemm~ we know tbat

Therefore, as u - 0+,

has a limit

Proof of the lemma. First, by the hypothesis, we know that the integrate

+co

1

-co

exists. Let F : R x SN -in<p: erds

N R be defined by (s, y) 1-+ <p,(y), then

{ er
iN

Thus if j is the embedding SN '-+ NR,

This proves the lemma.
(2) By definition, we know that

We discuss the various cases.

=

1.

RXSN

F·a.

Chapter 1.8.
(a) Ir 0:5 p :5 2N -I, as T -+00, by the latest relation before Lemma 5.a.l, we get

1. l
M

Jl

201

T

1

du
~-U

1. l+
M

Jl

1

co

du
~-,
U

which ie a locally integrab!e current.

(b) SetThen

We cOßsider this term by term.
(b.l) By the same relation as used in (a), as T - +c<\ we haveand(b.2) Since

we have

Hut for 0 :5 T :5 Te' I

202

Relative Bott-Chern w.r.t. Closed Immersions

we have

Therefore, put (b.i) and (b.2) together, we finally have

Hnece, by definition, we haveThus by (a) and (b) above, we have

Chapter 1.8.

203

So, by (1), we have

Now

Hence

Therefore, we have (2).

(3) By an argument similar to the proof of Theorem 7.1.c, we haveLexpct~I')'I'r.[NHeXP(-B')]=- (1 + t\dimN (21ri)dimN (td- 1j'(-Cv N)2)Tr[exp( -(V"7 )2)].
But for 0 < Re(s) < dimN I we have

1
r(s)

1+ ,t
1+
00

0

1
r(-,)

•

00

0

-t1Y1 2 dt

2I.

ex p (-2-)T =( IYI2) ,t'
dt r(dimN - ß)(l+t)dimN t = r(dimN) .

204

Relative Bott-Chern w.r.t. Closed Immersions

So, for 0 < Re( B) < dimN, we haveL(1:122)''fr, [NHexp(- B )]= _(211"i)dimN (td -1)' (_('v N )2)Tr(exp( -(V'J )2)) f( dimN - B)•f(dimN).

Each aide, of this equation extends to a meromorphic function of B which ia holomorphic at
O. ThusB=

L(2Log IY!- Log 2)Tr, [N Hexp( -8 2 ))= _(21ri)dimN (td- 1)'( "":(V N )2) Tr[exp( _(V"7)2)] f'(dimN)
f(dimN) .
Now by the fact that f(B + 1)

= f(8), we getf'(dimN)f(dimN)

which completes the proof.

,= r (1) +E k'dimN-1 1

Chapter 1.9.

205

Chapter 1.9.
Ternary Objects And Deformation To The Normal ConeIn the previous chapters, we gave the theory about the Bott-Chern secondary characteristic objects for various situations. Among others, the very important property for tbe
secondary cbaracteristic objects is tbat they measure the change of cbaracteristic objects.
Thu8 if we want 10 measure tbe change of secondary characteriBtic objects, naturally, weneed a kind of tbe so-called ternary objects.
Since we do not find any furtber application at the present time for tbe ternary objects
in general, in this chapter, we ooly deal with a special situation, which will be used in
tbe proof of the aritbmetic RiemannwRocb theorem in part II: We discuss the deformation
theory of relative Bott-Chern secondary characteristic objecta.
Before reading tbe first section, it may be helpful to look at tbe beginning of 1.9.2.
Also, if tbe reader is only interested in tbe deformation tbeory for the relative Bott-Chern
secondary characteriBtic forms with respect to smooth morphislIlS, he or she may just read
1.9.2. References bere are [BGS 91], [Fa 92], [We 91].

§I.9.1. Basic COßstructioßs And Facts
We recall tbe construetion of tbe deformation to the normal cone for certain closed
immersions i : X t-+ Y and 80me associated facts. Tbe advantage of thiB construction
is tbat, 10 study a property for an arbitrary closed immersion, we only need to study theproperty for a eection of a projective bundle. For topologists , this means that in a homotopy
cl 888 , one can find a good representative 80 that certain properties are very easy to check.
Tbe referenceB here are [SGA 6], [Ha 77] and [BFM 75].

1.9.1.a. Projective Bundles And Koszul Complexe8
Let i : Y c........ X be a closed embedding of camplex manifolds. Denote by Ii the ideal
sheaf of Y in X. It is a standard fact tbat if X ia of codimension one tben Ii ~ 0 X ( - Y)
aod there is a canooical inclusion Ox C Ox(-Y). Also, there is a canonical normal bundleI206

Ternary Objects and Deformation Theory

Ni associated with i. We then have the following canonical isomorphisms:

where N·, the conormal sheaf, is tbe dual of the sheaf of the sectiona of Ni. Moreover,
we have
Torfx (Oy, Oy) ~ /\iNi.
(The definition of Tor ia given in 11.1.)
Let <p : V - 0 x be a homomorphism of vector sheaves: Then tbe differential in tbe
assoeiated K08zul complex K(cp) := (AV) ia the complex defined byi"',el A ... /\ ei'- L./-1)

-':-1

~

cp(e,l:)el/\'" Ael: A ... /\ ei

.

,1:=1

Now let p : E - X be a vector bundle with [ the sheaf of holomorphic sections. Thereia a canonical homomorphiam L : p.[ - 0 dual to tbe tautological seeHon of p. E. This
bomomorpbism vanishes along the zero section 8 : X - E, and the assoeiated K08zul
complex K(t) := (/\'p. [.) is a resolution of 8.0 X. Thia fact hBS the followiog explaination:
Let p : P := P X (E EB C) - X be tbe projective bundle of E EB Cover X. On P, there
ia a universal exact sequence:o- 'H - p. [. EB Op - Op(l) - O.
As a divisor 00 P, P X (E) ia gi yen by tbe vanishing set of the map 0 p - 0 p (1) ind ueed
from tbe ind usion 0 pep·[. EB 0 p. That is, P X (E) is the loeus on whieh 0 p C 1l. Sinee
a line LeE EB C, whieh maps surjectively to C, is equivalent to a homomorphism C - E,
the complement of P X (E) is eanonically isomorphie to E.
On tbe other band, tbe map 0 : 11. - p. E- induced by the projection from p. E· fB Op
ie an isomorphism on E, and on E, if we compose 0- 1 with the homomorphism induced by
tbe negative of tbe second projectioo, we obtain L. Let cp : 1l - Op be the homomorphism
induced by tbe negative of tbe projectioo p. E- fB Op. By the fact that this map is surjectiveon P{ E) aod ia equal to L on E, we know that the BSSOciated Koszul complex K ( cp) is a
resolution of s.Ox, where 8 : X c....... E C P(E EI! C) is the zero section.

I.9.l.b. The CODatrnctioD Of The Deformation Ta The Normal ConeLet i : Y e-.. X be a closed embedding of complex manifolds of pure codimeosion n.
Denote by
7r: W:= Byx{co}X x pI _ X X pI,

207

Chapter 1.9.

where By)({oo}X X pI denotes the blowing-up of X X pi along Y X {CO}. Let P be theexeeptional divisor. Then the map qW : W - pi, obtained by eomposing 11" with the
projection q : X X pI - pi, ia fiat. For z E pi:

q

-I()
Z

for z # 00,= {Xl
PUB y X , for z =00.

Also pnByX ia the divisor at 00 on P, whieh may be identified with the exeeptional divisor
on ByX.
We know that P ~ P(NY)({OO}/X)(Pl) aod

where py : Y X {oo} -Y and Pco : Y X {oo} - {oo} are the projeetioDs. Thus we have

The bundle NOO/PI, while trivial, is not canonically trivial. Henee P ia the projeetive
eompletion of Ni (9 N;'}pl with the divisor P(Ni 0 N;'}pl) ~ P(Ni ).
In particular, we have the foUowing diagram:

P
1rp

1

Yx{oo}

l=. W = PUByX
11f
i oo
'--+

X X pi.

I.9.l.e. Deformation of The Resolution
At the beginning we should say that this aubsection will ooly be used in seetioD 4.
With the same notation 8S in subseetion c, let" be a vector sheaf on Y and let { -i." -obe a resolution of i.l1 by a bounded complex of veetor sheaves on X. We deforrn € throughea eomplex on W to a Koszul type resolution of 8.11 on P, where 8 : Y - Ni X N~ipl C P
is the zero seetion.
First we eonstruct the eomplex ({, di ). By the eananieal exact sequenee

0- Opl(-OO) - OPI - Oet:J - 0,
if we let KrxJ be the complex OPI (-00) - OPI with OPI of degree zero, we knaw thatp. ~ (9 q. K rxJ ia a resolution of i oo • 7J. Thus, by exp. VII, lemma 3.2 [SGA 6L if we let208Ternary Objects and Deformation Theory

be a bounded eomplex of veetor aheavea on Wandbe the veetor sheaf on P, we see that

Here, ?-lp(Q) denote the cohomologieal sheaves of Q. In partieular, it is loeally of projective
dimension one, by the fact that P ia a divisor of W. Heoee if [ is a veetor aheaf aod

ia an epimorphism, then Ker () ia also a veetor sheaf. As a eonsequenee, and by an induction
on i ~ 0, we know that Ker(df) ia a veetor sbeaf on W. With this, we may define as
folIows:e

For k ~ 0, eJ: := Ker (4) ®Ow Ow(koo). Here

Here q : X X. pI _ pI is a projeetion to pI, and qW := q 0 'f. Similarly, we denote tbe
projection X X pI - X by P and PW := po 11".
Next we rollSt give the definition of the boundary map8. Hy definition, we know that

where we identify Y E p'We i-I ( -00) wi th its image in pW ei under the natural iod usion.
Henee, (i is isomorphie to the fiber product of the diagram:ei-

p'Wei-l«i - 1)00)!

1PW€i(ioo) -pW€i-l(iOO).

4:€i - ei-I'

Tbe differential pwdi extends to a homomorphisrn
Then, the restriction of
Pw~ @ Idow(ioo) to fi has its image in PW{i-l«i - 1)00). Hut (~)2 = 0, so the image is

=

€i-l

contained in
C Pw{i-l«i - 1)00). On the other hand, 4lw-woa
pw(a1). So
vaniahes on W - W oo , and henee on W. In this way, we get a camplex (f, cJi) on W.

4:z

In addition, we know the following=

Corollary. Let U
X - Y. The restrietion of
aeydie complex on U x {oo} C B y X C W00'ee

to U x pIe W restriets to a split

Indeed, this eomes from the fact that if is acydie, then f is the puB-back, via 7T, a.
eomplex on X x pI, whieh is aeyclie when restricted to X X {O}.

209

Cbapter 1.9.

Now we discuss (i, ci) in more detail. Let I : Y x pi ~ W be the natural indusion.
Sinee I. Oy )( pi and Ow are Tor independent, the above eonstruction of eommu t~ wi th
the restrietioD to Y x pi C W. The restrietion of ~ to Y has loeally free homology:i.

00

1ip (i·€) =Tor~X(OY,11)~ Tor~x (Oy, Oy) 00 y 71 ~ rl Nt 00 y 1].

Henee, associated with i· €, we have two natural short exaet sequenees of veetor aheaves
associated with its kernel aod its image:

-

i· €i
Zi

-

Bi
1ii(i·€)

-+

0;

o.

Therefore, ii ly x pi is obtained by pulling back the extension from the ind usion Bi ( -00) ~Bi and twisting with O( ioo). Thus, restrieted to Y x {oo} C Y X pi, we find that sinee

Opi(oo)loo ~ N oo / p i,

Also tbe differential elf restriets to the map di : (x, Y) 1-+ h'i(Y), 0), wberet:>.. lfi-I

l i : 8 i 'CI JV oo/P I ~is the natural inclusion. So if we let

Zt:>.. lri-I

i-I '61 JV OO/Plr. be the split aeyclic complex withand the differential di(x, y) = (y,O), then we have an exaet sequence:

Before we go further, let U8 look at an example with 7] = Oy. Suppose that i : Y ~ X
is defined by equations %1
Zn
0 with the Xj part of a system of coordinates on
X. Let €. = K.(x) be tbe Koszul complex assoeiated with the map 01- -+ Ox which sends
a to ax. Here JI[ := (Zl,'" I zn). Then K.(x) ia a resolution of i.Oy. On W, we have an
epimorphism=... = =va, EB Ow(-oo) (a, b)

1-+

Ow(-P)
ax + b.

By considering tbe projective dimension, we know that tbe kernel of this morphism ia avector sheaf too.
Proposition. There is a eanonieal isomorphism of complexes
If' : PwI(oo) - Ow€- K.(lp) with

210

Ternary Objects and Deformation Theory
induced by tbe negative of the projection map 0"-, EB Ow (-00) -+ Ow ( -00). Furthermore, we have(1) {.I By X ia apli t acydic.
(2) €.I~-.o.JI-t ia the tautological Koszul complex.
• '0'

oo/pl

(3) { ia a resolution of I.Oyxpl.

Proof. The complex pwe. rs K.OO is tbe Koszul camplex aBsociated with the mapa ofsheaves .
~ Ow( -00) -+ Ow

Ow

induced by the indusion Ow (- P) C Ow. Hence the Koazul differential

is the comp08ition of the follewing maps: the canonical map

which haB the kernell\iI; the injective multiplicationand the natural indusion

Hence the indusienl\i(1) - l\i(OW EB Ow( -(0»identifies l\i(1) with Ker (df). In particular, we may identify l\i(1(oo» with fi. Next, we
check (1), (2) and (3).
The negative of the projection Va, EB Ow ( -00) - Ow ( -00) induces a surjective map
Ow(-OO) EB Ow - Ow,

and hence, a map 1(00) - 0w. On tbe other hand, there is an induced morphism between
Koazul complexes 1\'1 (00) - IV (Oa, (00) EB Ow) which gives the comrnutative diagram-Ai (1(00»1
I\i-l(0l'v) =€i-l-

l\i-l(I(oo»l\i(OW) = €i

- l\i(OW(oo) EB Ow) =1

l\i(OW(oo» EB l\i-l(OW(oo»1- l\i-l(OW(oo) EB Ow) =

by
a

-

(a, (-1 )idi(a»

1

1

da

(dj(a),O).

l\i-l(OW(OO» EB l\i-l(OW(OO»

IChapter 1.9.

211(1) Since Ow ( - P) ® Ow ( -00) = Ow (By X), we have an exact sequence

0- I(oo) - O?v(oo) EB Ow -+ Ow(ByX) -+ O.
The restrietion to ByX of the map OWIByX C (Ow(oo) EI) OW)IByX to Ow(ByX)IByX
vanishes. Therefore, on restricting to By X I(oo) splits as a direct SUffi
I

The Koszul complex associated with I( 00) - Ow therefore restriets to the Koszul complex
for the negative of the projection I 1 EB OByX - OByX, which is split acyclic.
(2) This comes from the fact that on P, 7(00) restricts to the kernel of the mapLe., to the analogue of the vector sheaf 1l above. Hence, with a similar diBcussion, we have(2)(3) It is sufficient to show that the map f -+ I.VYxPI, induced by the map

is a quasi-isomorphism of complexes. By tbe eorollary above, we may deduees it in a
neigbborbood U of Y x {oo} C W. Sinee e.lw-woo = Pw{. is a resolution of I.O yxc and·IByX is aeyclic, if we choose a loeal system of equatioDs for 00 in pI, then Z1 = ... =Znt 0 is a loeal system of equations for Y x {oo} C X X p1. Henee I is isomorphie to
tbe kernel of tbe map 0w+ 1 -+ Ow defined by

e

= =

n

( al , ... ,an, b) 1--+

L x + bt.
ai 1

i=l

Henee we may choose U SO that 7'-"",~, t iB part of a system of coordinates on U, with
Y x pI C U given by T
t = O. Tberefore tbe map Va, - O~+l sending(a I, ... I an) to (al, ... ,an I ai Xi /t) is an isomorphism onto 7(00). Composing wi th thenegative of tbe projeetion O~ 1 Oa, e Ow -+ Ow, we ean identify the Koszul eomplex
AI(oo) with tbe Koszul eomplex K'(T"'" =t), which is a resolution of j.OyxPl. Tbis
eompletes the proof.

= ... = =t =f.(=

Now we deal with the general situation.
Theorem. With the same notation as above, we have(1) is a reeolution of i.(pYf}).
(2) €IBYX is split acyclie.
(3) Tbere is a natural exact sequence of complexes of vector sheaves on P:

e

212

Teroary Objects aod Deformation Theory

Proof. (1) Thia ia a local problem on X. So we may suppose that ~ = (El1 (K(x) ~ V),
where ( is acyclic aod V is a vector sheaf on X such that j* V ~ 1]. Thua ~ (El1 K("';) ~ Pw V
with ( acyclic and K(x) is a resolution of I.OYxPI. Hence is a resolution of 0YxPI ~V ~ I.(py7]), which gives (1).

i

t

Pw

Note that {lw is ap!i t acyclic, and hence {.ll!Y X is split acyclic too. So by (1) of theprevioua proposition, we see that at least locally, ~.IByX is split acydic. However, by theconatruction, we get tlByX -ByXnp is split acyclic. So (2) comes from the fact that all the
splitting described above are compatible, aod uoiquely determined.
00(3) By the conatruction of
on P:e, we know that there is an epimorphism of vector sheaves

where 1{ is the kernel of Nt ~ N 00 I p I $ 0 P - 0 P (By X). Also by the expression in the
proof for (1), (2), we know that this process is compatlble with differentials. Hence ~ is amorphism of complexes and, 00 X, its kernel is <lp. On tbe other hand, by tbe indusion

we have a morphisffi of complexes (joo)*1fp7] - (joo)*(i (8l (Ow - Ow(co))). Hence it isenough to prave tbat the image,of this morphism ia contained in and ,is equal to tbe kernelof e. Again this is a local problem. Hence we may use tbe expression { ~ El1 K(x) 0 Pw V
and we see tbat ~ (IYxPI
KerE:, which completes the proof.

i

<

=

r.

SI.9.2. Deformation To the Normal Cone: Smooth Situations
In this section, we discuss tbe deformation theory for tbe relative Bott·Cbern secondary
characteristic forms with respect to smooth morphisms.

1.9.2.8 An Axiom
Let i : X c....... Z be 8 closed immersion over Y witb smooth structure morphisms ofregular arithmetic varieties f : X - Y aod 9 : Z - Y. Tben we bave the following diagram
for tbe deformation to tbe normal cone as stated in subsection l.b:x .....
!

X x {co}

!

p

r -

X X pIX-

X x {O}

X

r

{co}

joo !

-

BXx{oo}Z X pI = vV

L

pI

jo T

-

{O}.

C-...+-i oo

P(N EB Ox) + Bx Z = W oo

1
c.......
iot.-.+

Z X {O}

= Wo

!
T

(*)

Chapter 1.9.

213

Moreover, for 1 = 9 0 i, we may have the associated picturei_

X x {oe}lP(N ffi Ox)

c-

100 ,

l

/900

Yx{oo}I

X X ~lF'

lW

c.........-/G1

Y X pIio

X X {O}

Wo

~

10 ~

/90

Y X {O}

where 10 = Fa = I, 90 = Go = 9, i o = 10 = i aod 100' 900' i oo are the restrietions of
Foo , Goo,ICQ to P(NEBOx), respectively. Let 8WCQ:= w~nw~ with w~ := P(NffiOx)
and W~ := BxZ. Then w~ and w~ intersect transversally along awCQ' Choose hermitian
metrics on the normal bundle of i and relative tangent bundles of 1 and 9. Let (E, p) be ahermitian vector sheaf on X. Assume that
0- :F. - i.E - 0
is a vector sheaf resolution of i.E. Put hermitian metrics on :F. so that Bismut condition(A) is satisfied. Further, choose hermitian metrics for each pair so that Bismut condition(A) holds. By the results in Chapter 6 and Chapter 8, with certain acyclic conditions, wehave the following correspondences: (Here, for simplicity, we will omit the pull-back symbolby the projection p.)(1) With respect to the ,smooth morphism F, we have

ChBC(E, P; 100' PCQ) .-...-..
100.( ch(E, p) td(ICQ' Peo)) - ch(ICQ.E, Joo.p);
ChBC(E,p; F,PF) F.(ch(E,p) td(F, PF)) - ch(F.E, F.p)jchBC(E,p; Jo,po) . fo-(ch(E,p) td(Jo, Po)) - ch(Jo.E,Jo.p).
(2) With respect to the closed immersion I, we have

ChBC(E, P; ioo,Poo) td-I(Noo , Poo) ch(E, p)600 - ch(:Foo ., Poo.)

ChBC(t,p; [,PI)td-I(NI , PI) ch(E,p)6 I - ch(FI·,PI.)

ChBC(E, Pi i o, Po) . td-I(No, Po) ch(E, p)8o - ch(Fo., Po.).

214

Ternary Objeds and Deformation Theory(3) With respect to the smooth morphism G, we haveehBC(.:FCCI" PCCI'; gCCl' PCCI) gCCl. (ch(.FCCI ., PCCI .) td(gCCl' PCCI») - eh(gCCl. F CCI ., gCCl. PCCI .) ichBC(.F, p.; G, pa) G. (eh(:F. , p.) td (G, pa)) - eh (G.:F., G. p.) iehBC(Fo., Po·; gOI Po) go.(eh(:Fo., Po.) td(go, po)) - eh(go.:Fo.,go.Po.).
Obviously, by the eonstruction of relative Bott-ehern secondary ebaracteristie forms
witb respect to smootb morphisffiS, we know that, for (1), there is the following relation:

dcJC

1..1 VoglzI2]~bBc(E,p;F,PF)=ehBC(t",p; !o, Po) - ehBc(t",p;!CCI,PCCI)+ }plf [loglzI2] dcfehBc(f, p; F, PF).
Tbis may be thought of as a trivial deformation theory of the secondary eharacteristie
forms, since both sides now are just zero. Nevertheless, this sheds a light about the theoryof ternary eharacteristic objects. So basieally, we may hope to have the following relations:

dcJC ( Voglzj2] ChBC(f, Pi 1, pr)}pl=ehBc(f, P; i o, po) - ehBc(f, P; i CCl , PCCI)+ }pl( ~oglz12] dlfchec(E, P; 1, pI);dlf

f [loglzI 2] ehec(:F., p.; G, pa)}pl=chec(.ro., po·; gOI Po) - ehec(:FCCI ., PCCI'; gCCl' PCCI)+ }plf VoglzI 2]dcfchBC(:F.,p.i G,pa).
In tbis sectioo, we ooly study (3). We delay the detailed discusaion about (2) till the endof this chapter.
For (3), io order to make the situation simple, we will assume that the closed immersion
ia a codimen~lonal one closed immersion: In this ease, we have a natural vector sheaf
resolution for i.t", tbat ia,
o- IiE - E - i.E - 0,
which comes from the structure ex~ct sequence

O-I-Oz-Ox-O.

Chapter 1.9.

215

(On the other hand, for (2), since i co ia just the zero seetion of the projective .bundle, we
may use the Koszul complex to make apreeise calculation.)In the following, let E' be the complex I JE (B x Z) '-+ E(B x Z) of vector sheaves on t-v.
We endow E aod I := I/ with hermitian metrics in such a way that on an open neighborhocd
U of W~B x Z, the metrie of I I coincides wi th that on Ow. (This is possible sinee X x pI
does not meet Bx Z in W~) Henee we also have ametrie on I E.

=

With this, from the above discuSBsion for (3), we introduce tbe following

Axiom. (Upstairs Rule) Let i : X '-+ Z be a codimension one closed immersion. Let(E, p) be an F-aeyclic hermitian vector sheaf. Suppose that io.E (resp. ico.E) ia a 90
(resp. 9co)-acyclie vector sheaf. Then, with the same notation aB above, we havedyety [}plnchBC(E', PE'; G, PG) ogjzl2]=chBc(E'lwo,PE'lwo ;90,Po) - chBC(E'lw 1 ,PE-I W 1 ;9r'X),Pr'X»
00

oo

+ }pl
f G.(ch(E" ,pe.) td(Ta( -Iogoo), pa»)[10glzI2] - }plf ch(G.E·, G.PE-) ~oglzI2].
EBBentially, this axiom ia the downstairs rule for ternary objects (in this special content).
In general, by the fact that the secondary objeet may be thought of aB tbe one whichmeasures tbe change of the first level objeets, e.g. ehBc measures ehanges in eh via dd e , sofor tbe seeondary objects, we mayaiso want to construct a ternary objeet which measures
tbe change of them. But we do not want to go further here 8B there is no further application
now. Instead, let us say a few words for the structure of the supposed axioms: Hy comparing
Proposition 1.1.3, or better 1I.1.3.a, with axioms for the secondary objects, we may say thatthe axioms for ternary characteristic objects should have the same structure 8B these forsecondary objects, i.e. the dcJC equation to measure the change of secondary objects, thefunctorial rule, aod tbe uniqueness rule. Furthermore, one may use tbe so-caUed b-calculus
developed by Melrose and others to check the above axiom, with the special attention onawoo • But from my point of view, aU of this needs another bock. So we will not really doin this way. Alternatively, we go directly following Faltings.
To do BO, the first problem we meet is that the projection from W to Y X pI is not
smooth: How we can define the relative Bott-ehern secondary eharacteristic forms withrespect to tbis projection? Note that now we are working with a very special situation, i.e.
tbe deformation to the normal cone with a codimension one closed immersion, we see thatance we use tbe logarithmic relative tangent vactor sheaf at infinity, we may da the sametbing 88 wbat we did in Chapter 1.6. For more details, see the later part of this seetion.
Next we denote by F' the complex f.(IiE) '-+ /.(E) on Y. Then, on Y x pi, we havean augmentation v: F'(- G.(E'» - G.(E' ®"n~l)l whieh vanisbes on BxZ. (Note thatbere F' - G. (E') is in fact a quasi-isomorphism.) For each t E pI - {oo}, let Vt be therestrietion of v at t, Le. v, : F' - Ft-(E'lw,). Also let Voo ; F oo • - G.(E'lw,:.) denote theaugmentation at infinity.

216

Ternary Objects and Deformation Theory
Note that

so to check the axiom above, as in 3.7.b for the eonstruction of the classical Bott-Chern
secondary characteristic forrns, we need to show that for the section of tbe augmentation Vtof v at t E P 1 - {oo}, we have the following.

Theorem. With the same notation as above, thenwhere chBc(cone(v.)) is defined hy using the same process as in section 6.2 for themorphism v..

1.9.2.b The Proof

oe The Theorem

We prove the latest theorem in the following steps.
Step 1. Find A Uniform Coordinate System:
From the previoUB discussion towards tbe relative Bott-ehern object, we saw that asuitable local discusaion is necessary: For doing so, we need use the normal coordinate
system. So to prove the theorem, we should have a uniform way to choose such coordinates.
But this is not an eBBy task. In fact, when t goes to the infinity, the hermitian metricon tbe relative tangent bundle of G t becomes to singular near 8Wco := W~ n W~. So,
the natural choice for the metrics does not work directly. To solve this problem, as we said
before, instead of using the relative tangent bundle of G t , we consider these a.ssociated withthe logarithmic tangent bundle at infinity, Le., we take the dual ofOWly)(pl(logoo). In thisway, by the fact that the closed immersion is of codimension one, we know the resulting new
metric on Wt looks like a small perturbation of a translation-invariant metric on X x C.
Thus we see that now we ean take the limit at infinity with respect to this new metric.
Hence, one may hope that we ean start from the very beginnin"g to establish estimates withrespect to this new metric.
But by doing things like this, we meet other two problems: Tbe first is that, at infinity,
tbe metric is no longer Kähler; while the ~ond ie that BB t - 00, tbe volume of W t withrespeet to tbe new metric become infinite; therefore, we can~ot make the L 2 -estimate by justconsidering tbe usual sup-norm estimate aB in section 3.4. Fortunately, these two problems
only happen near infinity around a~v00' where the assoeiated complex is in fact split. Hence,the problems are not that serioUB.
More precisely, we go aB folIows: With r~pect to the new metric, we find a good finite
system of coordinate charts, uniform for eaeh W t in the sense of estimations, by the factthat locally W is isomorphie to the product of X - awco with the product of two unit
disks {(z, w) : Izi < l,lwl < l}, and the projection is given byzw, and hence if wet=Chapter 1.9.

217

use log(z) sod log(w) iostead, and identify the punetured dise via exp(2n-iT) with 1i/Z,
where 1i is tbe upper half-plane, then the metrie on W t looks like a small perturbation ofa translation-invariant metrie on X x C and similar for W~ - oWoo and W~ - 8Woo ' In
partieular, we ean define Sobolev-norms with them (see Chapter 3): For any positive integers ~ 0, H. denotes the eompletion of Cff' under the square-integration of all derivatives up
to order 8, aod H_. is ita dual. Also we see that the new metrie ia complete. Sinee Cff' js
dense in H_., there ia a uniform Garding inequality for the 8-Laplacian ß, and ß extends
to a self-adjoint operator. Therefore, all proeess are quite regular as t - 00. However, as we
said above, the price we pay for thia is that tbe volume of W t approaches infinity as t - 00,
so that the estimates in su~norm do not imply L 2 -estimates as easily as berore. Also an
integral operator need not to be a trace-class anymore.
Step 11. Conatruct The Relatiee Bott-Chern Seeondary Characteristie Forms ehBC.
As in section 6.2, we know that there is a family cf super-Laplacians A~, defined as
the limit of the ordinary Laplacians coming from blowing~up the metrie on Y together witha resealed factor u for the total metrie (ar better, for tbe fibre metrie). So finally we ean
define the ehBc-cl8B8 via the Mellin transform, i.e. the regularized integral

+00

1
o

du
Tr.[Nexp( -A~)]-.

uThen via the eone eonstruetion, witb this proeess for tbe augmentation Vt with t E pl, we
eould get CbBC(cone(Vt)). With this, what we intend to show becomes
limt_oochsc(eone(Vt»=ehBc(eone(vco ».

Step IH. Relate tbe Objects with respeet to the New Metric And tbe Old Metrie.
We then have tbe associated relative Bott-Chern secondary eharaeteristie forms withrespect to two kinds of metrics mentioned above. Now we diseuss their relations.
Under an infinitesimal change of metrics, the derivative cf tbe regularized integral

l

+co

o

du
Tr 6 [Nexp( -A~)]u

is tbe Schwartz-lirnit

wbere Q denotes the bermitian operator describing the change of metrics. Ir Q has support
in U, this vanishes because of the splitting, exeept for terms related to the augmentationv. Ir we replace v by si v, with a parameter 8 between 0 and 1, the derivative 81; ofthis dass is equal 10 tbe Sehwartz~limit li~~o+ of the derivative (described by Q) of
Tr.(NF exp( -A~)), where N F denotes the nu mb er-operator whieh is identically 1 on F'

218 .

Ternary Objects aad Deformation Theory

and vanisbes 00 E·. Integratiog we obtain that the st-derivative of the effect of the changeof the metrics is given by the difference in Schwartz-limitsli~::o+ Tr,(NFexp( -A~)),taken onee for the origianl Kähler metric and onee for the logarithmie non-Kähler metric.
But for the original metric-this term does not depend on sand is equal to the ehern form ofp, with the associated L:l-metric, while for tbe new metric the term converges nieely as wemove our 'family to the infinity along pl. (See the last step for more details.) The similar
result also holds for the integration against ~' . So, with respect to the change of the metrics,
the terms contributes a correetion-term which eooverges for the family move to the infinity
along pi to the correspondiog term at infinity, and thus does not affect the conclusion of
the theorem. In particular, we only theo need study the situation for the new metric. Thua,by definition; there ia no problem at finite places: Ooly the singularities around 8W00 cause
difficul ties.
Step IV. First Attack Around 8Woo •By definition, at infinity, there are two irreducible (smooth) camponents. i.e. W~,
whicb is a prajective bundle aver Y, and W~, whieh is the blowing-up of Z &long X. On
the other hand, pasaing to the limit above, ooly W~ is eoneeroed. Hence, we need take careof the intersectian of W~ and W~.

We now introduce sorne twisted objects: Define a second family of vector sheavesE··= (&(BxZ)::::: &(BxZ)), p'. = ((0) - (0)).

The advantage for using these twisted objects ia that, on one hand, the original theorem
holds directly for E'· aod p,. since everything is spllt and bence all ehBc-classes vanish,
while, on the other hand, on U, E' and E'· are isomorphie, and the augmentation v has asmall norm. Thus if ..6. defines a heat kernel exp( -tl~) on W~ - 8W00, by the faet that,for the new metric, the volume is not finite, we now have the problem that the kernel is
not of traee-cl888. Nevertheless, if we identify E· and E·· on U n W~I and eonsider thedifference of' tbe beat kernels exp( -u..6.) aod exp( -ua·), we expeet that the singularities
around 8W00 will eancel out. The same thing should also hold for the super-analogue, if
we eonsider the differenee chBct - cb BC ;: We shall see that the difference is integrable. In
particular, we eould iotroduee the eombination ehBe 00 - ChBe~ although the individual
terms are not defined.
I

Claim. With the above notation, if

then the original theorem holds.

Proof. Suppose we have the equality, then we ean repeat the same procedure for theemb edding X c........ W ~, whieh is the deformation of i : X '- Z, wi th res pect ta the original219

Chapter 1.9.

Kähler metric. Nevertheless, for X ' - W~, the deformation to the normal cone gives that

8011 fibers Wt are isomorphic to W~. Hence, the original theorem holds for it, since there
is no essential change. Moreover, our construction for two c10sed immersions X '--+ Z and
X '--+ W;" gives the same ch scoo - ChBC~ at infinity. Hence if we add up all the pieces, wehave the claim and hence the theorem.
So (or the proof of th~ theorem, it is suffieient to verify the following

Lemma. With the same notation as above, we havelimt_oo(chBct - eh Bc ;)= ch Bcoo - ChBC~'

Step V. Prove Of The Lemma.
To prove the lemma, we need same technical results from the very beginning. Since thesituation is quite similar for E'·, we only formulate them for E-.
Note that the cone eonstruction of Vf is associated with F' ffi Ft.(E· @ "n~fl), when we
study the chBC-classes in the sequel, we usually ignore the V-part, since it is on the base
aod bence it does not afl'ect our discUBSions seriously.
Now what we need to study is to get the limit near Woo wben t must disCUBS tbe following two CMe8:00.

So basically, we(1) Away from 8Woc;l'(2) Near 8W00'
We discu88 (1) first. In order to explain tbe idea, we start with a primitive case: Let
D := tJ + ä· + Vf + v; denote the Dirac operator, aod ß := D2 . So note tbe fact that now
we are working away from the singular part, 80 everything goes weil. In particular, we may
hope to have tbe following

Fact. Let Kf(z; u; y) be the heat 'kernel e- u .6 on Wt . Tben, uniformlyon aoy compact
subset of (Woo - 8Woo ) x ]O,oo[ x(Woo - 8WlXI ), the family [(f(X, y) converges to theheat kernel Koo(z, y) on W;" U W~, where K.(x, y) denotes tbe asymptotic expansionof K.(x; u; y) when u -.. 0+.

Proof. First, by that fact that for any sequence of {t n } where t n - 00, we can finda subsequence such that K •• (x; u; y) converges in the Coo~topology, uniformlyon compact
subsets of(Woo -8Woo )x )O,oo[ x(Woo -8Woo ), we only need show that this limit K(x;u; y)is just Koo(x; u; u).
For doing this, by taking limf_oo, as C oo ~functions on W00 - 8W00' we have8

(au + ß)K(x; u; y)=0.

Then, from the heat kernel analogue of the fact tbat for aB ..\,

1-

e->'u

_< ..\U,220

Ternary Objects and Deformation Theory

we have, for tP E Cr(Wt ..),

Therefore, for tP E C~(Woo - 8Woo ),liffiu_o [( (u)<,6 = <,6,which gives the initial condition. Thus, from the uniquness of the heat kernel, by the fact
that, for such a tP,
are finite, so by a cut-off pracess, we see that aB distributions, at infinity,

8(8u + ~)K(r, y)=o.

This completes the proof.
Witb this, we may state tbe results over W oo - awoo as the following
Theorem.. On compact subsets in W co - 8Woo , we have(a) For u - 0+, the asymptotic expansion of Kr(u,r,y) converges ta Koo(u,x,y)
uniformly.
(b) For u - +00, K,(u, x, y) decays uniformly exponentially.
In particular, we have a complete contraion W - 8Woo , uniform on any compactum.

Proof. Obviously, (a) is a direct consequence of the fact above by the local nature of
the expansion. For (b), we need to control the eigenvalues of tbe Laplacians on cane( Vt). In
fact, by a standard procesB, we see tbat (b) is a consequence of the following
Proposition. There exists a positive A > 0, such that, uniformlyon each fiber
f-l(y), y E Y, and for each t E pl - {oe}, tbe Laplacian on tbe cone cone(Vt) has aU
eigenvalues ai least A. Tbe same is true for cone( voo ).
Proof. First, let UB discUSB the situation for Vt. Obviously, the problem may only
happen near 00. SUPPOBe near 00, the assertion is wrong. Then there are a sequencet n - 00 and eigenfunctions /n on Wo := W t " of eigerivalues An SO that An -. 0 aod
li/nil = 1. By Sobolev estimates, aB in section 3.1, we see that Jn are uniformly bounded in
the CCO-topology. Hence, by the Rellich lemma, we may assurne that Jn converges on eachcompactum in W - awoo , and on Y. Tbe limit must be annibilated by ß~. Now we claimthat tbe limit of In is identically zero.
Indeed, near W~, we may suppose again that we have a sequence of eigenfunctions Inof eigenvelues An such that An - 0 aod Hfn 11 = 1. As above, we mayaiso assurne that In
converges to f aod that over U n Wn In converges to zero. In particular, I should vanish
1. Thus, with respect to the original Kähler metric. I
ideitically on U, u~1 0 and IIJII
is a non-trivial harmonie form. Thus, cone(voo ) should have non-trivial cohomologh, which==

Chapter 1.9.

221

contradictB tbe choice of F". Therefore, we may assurne that f n concentrate their massaround W~, where E· is metrically split. Thus, by a cut-off proceas, we mayaiso assume
that In is supported in U n Wn. Now, by the condition that An - 0, (that this limit rnust
be annihilated by ßfj,) we have IIßidnll < 1. On the other hand, since E' is split on U, so
there ßg is the surn of Id and a positive operator. As a direct consequence,

which offers a eontradiction. This completes the proof of the proposition, and heuce theproof of the last theorem.
With the above primitive situation, the super-analogue is not vary hard to obtained:
As in Chapter 6, using the perturbation expansion, the above assertions also hold for thesuper-ease A~ as weIl aB for the operators A~ +0 N U1 with 0 a smaH parameter. Moreover,
tbe result is Coo in o. Some eare is needed since for small u, A~ and Nu may contain
Grasamannian terms whieb seale with negative powers of U aB U - 0+. However I note that
we now also bave tbe right cancellation, we then could derive estimates for the asymptotie
expansions. Tbis completes tbe diseussion for (1).
Next, we diseuss ease (2), i.e., consider what happens in an open neighborhood U ofW~.

On U, we ean first identify the two complexes E" and E··. Then, tbe Laplacian ß on
E' is equal to the sum of a loeal operator ß·, which eoincides with the Laplacian for E··;and an integral operator tl', which comes from the augmentation 11 aod is determined by
global sections of F·. So, it offers a trace-dass norm O(E:) over a E:-neighborhood of W~ inthe original metric. Furthermore, this primitive picture also holds for the super-analogue,
since, for the super-analogue, the difference A~ - A: 2 ia a sum of linear terms in U withcoefficient an integral operator of Grasmannian degree zero aod a constant term of U with an
integral operator of Grassmannian degree ;::: 1: The linear term comes just as before; while
the constant term is made up from covariant derivatives (in the Y-direction) of elements of
F·. Thua note that now we use the logarithmic metrics, 80 the above difference vanishes
on 8Woo ' Hence again the difference operator has trace-cl8BB norm O(e). Next we use thisconclusion to deduce the assertion in the lemma. 'First we give its primitive form. Note that now we need to consider the kernel forthe difference of the associated generalized Laplacians, we begin with an expression of thisdifference.
Let '{J. be a cut-off function with itB support in a 2E:-neighborhood of W~, which is 1 at
the points with the distance at most ! from aw00' Also 8BBume that '(J. has support in U,
that the COO-norm of 11'. with respect to the logarithmic norm ia uniformly bounded, andthat "Pe acta 8B zero on F·. With this, since, on U, we may identify E' and E··, it makes
sense to consider
Lt(u) := '(Je (Kt(u) - K;(u)) cp.
88 an operator on F,.(E·· ® I\n~~/Y)' where, aB uaual, ]( denotes the heat kernel.

Thus,

222

Ternary Objects aod Deformation Theory

we have(~. + :u)L,(u) =~·'Pt(Kt(u) - K;(u))'Pt + :uLt(u)=~·'Pt(Kt(u) -+ !Pt

K;(u))!Prr(:u u) -:u
Kt(

K;( U»'Pr=~"'Pt(Kt(u) -K;(u))'Pt-!pt(LlK,(u) -~" K;(u»lpr=[Ll" I 'Pt]lP2t(K,(u) - J(;(u»cpt- V't~' K,(U)tptl
where 'i't~' has trace-dass norm O(E:).
From this relation, by the initial condition that liIIlu -0+ L, ( u) = 0 I we haveLt(u) =uJ.
0+~=1K;(au)[~·,'Pt]'P2t(K,(bu)- Kt.(bu»'Pr da- u J.

K;(au)'Pt Ll1 Kt(bu)'Pt da .

0+~=1

Wi th this expression for L, ( u) I we may deduce the stureture of L, ( u) as folIows: Tbe second
term is of trace dass norm O(E:ue->'U) uniforrnly in t for u large, for small u , hence is O(e)in the aaymptotic sense, i.e. all terms in the BSymptotic expansion will be of trace-dasa
norm O~e). To determine the first term, we apply the same procedure from the right, Le.
apply (~ + Ä·) from the other aide, then we have tbe foUowing expression for Lt ( u)L,( u) =u 2 J.
o+.+c=1{K t• (au)[6" ,'Pt]'P2t(I<t(bu) - K; (bU»1p?2t [V't, ~ ·]K;( cu) }dadb+ O(eu 2 e->'U).
SimilarlYI when we introduce tbe number operator, for Tr.[N" Lt(u)], we could have thefollows:

Tr,(N" L,(u»=u 2 J.

Tr, (NuK; (au)[Ll" tpr ]tp2r{K t (bu) - K;(bu»tp2t: [tpt:, Ll ·]K; (cu)]dadb
Io+b+e=l+ O(eu 2 e->''')=u 2 J.
o+b+e=1

Tr,[K; (cu)N"K; (au)[Ll" ,tpt: hp~(KI( bu) - K;(bu ))tp2r [lfTr, 6 ·]]dadb+ O(E:u 2 e->"")= - uJ.

8Tr"[a0+6:=1+ O{eu 2 e->"").

Q[e-au(o"+ClN)]Q=o[~",CPt]'P2t(Kt(bu) - K;(bu»'P2f![IfT"ß"]]daChapter 1.9.

223

So the above trace is an integral over Supp(d'Pe )1. Thus, for a fixed!, we are in a eompaet setc,>f W - 8Wco, so everything has good aaymptotie expansions and deeays nicely aB U- +00,
ete. In partieular, it makes sense to use the Mellin transform, i.e. to form the regularized
integralFurthermore, by the above diseussion, it follows that up to a term O([), aB t eonverges to the corresponding integral at 00.

00,

this

By taking a limit for tbe blowing-up of the base metrie on Y, we mayaiso get similar
asaertions for the super·analogue aa we did before: We need to replaee au~" by aA: 1 , ete.
Thus, up to a term O([), aB t ...... 00, the regularized integral eonverges to the eorresponding
reguJarized integral at 00. But these integrals have their eontributions to ehBct - eh Bc ; ,which comes from integrating the appropriate kernelover Supp( 'Pe). So, finally, we have

which completes the proof of Theorem a.

§I.9.3. Euler.. Green Currents
From now on, we study tbe deformation theory of tbe relative Bott-Cbern secondary
characteristic currents with respect to closed immersions. Before doing so we give one
example for such a eurrent, ODe whieh will also be used in the deformation theory.
Let M be a complex manifold aod (E, p) a hermitian veetor bundle on M. Denote thetotal space of E by MB, and let i : M c.......+ MB be the natural embedding. Then, the Koszul
complex (I\E" i~) gives a resolution of i. 0 M on ME aod the normal bundle to M in MB
is exactly E. Let g8 be tbe metric on I\E· induced by p. We are going to calculate therelati ve Bott-Cbern seeondary characteristic eurrent ChBC (E, Pi i, Pi) on M 8 .
The basic idea is as folIows: Let e be the Euler ebaraeteristic form. Then we know thateh = e td -1. Tberefore,

Here we look everytbing on tbe total space ME. Thus it is natural for us to consider how
to measure the difference 6M - e(E,p). Tberefore, we introduce tbe Bott-Chern secondary
cbaracteristic current witb respect to this difference, say eBc(E,p). Tben we try to construet
it. Thus finally, we may use it to deal with the original relative Bott-Chern seeondary
characteristie currents. The main reference here is [BGS 91].

224

Ternary Objeets and Deformation Theory

1.9.3.a. The Bott-Chern Secondary Characteristic Current of A Koszul Complex.

Let M be a eomplex manifold of dimension land (E,p) be a hermitian vector bundle ofrank Je on M. Denote the total spaee of E by ME, p : ME - M the natural projection, aod
i : M ~ ME the natural ~mbedding. Let E· be the dual of E, and let I\E· = $1=0 I\i E·
be the exterior algebra of E·. For aoy y E E, the interior multiplieation operator i y aets
oaturally·on (I\E·)p(y)' Then the K08zul eomplex (I\E·, i,) gives a resolution of i.O M on
ME, i.e. we have tbe exact sequenee of sheaves

Let gE be the metric 00 I\E· indueed by p. Denote by V E tbe eanonieal eonneetion on bothof them. Thus the adjoint i; of i y is fjl\. Sinee the normal bundle N to M in ME is exactlyE, we know that Bismut eondition (A) is automatieally satisfied in this special situation.
Even though ME ia ooneompact, we ean talk of a eurrent 00 ME whieh comes from a.
smooth form with a eompact support on ME. Also we see that the results in the laat ehapter
are valid on ME. In partieular, we have eh BC (E, Pi i, p) tbe Bott-ehern secondary eharacteriatic eurrent on ME assoeiated wi th (1\ E· , i l1 ). So by the fact that ehBc (E 1 Pi i, Pi) E&
depends only on p. For short, we denote it aB ehBc(E,p).

pU

Let e be the ad- invariant polynomial on (Je, k) matriees e : A 1-+ DetA. Then e( E , p)
and td( E, p) are smooth forms on M. We eall e( E, p) the Euler characteristic form of(E, p). Hy the cl888ical theory for characteristic forms, we know thateh = e td- 1 ,so when lifting such forms to ME, we haved<fehBc(E,p) = td- 1 (E,p)(6 M - e(E,p».
Next, we give a precise description for ehBG. Ta da so, the basic idea is to use a locallyintegrable eurrent , whicb comes from td-1(E,p) aod the difference 6M - e(E,p) in the senseof dcJC.

If we let w(E,p) be tbe restrietion of ehBc(E,P) to MB - M, we koow that w(E,gE)
is sffiooth 00 MB - M and by the finite result in section 8.5, we know that w(E, p) entirely
determines ebBc(E, p). We take this as a start point.
The canonieal connection 'V E definea a horizontal subspace T H ME in TM E so that
E. As usual, we let OE = (V E )1 be its eurvature.

TM E = T H ME $Next we introduce Mathai-Quillen's eonvention. Ir Y E TRM E , let yV be the eomponeut in ER with respect to abave splitting. If A is an antisymmetric tensor in End ER, we
identify A with the 2-form on Ta M E:

Y, Z E TalJE 1---+< y V ,AZv > .

225

Chapter 1.9.

We also denote this 2-form by A and its exterior powere by A2, ... A k • If A is invertible,
we mayaiso define the fonns A -I, ... , A - t . If Pf( A) is the Pfaffian of A, the forms
Pf(A)A- 1 , ••• , Pf(A)A- k are rational funetions of Al whieh may be extended by eontinuity
to any arbitrary A, which may not be invertible. In the following we still denote them as
above, even if A is not invertible. Similarly, we have the eomplex analogue: Let JE..;=TIE
be the eomplex structure ?f ER with IE the identity map of E, then for any j ;::: 0,
I

l

=

11 E
11 E
.
det(-. + bIE)( -2. + bJE)-]
211'1
11'1
are defined, aB forms on ME.

Theorem. (1) On ME - M, we havew(E, gEl

8OE= -8b[det(1E- exp(-. + bIE))
211"1

IYI 2

OE

Log(-2- + (211'(211"i + bJE))-I)1b=O.

(2) Let I(E, gE) on ME be defined by I(E, gEl := b;8 chBc(E, gEl. Then I(E, gE)
is a locally integrable current on ME and=Proof. 1. This ia astandard argument. Let NB be tbe number operator, V := iy+i;
i y + Y 1\ . Ir Au := 'V E + y'UV I tben by the proof of tbe seeond relation in Theorem 7.1.e,
we have
Tr,[NHexp(-A~)]8= 8b[det(IE- exp(n + bIE))
E

IYl2 + (E
exp (-u(-211 + b1E )-1»)] 6=0·
Henee 00 ME - M, if 8 E C with Re(s) > 0,

AB a eoosequenee,
1
[-()

r 8

].00 u'Tr,{NHexp( -A~)]-]'(O)
du
0

.

U

2

8
IY1 (nEb )-1)] 6=0,
p (rl E +bIE))Log(-2-+=- 8b[det(IE-ex.u + IE

226

Ternary Objects and Deformation Theory

which gives (1).
(2) Hy Theorem 8.5.b, the finite part theorem, we know that the current ,(E, gE) ia
locally integrable. Let
frxJ[. r::2 du
1]:=
Tr, vuVexp(-Au)]2u'

Ja.
Then definea a locally integrable current on

l\1 E , andOn the other hand, by the fact that iy(l1 E )-l= _(OE)-ly, we know that1]

= -iyTr,[exp(-A~)]

Tr,[JUVexp(-A~)]

.

Ilfj2

= - iydet(l - exp(l1 E ))ex P(-u(-2- + (OE)-l)),j Yl2=- u det(I - exp(OE)) (OE)-ly exp(
-u(-2- + (OE)-I)).

So we get1]

= - ~det(I - exp(OE)) (OE)-1 Y (1~12 + (OE)-l )-1.

But the form (OE)-1 Y ia of formal degree -1. Thua

which completes the proof.

I.9.3.b. Several Intermediate Results
We start witb a description of exterior differentiation aeting on smooth sections ofA(TiM E ) on ME. From the decomposition of TRME indueed by the canonical connection
VB, we know tbat
A(TRAf E ) ~ A(TRA1)c,g,(AEÄJ.
Hence, by antisymrnetrization, we ean define an operator cHgE aeting on the smooth sectionsof A(TRM E ) by letting 4V E ß be the eorrespondiog j + 1 form on ME, aod

where 0', ß are smooth sections on ME of A(Tä1\1) and Ai ER, respectively. On the other
hand, OEy is a 2-form on Al with the values in ER C TaME. So the operator iOs y aetson ATiM E .

227

Chapter 1.9.
Lemma. We have the following relation
d =0 1;JE + ins y .

Proof. We denote the lifting of 1;JM on T H AlE also by 1;JM for any torsion freeconnection on TRM. Thu~ the connection V$ := tyM EB 'V E defines a connection on TRME
with torsion T. Hence if Y E ME U, U' E (TäME)y, thenlTy(U, U')= nE(p.u, P.U')Y.

The connection '\7$ induces an operator 0V'ED on ATiME similarly as above, we haved =0 V"$ + i T .

Now tbe result comes by considering the actions on a,ß as above. In fact, sinee nEy takes
its values in E 1b / R , i n 8ya 0 and 80 OVa da. Similarly, we may get «(IV +inEy)ß dß,
aod benee the lemma.

=

=

=

We ean now introduee several families of differential forms on ME. For any u > 0, letThen we have
Theorem. (1) For any u> 0, the form a u is closed and lies in pM
is integrable.
(2) (Double Transgression Formula) For any u > 0,

8

lS
•

Furtherrnare, a u1-8
a u =- -2.dbui
u
11'1

8-8

bu=~cu.

In particular,{)
1-8 a u = --dcfcu ·uuProof. (1) Hy the definition, we koow that G u ia a form of type (k, k).

=

Hy Bianchi's identity, we know that 0V'En E O. So 4V'$(OE)-1
a form of degree -1 taking values in E. Thus by tbe lemma above,= 0, aod (OE)-l Y is

228

Ternary Objecta aod Deformation Theory

So

AB a corollary, dau= O.

(2) From the proof of (1), we know that

..

Hence

IYl2
="2 det (- 21l"i)d(OE)-IYexP(-u(-2+ (OE)-I)

aloE

8u a u=d( !({lE)-ly a u ).

2
Let 9 be the Kähler form of ER. Ir X, Y E ER, 8(X, Y) =< X, JEY > and tbe element in
E- corresponding to X ia given by

If for any invertible skew-adjoint matrix A in End (E), let 8A . be the (l,I)-forrn on E definedby
we have as an element in ER

On the other hand, if dE

Clearly (8f)2=aE + aE is the exterior differential on E, set=0, (a~)2 = O. Hut the Lie derivative with respect to AY is given by

Hente by the fact thai L Ay 8 A= 0, we have

Note that Blnce 8A. ia hotb EJE- andaB -closed, we have

Chapter 1.9.

229

In particular,

8~ ( IY 1 + A - 1) = 0, [)~ ( IY 1 + A -1 ) = O.
222

Therefore2det( -~)( _A)-l Y exp( -u( IYI

2A+ A -1))
IYI 2

21rt

2

2rt

2

_

=27T(8f - 8~)[det(--.)OA exp(-u(- + A- 1 ))].
However, if A is not invertible, the form

-A
IYI 2
det(-.)OAexp(-u(- + A- 1 )27Tt
2cannot be extended by continuity to a well-defined form. Nevertheless, the form
8
A
IYI 2
A
8b[det(-(27ri + bIE))exP(-u(-2- + (21r(27T + bIE»)-I»)h=o

lJ
A
IYl 2
1
= 8b(det(-(27ri + bIE))h=oex p(-u(-2- + A- ))2- udet ( - -A). 21f9 A exp ( -u ( -IY 1 + A -1))
21ft
2
may be extended by continuity for arbitrary A. Thus by the fact that
1
{)
A
IYI 2
ab [det( -( 21ri + bIE))]b=o exp( -u( ~ + A - ))-E
ia 8A.E , {)A.
-closed, we havedet( -~) (_A)-ly exp( -u( IYl2 + A- 1)

= a

2

2~t

E
2
1 -E
A
IY
j. A
-1
A - 8 A {)2'
8b[det( -( -2. + bIE))exp( -u(-2- + (21rt( -2. + bIE )) ))]6=0.

tJ

11"1

'1"1

11"1

In particular, now both sides cau be extended to· arbitary A. So by the lemma above, wehave the first two relations, while the last one is a direct consequence of these two.
Next, we establisb tbe convergence of the above currents. For this, we have

Theorem. For any n E N, there exists a constant C > 0 such that if JJ is a smoothdifferential form on ME with a compact support in B n := {Y E ME : IYI :5 n}, then,for u ~ I,

230

Ternary Objects aod Deformation Theory
Furthermore, if U, r, VJ, mare taken wi th respect to the embedding i : M <:..:+ ME as in
7.1.d, there exists C' > 0 such that for u 2: 1,

Proof. _We onIy prove the first part, as the proof of the second part ia similar to theproof of the correspondine part of Theorem 8.1.
Let Tu be the map Y 1-+ .jUY. Then

Let (Tu := T;; I. We haveHence, aB U -+co, we have

Now the result follows from the facts that

l

al

= 1,

l

bl = 0,

l

Cl

= O.

Chapter 1.9.

231

1.9.3.c A Special Bott-Chern Secondary Characteristic Current: The EulerGreen Current

Hy the eonvergenee of section 3.b, we ean use the Mellin transform to go further: For
sEC,O< Re(s) <let 'PE.,B(s) be the eurrent on ME, defined byt,

From Theorem 3.b, we know that 'PE,p(S) aod <fJ(E, p) are well-defined. Further , 'PE,p may be
extended to a meromorphie function of s on the whole complex plane, which is holomorphie
at S O. Henee CPe,/O) existe and ie equal to

=

1
1

o

du
(cu - co)- +
u

1+
1

00

du

Cu -

- r'(I)co.

U

Now the Euler-Green eurrent, a special Bott-Chern secondary charaeteristie form, denotedby eBC, is defined by

Sißee ·E is tbe normal bundle to M in ME and Co is a closed form, we have the following
Theorem. (l) The total degree of <fJ( E, p) is 2 dimE - 1. The eurreot eBC (E, p) is ofeomplex type (dimE - 1, dimE - 1).
(2) The wave front sets of the currents 'eBc(E, p) and <fJ(E, p) are eontained in Ei.
Moreover, we have eBc(E,p) E P:f~,(3) The following equations of eurrente hold on ME:

18-8

ljJ(E,p) =2 21ri CPe,p(O);dljJ(E, p) =e(E,p) - fJM.
In partieular,

Moreover, we know that tbe singularities of eBc(E, p) and IjJ(E, p) are given by thefollowing

232

Ternary Objects aod Deformation Theory

Theorem. (1) The eurrents eBc(E, p) and q,(E, p) are loeally integrable.
(2) The following equations holdIn particular, if dimE = 1,

Proof. First of aH, we haveaulYI 22
ab

OE+ bIE))
21f'%

Cu = exp(---)-[det(-(-.
dimE
~

nE
~,

L,; (( -21f'i( ~j=Oj+ bIE))- 1Y·u""7j'"lo=o.
J.

tri

Then, we know that in the surn, the last index ia dirn E - 1, not dirn E. Moreover, we see
that
1
+00
(u!YI 2 ) /c du{ C(1 + Log ), if k = 0;
exp - - - u - <1
2
u IYI-2/C,
if k > O.

rvr

1

But, LoglYI ia locally integrable on ME and for 1 :5 k :5 dirn E - 1, the funetion jYI- 2 /C ia
locally integrable. Henee esc(E, p) ia loeally integrable. Sirnilarly, we ean show that q,(E, p)
ia loeaHy integrable. Thia cornpletes the proof for (1). For (2), we ooly need to do the samething as what we did for the proof of Theorem 3.a.
1

2. It also follows from the above discussion that 2 holds. Furtherrnore, if dirn E = 1,

then

Chapter 1.9.

233

I.9.3.d An Explicit Formula For chBC(E,p)
We now compare the current ChBC(E, p) with eBc(E, p).

Th.eorem. We have tbe following relation

Proof. We can use the Mellin transform to obtain tbe following: For SEC, 0 <Re(s) < t, 6(s) ia defined by

(+oo

1

8(8) = r(s) Jau'([211'i](Tr,[NHexp(-A~)])- td-1(E,gE)c u + (td- 1)'(E,gE)8M )du.
u

Tben, by the asymototie expansion orTr, [N Hexp( -A~)], we know that 8(s) ia a well-definedcurrent on ME, wbich extenda to a current 80 that it is a meromorphic function of sEC.
Furtbermore, the function is holomorphic at s 0 aod=

On tbe otber band, by the first equation in tbe proof of Theorem 3.a, we know that(211'iJTr, [NHexp( -A~)]8=8b [td-1

OE(-( 21ri

IYI 2

OE

+ bIE )) det( -( 21ri + bIE))OE

ex p (-u(-2- + (27ri(21ri + bIE)-l)))]o=o.
Tberefore[2ri]('I'r,[NH exp( -A~)]) - td-1(E, p)eu
]n particular I for 0 < Re( s) <= -(td-1)'(E, p) at, we have

Hut, by Theorem 3.b.!,So, for 0 < Re(,,) < i,

8(8)

1+

= (td-1)'(E,p)dtf[r( 8+1 1) 0

00

du] .
u'c u -

u

u'

234

Ternary Objects and Deformation Theory

Since

1
(
)
fs+l
and 'PE,p(O)

1+

00

0

J

du _ 'PE,p(S),usu cu -= Co ia a dosed form, we haveHence

In particular,

Thus, the theorem comes from the fact that the wave front ofthe current 'P~,p(O) ia contained
in E.
.
The advantage of this result ia that we may chooee a locally integrable representative
current
td- 1 (E, p) esc(E, p)
in the dass of the non-Iocally integrable current ch BC (E, p). For certain purposes, this resul t
is very useful.
Next we conaider esc(E, p) aB a function of p. Hy the fact thatf [Loglzl~]dJJ = 0,}pl

we make the pi-deformation and have the following

Theorem. In pMS/pMPJ,O
M
M

I

I.9.3.e. Compatibility With Sections
We consider now the compatibility of the above process witb a certain kind of sections,which will be used in Part 11. Let 8 be a holomorphic section of E on 1\1 which is transversal
to M in ME. Namely, we assurne that if x E M ia such that s(x) = 0, and d(s) ia thedifferential of s at X, then Im [ds(x)] = E. Let M' := {x E M : s(x)Ol, then on M', ds
identifies E]M' with the normal bundle N to M'. Let i be the embedding M' e......., M. Then

=

235

Chapter 1.9.

=

the chain complex (I\E·, i.)s· (I\E·, i~) provides a resolution of i.O M'. Hence we canconstruct chBC(E,gE;s) on M associated with (I\E·,s). Sy axiom 2, we know that

Also, the current eBc(E,gE) can be pulled back by the section s. Then, s·eBc(E,gE) is acurrent on M which lies iI1 pff, andFurthermore, if E, E' are two holomorphic vector bundles on M, we can consider E andE' as sub-vector bundles of E $ E'. In the same way, tbe manifolds ME aod ME' are seenaB submanifolds of MEfIJE' which intersect transversely, aod ME n ME' = M. The vectory, rr (z) y'.
bundles E and E' Hit naturally to M EfIJ E '. If Z (y, y') E E EB E' , set 0-( z)Then u,u' are holomorphic sections of E and E', which vanisb exactly on ME' and ME,

=

=

=

respectively.
Let p and p' be hermitian metrics on E aod E'. We equip E EB E' with the metric
By above discusaion, we know tbat (1. eBe (E, p) and u'. eBe (E' ,p') are
weJl-defined currents on M EfJ)E'. Ir we imitate the proof of the axiom 4 for the relative
Bott-Chern secondary characterietic current, we have the followingpfJ) := p EIl p'.

eec(E EB E', pfJ)) =e(E' ,p')u·eBc(E, p) + eBc(E', P')6 M8 ,=e(E, p)u'• fBC(E', p') + eBc(E, p)6 M 8.
Accordingly, we let ", s' be the holomorphic sectione on M of E, E', respectively, chosenaB above, and let

M' := {z E M : 8(X)=Ol, M':= {z E M : S'(Z) =O}.

Then the section ,," := (s, s') of E EIl E' is chosen for E EIl E'. Let M" := M' n M'. Then,in P~'UM'/~:~MI from the above theorem,§I.9.4 Deformation of Relative BoU-ehern Secondary Characteristic Currents
In this sedion, we give the deformation theory for the relative Bott-ehern secondary
characteristic currents with respect to closed immersions. We will use tbe same notation
aB above. We know that there may be have two different relative Bott-Chern secondary

236

Ternary Object.s and Deformation Theory

characteristic currents with respect to i o and i eo respectively. As before, a quite natural
question is how we can measure the difference and this leads us to introduce certain ternary
objects. But aB in the case of smooth morphisms, we only prove a weak result in thisdirection. We consider this more carefu1Iy. For short, we let N = Ni, N' = Ny x plI W •<.p = pz : W - Z and let tP = 1fp : P - X be the restriction to P. We fix hermitian metrics
gN. g" on N, Tl and ge on €, which satisfy Bismut assumption (A). Ir we put the standard
Fu bini-Study metric on Oiu ( -00), then by the fact that
JVYxPl/W ::::: PxJVx/z 0 qxOpl( -00),

we have a hermitian metric gN ' on N'. We also choose hermitian metrics gi on (, which
satisfy Bismut assumption (A) with respect to g" aad gN ' . Since BxZ does not meetX x {oo} at W eo , we may assurne that the restriction of gi to Wo = Z x {O} coincides withge, and that the restrietion of
to B x Z C Weo is split acydic. Hence 88 a holomorphic
herrnitian vector bundle, the normal bundle of X in W00 coincides with N with the metricI

gN.

Note that I, defined in 2.a, is also a dosed immersion, and hence, there are associatedrelative Bott-ehern secondary characteristic currents. (As before, we ornit the notation of
the puIl-back from the projection.) That is, we have a current
chsc( Tl, 9'1; I, gl;e, gi).

On the other hand, we can also consider the natural current Log lz1 2 on W, which may be
defined as the pull-back of Log Izl 2 from pI with z the standard coordinate of pt. Near
P(N), we have the equation Z-1 = ~Yi aad we know that Log Izl 2 ia integrable on W. Thus
by Theorem 8.2.4 of [Hö 86L since WF(chBc(Tl,9,,; I,gl;(,gi)) C N'it., and qw : W _ pI
ia a submersion near Y x pI C W, we have that
WF(chec(Tl, 9,,; I, gl ;(, gi)) n WF(Log Iz1 2 ) = 0.
Hence by Theorem 8.2.10 of [Hö 86], tbe product of currents Log Izl 2 (ch ec ( 7], g,,; I. gl; e, gi))
ia well·defined. The usual rules of differential calculua apply to thia product. In particular,aa.

fJ{J.(Loglzi 2 )( ch ec(Tl,9",I,g.
1.,{,9- f )) - Log Iz [2 -2.(chBc(7],g",I,g I.,{,g- i) )
-2
~1

1f1

8
=~((8LoglzI2)chBc(7],g,,;
I,gI;{,gf)) + 2 .(LoglzI 2 (ach ec (7],g"jI,gI;e,gi))) E pW,o.
211'1
1rt

On the other hand, we haveaa

-2. Log 1z 12 = 6wo - 6w... ,11'1

so the restrictiona of the current ch ec ( 7], g,,; I, gl ; (, gi) to Wo and W00' respectively, arewell-defined withI

Tberefore
ChBC('1,g,,; I, gl; l, gi)c5 wo - chec('1,p;g,,; I, gl ;l, gi)c5 w_-LoglzI2(td-l(N',gN')ch('1,9,,»c5xxpl - cb(l,g€» E pW,o.
But, if R is the curvature o..f the canonical connectioD on 0(-1), from the definition,Thus, by integrating aIong the fibers of tp : W - Z and noting tbat

we havechsC(111 g" i i , gi i~, g€) - "p. [chac(11, g"j 1I gl;e, gi)6w_l+ ep.[Log Izl2cb(e, gi)l - [( Loglzl 2( -2 ~)] (td- 1 )'(N, gN) cb(11, g")c5y
}pl

·ä

11"1

-

8
= -2.<p.«8I:,og Iz1 2 ) cbac (11,9,,; 1, gl;{,gen + -2./f'.(Log Izl 2 (8cb BC (1119"i I, gl ;(,gi»).
~I

11"1

So in order to inve8iigate the weak deformation theory ai ~he level P/ pO, we have to evaluate

But tbis is about ibe deformation at infinity and it is natural for us to use the Koszul
complex to study it. So we need to recall a few facts from Section 1:

Let H be the B880Ciated vector bundle of 7-l on P. Then, by the indusion 7-l t...+4>- (lvxl z fB N~/P' ) we have an ind uced metric U. H on H. On K j (tp) = Al H, we take themetric induced from gH. Lei tF be the canonical section of H· and u-(esc(H-, gH-» theI

correeponding Euler-Green current on P.
For each j ~ 0, we have the exact sequence of vector sheaves:

AJ : 0 -1'p.c. - J = j~e - K.(cp) 011'p11- O.

Let L be the 8880Ciaied vector bundle of

We can UBe the orthogonal direct surn of the induced metriC8, 80 that tbe complex L attached
to l beeomes split acyclic, &8 a cornplex of hermitian holomorphic vector bundles. So we
metrize the complex Aj and we get a smooth current cbBc(Aj, PAj) on P.

238

Ternary Objects aod Deformation Theory
Now we have

Claim. In pi/ p;'o ,tp*( chec(TJ, g,,; I, gl;€, gi)6w_)m=[t/J.[td- 1 (H*, gH")u*(eec(H*, gH·))]eh(TJ, g'1) - t/J*(E( -11 chec(Aj, PAj )]]6x.
j=O

Wben the claim is proved, or equivalently, we have the following

Theorem.. (1) As current.s, modulo tbe 8- and ä-exact currents, we have the following
relation eh ec ( TJ, g'1; i, gi; ~,g~)=[4'* [td- 1 (H· ,gH· )u* (eec( H· ,gH·) )]eh( '1, g'1)
m

2

- tp.[LoglzI cb(€, gE)] - 4'.[L(-lY CbBC(A PAj)]]6y.
j ,

j=O

(2) The integral 4>- [td- 1 (H- ,gH*) u- (eec( H- ,gH*»] along the fibers of 4> is a smoothclose<! differential form on X I whose cohomology class does not depend on the choiceof the metrie gN.

Proof. (1) By the earlier diseussioD, the first statement is a eonsequence of the certain
statements for WRve front sets: By Theorem 8.2.13 [Hö] and the fact that tp is a compositioDof an immersion and a submersion, if w La a current on W, then

WF(tp_w) C {p E TR. - {O} : tp.p E {O} UWF(w)}.
Thus it is sufficient to show that the wave front sets of

aod
2

-

1 -

f

Loglzi (8ch ec (TJ,g,,;I,g ;~,g»are in tbe surn of conormal bundles to W 00 and to X x p l in W. This, by definition, ia theconorma.l bundle to Woo on P(N), which is a direct conaequenee ofTheorem 8.2.13 [Hö 86].
Before going furtber, we now we prove the claim. Frorn the very beginning, we use thesupereonneetion formaliam für :-system. We have Äu , etc. Let k : Woo " - ~V. Near P(N),Woo is the union of two smooth manifolda intersecting transversely along P(N). If a ia asmooth form on W, the form k*a is unambiguously defined on Woo - P(N), and defines
an integrable current on Woo ' Furthermore, as a current on W, k·(a)6w_ is exactly the
product of the currents 0' and 6w_.

239

Chapter 1.9.
For 0 < Re(8) < 1, let<r (8) be tbe current on W defined by00

Si nce the forms Tr, [N H exp( - Ä~)] decay exponentially as u - +00 on compact su bsetsof Woo - X, aod in particular oear P(N), the finiteness theorem given in chapter 1.8.5
shows that the above process is well-defined. Furthermore , this function of 8 extends to ameromorphic function, whicb is bolomorphic at 8 O. So we may introduce the following=

Lemma. (a) With the same notation as above,(b) 'r'(O)=0 on Woo - P.

Proof of the Lemma. (a) We show this fact as follows: First, replace on both sides theintegration from 0 to 00 by integration from 0 to a finite T, aod then equality is an obvious
consequence of the previoUB cODsiderations. FinallYI let T - +00. Since the truncatedintegrals approximate cbac(11, g17; I, glg€) 6w... in 'ON'"R (W), and the multiplication by
6w _ ia a continuous map of 'D~,~ (W) into V'(W), we have the above equality.

;i,(b) This follows since on Woo - P, the complex (t, iI) splits as a hermitian complex,

Now we continue the proof of the claim or (1) of the theorem. The support of thecurrent (;0'(0) is contained in P. More precisely, the restriction of (r'(O) to P is exactly
tbe singular current associated with the complex €Ip of hermitian vector bundles. This
provides a resolution of the direct image 8.11 of 11 by the immersion s : Y - P.
Now cODsider <i'(O) as a current on P. By the exact sequence

we wee that OD P, r; ie eplit acyclic even with the metries. Then, by axiom 3, we know thatm

[21riKr'(O) = cbac(K.(<p) ~ 7Tp'1, g') -L(-lY chec(Aj,PAj)
j=O

in

P:/

~,O. Here g' denotes the induced metric.

240

Ternary Objects and Deformation Theory
Hy Theorem 3.d, we know thatlies in Pk'o I which is the claim. Thia completes the proof of (1) too.
(2) We now use the Cltern·Weil theory to prove that t/J, [td -1 (H' , gH- )0" (eBC( H' ,gH-) )]1
ia closed. Let Q be the bundle of unitary frarnes in N. Then Q ie a U(e )-principal bundle,
whieh weOequip with the connection 'V N. With the canonieal metric on ce, there ie a naturalaction of U(e) on P(C e EB 1) as a group of holomorphic transformations and

P = P(N EB 1)= Q XU(e) P(CI! e 1).

On the 'fiber' we ean form P(C e EB 1) the holomorphic hermitian veetor bundle Ho, with amorphism <Po : 'Ho - OP(C'l!Il)' and hence, obtain the holomorphic hermitian Koszul chain
complex AHo = K(<po). The group U(e) acts naturallyon Ho aB a group of holomorpbic
unitary transformations, which preserves the map <Po. So

H= Q XU(e) Ho, K(<p) = Q XU(e) K(c.po).

But the conneetion 'V N induces a connection on the fibration P - X. In particular, thecurvature T of P - X is obtained by lifting the action of (V N)~ on the fibers N to P.
Hence T lifts a 2-form T on X with values in the infinitesimal unitary transformation of
H along the fibers. Let TH be the horizontal part of with respect to V H. Then TH is a
2-form on X with values in the skew-adjoint endomorphism of H, so that T = - Vif. + TH .

t

The connection V N induces a splitting
TRP = t/J'TRY E9 TKp,

aod if R is the restriction of (VH)2 to vectors of T~ P, then ('Q'H)~ = R + TH. Therefore,
we have the follows.
(a) On T;{ P, ('Q'H)~ coincides with R;(b) On the horizontal, ('Q'H)~ coincides with TH;(c) If U E ,p'TRY aod V E T;{P, theo (V'H)1(U, V) = O.
We now make the followi ng changes: In the Chern-Weil formula for td( H, gH), and in
the formula for eBc(H, gH), we replace ('Q'H)2 by &rH • We let Ua be a unitary frame inN, which mayaIso be thought aB a linear isometry from ce into N. The above discussion
implies that for A E U (e ): the Lie algebra of U(e), there exists a smooth form w (A) on
P(C e EI;) 1) with the following properties:=(a) The map A f-+ fp(C.$l) w(A) iB ad-invariant;(b) c.p_[td- 1 (H',gH-)u·(eBc(H',gH-))]= fp(C'$I)W(uä 1('Q'N)2 uo ).

Thus by the Chern- Weil theory, we know that t/J, [td - 1 (H' , gH - ) 0'- (e BC (H\ gH - ))] isclosed. The rest is rather simple.

Re ferences

REFERENCES..

[ABP 73] ATIYAH, M., BOTT, R., PATODI, V.K.: On the heat equation and the index
theorem, Invent. Math. 19 (1973), pp. 279-330(Ar 74] ARAKELOV, S.J.: Intersection tbeory of divison on an arithmetie surface,
Math. USSR Izvel!Jtija 8, 1974, pp. 1167-1180(Ar 75J ARAKELOV, SJ.: Theory of interseetions on an aritbmetie surfate, Proc.
Intm!. Cong. cf Mathematic:ianJ!I, Vancouver, 1975, Vol. I, pp. 405-408(BC 6fBOTT, R., CHERN, S.S.: Hermitian vector bundles and the equidiatribution
ofthe zeroes of their hol~morphic cr~section, Acta Math., 114 (19~ 71-112(BFM 75] BAUM, P., FULTON, W., MACPHERSON, R.: Riemann-Roch for singularvarieties, Publ. Muh. IHES 45 (1975), 101-145[BGS 88] BISMUT, J .M., GILLET, H., souLE, CH.: Analytie torsion and holomorphie
determinant bundles, I, 11, III, Comm. in Math. Physies 115, (1988)[BGS 90] BISMUT, J .M., GILLET, H., SOULE, CH.: Bott-ehern currenta and complex
inune.rsions, Duke Math. Journal, Vol. 60, No.1, 1990
[BGS 91] BISMUT, J.M., GILLET, H., SOULE, CH.: Closed Immersions And Arakelov
Geometry, Grothendieck Festschrift I, (1991)(BGV 92] BERLINE, N., GETZLER, E., VERGNE, M.: Heat kernels &Ud the Dirac
operator, Grund. der Math. 298, Springer-Verlag, (1992)[Bi 86] BISMUT, J .M.: The Atiyah-Singer index theorem for families of Dirac oper~
ton: two heat equation proofs, Invent. Math. 83, 1986
[Bi 90] BISMUT, J.M.: Superconnection currents and complex immenions, Ivent.
math. 99, 59-113 (1990)(Bi 90b] BISMUT, J .M.: Kos:&ul complexes, harmonie oscillatolS, and the Todd dass,
J. AMS., Vol.3, No.1, 1990

(BS 58] BOREL, A., SERRE, J.P.: Le theoreme de Riemann-Roch (d'aprea Grothendieck),.
BuU. Sec. Math. de France 86 (1958), pp. 97-136

References(De 70) DELIGNE, P.: Equations differentielles a pointe singuliers reguliers, Springer
LNM, no. 163, 1970

[Oe 87] DELIGNE, P.: Le determinant de la eohomologie, in Current bande in Arithmetie Algebraie Geometry, Contemporary Math., 67 (1987), 93--178[Fa 84) FALTINGS, G.: Caleulus on arithmetie 8urfaces, Ann. Math. 119 (1984), pp.
387-424.
[Fa 91a) FALTINGS, G.: Diophantine approximation on '"abelian varieties, Ann. Math.,
133 (1991), 549-576[Fa 91b] FALTINGS, G.: Tbe general ease of Lang's eonjeeture, preprint, Prin ce ton,
1991
[Fa 92] FALTINGS, G.: Leetures on the arithmetie Riemann-Roeh theorem, noted by
S. W. Zhang, Ann. of Math. Study 127 (1992)[Fu 84] FULTON, W. Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, Band 2, Sprinser-Verlag (1984)[Ge 86] GETZLER, E.: A short proofofthe Atiyah-Singer theorem, Topology 25 (1986)
pp. 111-117[Gi 84) GILKEY, P.B.: Invariant Theory, the heat equation, and the Atiyah-Singer
index theorem, Math. Lecture Serie8 11, Publish or Perish, Ine., 1984.
[Gil 81] GILLET, H.: Riernann-Roeh theorems for bigher algebraie K-theory, Advaneed
in Math., 40 (1981) 20~289[GS 90) GILLET, B., SOULE, CH.: Arithmetie Intersection Theory, PubI.
lHES 72 (1990)

Math.

[GS 91a] GILLET, H., souLE, CH.: Analytie torsion and tbe arithmetie Todd senus,
Topology 30, (1991)[GS 9Ib] GILLET, H., SOULE, CH.: Characteristie c:lasse8 for algebraie vector bundles
witb hermitian metric, Annals of Math., 131, 163-203, 205-238 (1990)[GS 92] GILLET, H., souLE, CH.: An arithmetie Riemann-Roeh theorem, Invent.
Maib. 1992
[Ha 77) HARTSHORNE, R.: Algebraie geometry, GTM., Springer-Verlag,1977[Hi 64) HIRONAKA, H.: Resolution of singularities of an algebraie variety over a fieldof eharacteristic lero, Ann. Math. 13 (1971) pp. 109-326[Hir 56] HIRZEBRUCH, F.: Neue topologische Methoden in der algebraischen Ge0metrie, Eigebnisae der Math. und ihrer Grenzgebiete, Springer-Verlag, 1956

Referente8[Hö 83] HÖRMANDER, L.: The analysis of linear partial differential operators I,
Grund!. der Math. Wiss., Band 256, Springer, 1983

[Ki 74] KING, J.: Global residues and intersections on a complex manifold, TraDI.
Am. Math. Soc., 1,,92 (1974) pp. 163-199[Me 92] M;ELROSE, R.B.: The Atiyah·Patodi·Singer index theorem, MIT manuscript,
1992.
[MQ 86] MATHAI, V., QUILLEN, D.: Superconnections, Thom cl888e8 and equivariant
differential farms, Topology 25 (1986), 85-110[Qu 73] QUILLEN, D.: Higher algebraic K·theory I. Lecture Notes in Math. 341.
Springer 1973
[Qu 85a] QUILLEN, D.: Superconnections sad. the Chern charactet, Topology 24, 1985,
89-95[Qu 85b] QUILLEN, D.: Determinante cf Cauchy·Riemann operators OVer a Riemannsurface, Funet. Anal. Appl., If>:JS, 31-34[Ra 72] ROBEIITS, J.: Chow's moving lemma, in Aigebraic Germoiry, Oslo 1970, F.
Oort (ed.) Woltera-Noordhoff Publ., Groningen (1972), 89-96[RS 73] RAY, D.B. SINGER, I.M.: AnlJytic torsion for complex manifolds, Ann. Muh.
98, 1973, 154-177[SGA 6] BEIITHELOT, P., GROTHENDIECK, A., ILLUSi~;· r.., et &1.: Theorie dee
intenections et theorthe de Ricmann·Roch, Springer Lec~~n.·~ Notes 225, 1971

[So 90] C. SOULE: Arakelov Geometry, Huvard lecture notes[Sr 91] SRINIVAS: Algebraic K-Theory, PM 90, Birkhäuser, 1991

[VO 91] VOJTA, P. Siegel's theorem in the compact caBe, Ann. Math., 133 (1991)[We 91] L. WENG: Arithmetic Riemann-Roch Theorem for smooth morphism: An approach with relative Bott-Chern sec:ondary characteristie forms, preprint 1991

[WG 89] WANG, Z.X., auo, D.R.: Special Functions, World Scientific, 1989