ON MOTIVIC PRINCIPAL VALUE INTEGRALS

Willem Veys
Abstract. Inspired by p-adic (and real) principal value integrals, we introduce motivicprincipal value integrals associated to multi-valued rational differential forms on smooth
algebraic varieties. We investigate the natural question whether (for complete varieties) thisnotion is a birational invariant. The answer seems to be related to the dichotomy of the
Minimal Model Program.

Introduction
0.1. Real and p-adic principal value integrals were first introduced by Langlands in the
study of orbital integrals [Lan1] [Lan2] [LS1] [LS2]. They are associated to multi-valued
differential forms on real and p-adic manifolds, respectively.
Let for instance X be a complete smooth connected algebraic variety of dimension
n over Qp (the field of p-adic numbers). Denoting by ΩnX the vector space of rational
differential n-forms on X, take ω ∈ (ΩnX )⊗d defined over Qp ; we then write formally ω 1/dand consider it as a multi-valued rational differential form on X.
We suppose that div ω is a normal crossings divisor (over Qp ) on X; say Ei , i ∈ S,P
are its irreducible components. Let div ω 1/d := d1 div ω = i∈S (αi − 1)Ei , where thenthe αi ∈ d1 Z. If ω 1/d has no logarithmic poles, i.e. if all αi 6= 0, the principal valueR
integral P V X(Qp ) |ω 1/d |p of ω 1/d on X(Qp ) is defined as follows. Cover X(Qp ) by (disjoint) small enough open balls B on which there exist local coordinates x1 , . . . , xn suchthat
all Ei are coordinate hyperplanes. Consider for each B the converging integral
R
s 1/d
|x
|p for s ∈ C with R(s) >> 0, take its meromorphic continuation to1 x2 · · · xn |p |ω
B
C and evaluate this in s = 0; then add all these contributions. One can check that theresult is independent of all choices.
In the real setting we proceed similarly but then we also need a partition of unity,
and we have to assume that ω 1/d has no integral poles, i.e. the αi ∈/ Z≤0 . Here theindependency result is somewhat more complicated; it was verified in detail in [Ja1].
0.2. These principal value integrals appear as coefficients of asymptotic expansions ofoscillating integrals and fibre integrals, and as residues of poles of distributions |f |λ or
Igusa zeta functions. See [Ja1, §1] for an overview and [AVG][De2][Ig1][Ig2][Ja2][Lae] formore details.
1991 Mathematics Subject Classification. 14E15 14E30 32S45 (28B99).
Key words and phrases. Principal value integral, multi-valued differential form, motivic zeta function,birational geometry.

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WILLEM VEYS

0.3. Last years (usual) p-adic integration and p-adic Igusa zeta functions were ‘upgraded’
to motivic integration and motivic zeta functions in various important papers of Denefand Loeser (after an idea of Kontsevich [Ko]). We mention the first papers [DL1] [DL2]and surveys [DL3][Lo][Ve4].
In this note we introduce similarly motivic principal value integrals. It is not totally
clear what the most natural approach is; however the following should be satisfied. Re◦
turning` to the◦ p-adic setting of (0.1), we denote EI := (∩i∈I Ei ) \ (∪`6∈I E` ) for I ⊂ S. So
X = I⊂S EI . Then, if suitable conditions about good reduction mod p are satisfied, asimilaras for Denef’s formula for the p-adic Igusa zeta function [De1] yields thatR proof 1/d
P V X(Qp ) |ω |p is given (up to a power of p) by
X](EI◦ )Fp

I⊂S

Y p−1
,
pαi − 1i∈Iwhere ](·)Fp denotes the number of Fp -rational points of the reduction mod p. Since
motivic objects should specialize to the analogous p-adicR objects (for almost all p), any
decent definition of a motivic principal value integral P V X ω 1/d associated to analogous
X and ω 1/d (say over C) should boil down to the formula
X
I⊂S[EI◦ ]Y L−1
Lα i − 1i∈I(up to a power of L). Here [·] denotes the class of a variety in the Grothendieck ring ofalgebraic varieties, and L := [A1 ], see (1.5). Note also that this is precisely the ‘userfriendly formula’ (in the terminologyof [Cr]) for the converging motivic integral associatedP
to the Q-divisor div ω 1/d = i∈S (αi − 1)Ei if all αi > 0. We will use (evaluations of)motivic zeta functions as in [Ve2] or [Ve3] to introduce this desired motivic principal value
integral.
0.4. Remark. (1) Remembering the origin of principal value integrals, we mention that
Hales introduced motivic orbital integrals, specializing to the usual p-adic orbital integrals[Ha1][Ha2].
(2) As in the p-adic case, the study of motivic principal value integrals, especially theirvanishing, is related to determining the poles of motivic zeta functions, and hence of thederived Hodge and topological zeta functions. A nice result about the vanishing of realprincipal value integrals, and a conjecture in the p-adic case, is in [DJ].
0.5. Since a multi-valued differential form ω 1/d is in fact a birational notion, it is a natural
question whether the motivic principal value integral is a birational invariant. In otherwords, if X1 and X2 are different complete smooth models of the birational equivalenceclass associated to ω 1/d such that div ω 1/d is a normal
crossings divisorand ω 1/d has noRR
logarithmic poles on both X1 and X2 , is then P V X1 ω 1/d = P V X2 ω 1/d ?
This appears to be related to the dichotomy of the Minimal Model Program. We show
by explicit counterexamples that the answer is in general negative when the Kodairadimension is −∞. On the other hand, when the Kodaira dimension is nonnegative, weON MOTIVIC PRINCIPAL VALUE INTEGRALS

3

prove birational invariance in dimension 2. In higher dimensions, we explain how the motivic principal value integrals yield a ‘partial’ birational invariant, assuming the Minimal
Model Program. Here some subtle problems appear, which we think are interesting toinvestigate.
0.6. We also introduce motivic principal value integrals on a smooth variety X if div ω 1/d
is not necessarily a normal crossings divisor, facing similar problems. (For real principal
value integrals this was considered by Jacobs [Ja1, §7].)
0.7. We will work over a field k of characteristic zero. When using minimal models
we moreover assume k to be algebraically closed. In §1 we briefly recall the necessary
birational geometry, and the for our purposes relevant motivic zeta function. In §2 we
introduce motivic principal value integrals on smooth varieties. We first proceed on thelevel of Hodge polynomials to show that our approach with evaluations of motivic zetafunctions can really be considered as the analogue of ‘classical’ real or p-adic principal
value integrals. Then in §3 we consider the birational invariance question.
1. Birational geometry
As general references for §§1.1–1.4 we mention [KM] and [Ma].
1.1. An algebraic variety is an integral separated scheme of finite type over Spec k, where
k is a field of characteristic zero. A modification is a proper birational morphism. A log
resolution of an algebraic variety is a modification h : Y → X from a smooth Y such thatthe exceptional locus of h is a (simple) normal crossings divisor.
Let B be a Q-divisor on X. Then a log resolution of B is a modification h : Y → X
from a smooth Y such that the exceptional locus of h is a divisor, and its union withh−1 (supp B) is a (simple) normal crossings divisor.
1.2. Moreover, let X be normal and denote n := dim X. A (Weil) Q-divisor D on X iscalled Q-Cartier if some integer multiple of D is Cartier. And X is called Q-factorial ifevery Weil divisor on X is Q-Cartier.
The variety X has a well-defined linear equivalence class KX of canonical (Weil) divisors. Its representatives are the divisors div η of rational differential n-forms η on X.
Denoting by ΩnX the vector space of those rational differential n-forms, we can consider
more generally elements ω ∈ (ΩnX )⊗d for any d ∈ Z>0 , and their associated divisor div ω.
Then we write formally ω 1/d , considered as a multi-valued rational differential form on X,
and put div ω 1/d := d1 div ω. Since div ω represents dKX , we can say that the Q-divisor
div ω 1/d represents KX .
One says that X is Gorenstein if KX is Cartier, and Q-Gorenstein if KX is Q-Cartier.
1.3. For a Q-Gorenstein X, let h : Y → X be a log resolution of X, and denote by
Ei , i ∈ S, the irreducible components of the exceptional locus of h. One says that X is
terminal and canonical if in the expression
X
KY |X := KY − h∗ KX =ai E ii∈S

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WILLEM VEYS

all ai , i ∈ S, are greater than 0 and at least 0, respectively. (These notions are independentof the chosen resolution.) Such varieties can be considered ‘mildly’ singular; note that asmooth variety is terminal.
1.4. Let k be algebraically closed.
(i) A minimal model in a given birational equivalence class of nonnegative Kodairadimension is a complete variety Xm in this class which is Q-factorial and terminal andsuch that KXm is nef. This last condition means that the intersection number KXm ·C ≥ 0
for all irreducible curves C on Xm .
The existence of these objects is the heart of Mori’s Minimal Model Program. This is
now accomplished in dimension ≤ 3 (and there is a lot of progress in dimension 4). In
dimension 2 it is well known that there is a unique minimal model, which is moreoversmooth, in each birational equivalence class. Also, each smooth complete surface in theclass maps to the unique minimal model through a sequence of blowing-ups. In higher
dimensions, two different minimal models are isomorphic in codimension one. Here eachsmooth complete variety in the class maps to a minimal model through a rational map(which is a composition of divisorial contractions and flips).
(ii) In a given birational equivalence class of general type (i.e. of maximal Kodairadimension), a canonical model is a complete variety Xc in this class which is canonicaland such that KXc is ample. This object is unique and there is a morphism from every
minimal model in the class to it.
1.5. Here, by abuse of terminology, we allow a variety to be reducible.
(i) The Grothendieck ring K0 (V ark ) of algebraic varieties over k is the free abeliangroup generated by the symbols [V ], where [V ] is a variety, subject to the relations[V ] = [V 0 ] if V is isomorphic to V 0 , and [V ] = [V \ W ] + [W ] if W is closed in V . Itsring structure is given by [V ] · [W ] := [V × W ]. (See [Bi] for alternative descriptionsof K0 (V ark ), and see [Po] for the recent proof that it is not a domain.) Usually, one
abbreviates L := [A1 ].
For the sequel we need to extend K0 (Vark ) with fractional powers of L and to localize.
Fix d ∈ Z>0 ; we consider
K0 (Vark )[L−1/d ] :=K0 (Vark )[T ](LT d − 1)(where L−1/d := T̄ ). We then localize this ring with respect to the elements Li/d − 1,i ∈ Z\{0}. What we really need is the subring of this localization generated by K0 (Vark ),L−1 and the elements (L − 1)/(Li/d − 1), i ∈ Z \ {0}; we denote this subring by Rd .
(ii) For a variety V , we denote by hp,q (Hci (V, C)) the rank of the (p, q)-Hodge component in the mixed Hodge structure of the ith cohomology group with compact support of
V . The Hodge polynomial of V is
X¡X¢
H(V ) = H(V ; u, v) :=(−1)i hp,q (Hci (V, C)) up v q ∈ Z[u, v].
p,q

i≥0

Precisely by the defining relations of K0 (Vark ), there is a well-defined ring homomorphism
H : K0 (Vark ) → Z[u, v], determined by [V ] 7→ H(V ). It induces a ring homomorphism
H from R to the ‘rational functions in u, v with fractional powers’.

ON MOTIVIC PRINCIPAL VALUE INTEGRALS

5

1.6. In [DL1] Denef and Loeser associated a motivic zeta function to a regular function
on a smooth variety. In [Ve2] and [Ve3, §2] we considered several generalizations; wemention here a special case of [Ve3, (2.2)].
(i) Let X be a canonical variety and D any Q-divisor on X. Take a log resolution
h : Y → X of D and denote by Ei , i ∈ S, the irreducible components of the union ofh−1 (supp D) and the exceptional locus of h. For each i ∈ S let νi − 1 and Ni denotethe multiplicity of Ei in KY |X and h∗ D, respectively. Note that all νi ≥ 1 since X is
canonical. We also put EI◦ := (∩i∈I Ei ) \ (∪`6∈I E` ) for I ⊂ S. We associated to D on X
the zeta function
X
Y
L−1
.
ZX (D; s) := L−n[EI◦ ]
νL i +sNi − 1
I⊂S

i∈I

−s

Here L is just the traditional notation for a variable T . So ZX (D; s) lives, for example,
in a polynomial ring ‘with fractional powers’ in a variable T over some ring Rd , localizedwith respect to the elements Lν T N −1 and Lν −T N for ν ∈ d1 Z (and ν ≥ 1) and N ∈ Q>0 .
(We verified that the defining expression does not depend on the chosen resolution using
the weak factorization theorem [AKMW][WÃl].)(ii) One can specialize ZX (D; s) to the Hodge polynomial level via the map H, obtainingYX
uv − 1
H(EI◦ )ZX (D; s) = (uv)−n,(uv)νi +sNi − 1
I⊂S

i∈I

where now (uv)−s is a variable.
(iii) For any constructible subset W of X, we can consider more generally zeta functions
ZW ⊂X (D; s) and ZW ⊂X (D; s), using EI◦ ∩h−1 W instead of EI◦ in the defining expressions.
Note. Here we re-normalized the zeta functions of [Ve3, §2] with a factor L−n and (uv)−n ,respectively.
2. Motivic principal value integrals on smooth varieties
2.1. Let Y be a smooth algebraic variety of dimension n. Let ω 1/d be a multi-valued
differential form on Y , such that div ω 1/d is a normal crossings divisor and ω 1/d has nologarithmic poles.
Denote by Ei , i ∈ S, the irreducible componentsPof E = supp(div ω 1/d ), and let αi −1 be1the multiplicity of Ei in div(ω 1/d ). So div ω 1/d = i∈S (αi − 1)Ei , and the αi ∈ d Z \ {0}.
`For I ⊂ S we put EI◦ := (∩i∈I Ei ) \ (∪`6∈I E` ). Note that Y = I⊂S EI◦ .
2.2. We start by giving two equivalent definitions for the principal value integral of ω 1/don Y on the level of Hodge polynomials. The first one is analogous to the classical real andp-adic situation, and the second one will turn out to be a specialization of the definitionon the motivic level.
(1) Consider for s ∈ Z, s >> 0, the ‘motivic integral on Hodge polynomial level’Z
1/d
I(s) :=(uv)− ordt (div ω +sE) dµ,
L(Y )

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WILLEM VEYS

where L(Y ) is the arc space of Y and ordt (·) denotes the order of the given divisor along
an arc in L(Y ); see e.g. [DL2][Ba][Ve2]. This is the analogue of the converging integral
for s >> 0 in the classical case. By [Ba] or [DL2] also I(s) converges for s >> 0 (in fact
if and only if αi + s > 0 for all i ∈ S), and thenI(s) = (uv)−nX

H(EI◦ )I⊂S

Y (uv − 1)(uv)−si∈I(uv)αi − (uv)−s.

Consider now the unique rational function Z(T ) over Q(u1/d , v 1/d ) in the variable T ,yielding I(s) when evaluated in T = (uv)−s for all s >> 0; it is given byZ(T ) = (uv)−nX

H(EI◦ )I⊂S

Y (uv − 1)Ti∈I(uv)αi − T.

Hodge level definition 1. The principal value integral of ω 1/d on Y is limT →1 Z(T ) =evT =1 Z(T ) and is thus given by the formula(uv)−nX

H(EI◦ )

I⊂S

Y
i∈I

uv − 1.
(uv)αi − 1

Note that this proces is indeed analogous to the classical case, where we take the limit
for s → 0 of a meromorphic continuation.
(2) We consider the zeta function ZY (div ω 1/d ; s) of (1.6(ii)). Since here div ω 1/d isalready a normal crossings divisor on the smooth variety Y , we have
ZY (div ω 1/d ; s) = (uv)−nX

H(EI◦ )

I⊂S

Y
i∈I

uv − 1(uv)1+(αi −1)s − 1.

(If you don’t like fractional powers of T = (uv)−s , just consider T 1/d as a variable withinteger powers −d(αi − 1).)
Hodge level definition 2. The principal value integral of ω 1/d on Y islim ZY (div ω 1/d ; s) = evs=1 ZY (div ω 1/d ; s).

s→1This means of course evaluating in T = (uv)−1 , and yields the same formula as in theprevious definition.
Remark. Alternatively, we could have taken the (re-normalized) zeta function of [Ve2],
associated to the effective divisor aE and the sheaf of regular differential forms O(aE) ⊗ω 1/d for some a >> 0, which is given by the formula(uv)−nX
I⊂S

H(EI◦ )

Yi∈I

uv − 1(uv)a+αi +sa − 1,

ON MOTIVIC PRINCIPAL VALUE INTEGRALS

7

and evaluate it in s = −1.
2.3. On the level of the Grothendieck ring, we cannot use the first approach since zero
divisors may occur. The second approach however generalizes and yields the desired
formula.
Definition. The motivic principal value integral of ω 1/d on Y is the evaluation ofZY (div ω 1/d ; s) in s = 1; it is given by the formula
L−n

X
I⊂S

[EI◦ ]Y L−1,
Lα i − 1i∈Iliving in the ring Rd of (1.5). We denote it by P VRYω 1/d .

Remarks. (1) One easily verifies that evaluating ZY (div ω 1/d ; s) in s = 1 (i.e. in T = L−1 )indeed yields a well defined element in Rd .
(2) In the special case that all αi > −1, we can just use the converging motivic integralR
1/d
L− ordt (div ω ) dµ, given by the same formula, but then it is only well defined in aL(Y )completion of K0 (Vark )[L−1/d ], see [DL2] and [Ve2].
R(3) For a constructible subset W of Y we can consider more generally P V W ⊂Y ω 1/das the evaluation of ZW ⊂Y (div ω 1/d ; s) in s = 1, see (1.6(iii)). This corresponds morally
to ‘classical’ p-adic and real principal value integrals involving a locally constant function
and C ∞ function, respectively, with compact support.
2.4. Let now X be a smooth algebraic variety of dimension n, and ω 1/d any multi-valued
differential form on X. Is it possible to associate a well defined ‘natural’ principal value
integral to ω 1/d ? Of course in some sense logarithmic poles will have to be excluded.
A first natural idea is to consider (the pull-back of) ω 1/d via a modification h : Y → X
such that the divisor of h∗ ω 1/d is a normal crossings divisor on Y . If there exists a
modification for which h∗ ω 1/d has no Rlogarithmic poles on Y , then the desired principal
value integral could be defined as P V Y h∗ ω 1/d . Of course the point here is whether thisis independent of the chosen modification.
Looking at the zeta function approach in Definition 2.3, another natural idea is justto define the principal value integral of ω 1/d on X as the evaluation of ZX (div ω 1/d ; s) in
s = 1, if this makes sense. Here no choices are involved.
We verify that the first approach works and yields the same result as the second
approach if there exists a modification h : Y → X satisfying a slightly stronger condition
than above. Then we indicate some subtle problems concerning the ‘naive’ first approach.
2.5. We say that a modification h : Y → X is good if it is a log resolution of D := div ω 1/d
on X, for which div(h∗ ω 1/d ) has no logarithmic poles on Y .
Note that there could exist modifications h : Y → X for which div(h∗ ω 1/d ) is a normal
crossings divisor and h∗ ω 1/d has no logarithmic poles, but such that h∗ D is not a normal
crossings divisor ! Indeed, it is possible that h∗ ω 1/d has multiplicity zero along somecomponent of h∗ D, meaning that this component does not occur in div(h∗ ω 1/d ). For

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WILLEM VEYS

example, let D (locally) be given by 12 D1 − 23 D2 , where D1 and D2 are smooth curves,intersecting each other in a point P with intersection multiplicity 2, see Figure 1. Let
h be the blowing-up of X with centre P . Then h∗ ω 1/d has multiplicity zero along theexceptional curve D3 of h; so div(h∗ ω 1/d ) is (locally) a normal crossings divisor but h∗ D
is not.
D1............ ............ D2... ...
.....
....
..
.
......
.
. ...
.
.
.. .....
.
.
.
....
...
...
....

h
←−

•P

D1........

.
...
...
...
... ....
... ...
......
... ....
.
.
. .....
.
.
...
..
...
...
..
..

D2
D3

Figure 1
2.6. Proposition. Suppose that there exists a good modification h : Y → X. Denoteby Ei , i ∈ S, the irreducible components of h−1 D, and let αi − 1 be the multiplicity of
Ei in div(h∗ ω 1/d ). Put EI◦ := (∩i∈I Ei ) \ (∪`6∈I E` ) for I R⊂ S. Then the evaluation ofZX (div ω 1/d ; s) in s = 1 is well defined, it is equal to P V Y h∗ ω 1/d , and is given by thePQformula L−n I⊂S [EI◦ ] i∈I LL−1
αi −1 .
Note. Here αi = 1 could occur, meaning that Ei does not appear in div(h∗ ω 1/d ).
Proof. Denote the multiplicities of Ei in KY |X and in h∗ D by νi − 1 and Ni , respectively.
Since h is really a log resolution of D, we can express ZX (D; s) as
L−n

X
I⊂S

[EI◦ ]

Yi∈I

L−1
Lνi +sNi − 1.

Now D = div ω 1/d and div(h∗ ω 1/d ) are representatives of KX and KY , respectively.
Hence div(h∗ ω 1/d ) = KY |X + h∗ D, meaning that αi = νi + Ni for all i ∈ S. Sincediv(h∗ ω 1/d ) has no logarithmic poles, all αi 6= 0. So indeed evaluatingR ∗ 1/dZX (D; s) in s = 1
makes sense and yields the stated formula, which is just P V Y h ω . ¤We define the principal value integral of ω 1/d on X as given by Proposition 2.6. For
completeness we recall all data.
2.7. Definition. Let X be a smooth algebraic variety of dimension n and ω 1/d a multivalued differential form on X. Suppose that there exists a log resolution h : Y → X of
div ω 1/d on X, for which h∗ ω 1/d hasR no logarithmic poles on Y . Then the principal value
integral of ω 1/d on X, denoted P V X ω 1/d , is given by one of the equivalent expressions
in Proposition 2.6.
Remarks. (1) Also here we could proceed alternatively using the zeta functions of [Ve2].
(2) We can proceed more generally, involving a constructible subset W of X, just as
in Remark (3) after Definition 2.3.

ON MOTIVIC PRINCIPAL VALUE INTEGRALS

9(3) For real principal value integrals, Jacobs gave a similar definition [Ja1, §7].
2.8. We return to the first approach. Suppose now that there exists a modificationg : Z → X such that div(g ∗ ω 1/d ) is a normal crossings divisor and g ∗ ω 1/d has no logarithmic poles on Z. (Recall that then g is not necessarily good.) Two subtle questions
impose themselves here.
RI) Suppose that there exists at least one good modification of X; so P V X ω 1/d isdefined. Is this principle value integral P
then equal to the obvious formula associated tog ∗ ω 1/d on Z ? I.e., with div(g ∗ ω 1/d ) = i∈SZ (αi − 1)Ei and EI◦ for I ⊂ SZ as usual, isZω 1/d = L−n

PV
X

X

[EI◦ ]

I⊂SZ

Y L−1
?
Lαi − 1i∈I

If there exists a modification π : Y → Z such that h := g ◦ π : Y → X is good, then theanswer is yes. This can be verified using the zeta function ZZ (div(g ∗ ω 1/d ); s), for which
we use the defining expressions on both Z and Y . One easily computes that evaluating in
s = 1 yields the right and left hand sides above, respectively. The point is that ‘deleting
irreducible
components of h∗ (div ω 1/d ) on Y with α = 1’ does not change the formula forR
1/d
PV X ω
in terms of h.
When there does not exist such a modification Y → Z, we do not know the answer.
II) Suppose on the other hand that no good modification of X exists. Are the expressions
Y L−1
X[EI◦ ]L−n
Lα i − 1I⊂SZi∈I

for modifications Z → X as above independent of the chosen Z ?
2.9. The principal value integrals of Definition 2.3 and the more general Definition 2.7
satisfy a ‘Poincaré duality’. Bittner [Bi] showed that there exists a ring involution D of
K0 (Vark )[L−1 ] satisfying D(L) = L−1 , and characterized by D([X]) = L− dim X [X] when
X is a complete connected smooth variety. (It is in fact a lifting of the usual duality
operator on the level of motives to the level of varieties. And specializing D to the levelof Hodge polynomials is just a reformulation of Poincaré and Serre duality.) It extends
to the rings Rd via D(L1/d ) = L−1/d .
Proposition. The principal value integrals of Definitions 2.3 and 2.7 satisfy
Z
D(P V

Z
ω

1/d

)=L

− dim Xω 1/d .

PV

X

Proof. This follows from the concrete formula for P V
as in e.g. [Ba], [DM] or [Ve3]. ¤XRXω 1/d by the same computation

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WILLEM VEYS

3. Birational invariance ?
Here we assume k to be algebraically closed.
3.1. Actually, a (multi-valued) differential form is a birational notion. When we consider such a form ω 1/d on a variety X and its pull-back h∗ ω 1/d on Y via a modificationh : Y → X, this is just a matter of notation : ω 1/d and h∗ ω 1/d are in fact the sameelement in a one-dimensional vector space over the function field of X.
Fix a form ω 1/d for which there exists a smooth complete variety X such that div(ω 1/d )on X is a normalR crossings divisor and ω 1/d has no logarithmic poles on X. Then wecan consider P V X ω 1/d , and it is a natural question whether this notion depends on thechosen such model X. In other words : is the motivic principal value integral a birational
invariant ?
3.2. Remark. A necessary condition is of course that, if π : X 0 → X is the blowing-up
1/dof an X as above in a smooth centre that has normal crossings
with div(ω) and such
R
R
1/d
0
1/d
1/d
that ω
has also no logarithmic poles on X , then P V X ω= P V X 0 ω . This canbe verified by straightforward computations as in [Ve1], [Ve3, Lemma 2.3.2] or [Al].
We should remark that this is however not sufficient to derive birational invariancewith the help of the weak factorization theorem. Indeed, on some ‘intermediate’ varieties
connecting two such models the form ω 1/d could have logarithmic poles.
3.3. Note that in dimension one there is only one smooth complete model in a given
birational equivalence class. So from now on we work in dimension at least two.
First we show that when the Kodaira dimension is −∞, the answer is in general
negative.
3.4. Example. We work in the class of rational surfaces and take ω 1/2 on P2(X:Y :Z) , givenby ω 1/2 = y −3/2 dxdy on the affine chart A2(x,y) . Then on the chart A2(y,z) we have that
ω 1/2 = y −3/2 z −3/2 dydz. Denoting C1 := {Y = 0} and C2 := {Z = 0}, we see thatdiv(ω 1/2 ) = − 23 C1 − 32 C2 is a normal crossings divisor on P2 and that no logarithmic
poles occur.
Consider now the birational map π : P2(X 0 :Y 0 :Z 0 ) − → P2(X:Y :Z) given by A2(x0 ,y0 ) →A2(x,y) : (x0 , y 0 ) 7→ (x0 , y 0 −x02 ). On P2(X 0 :Y 0 :Z 0 ) our form ω 1/2 is given by (y 0 −x02 )−3/2 dx0 dy 0
and (y 0 z 0 − 1)−3/2 dy 0 dz 0 on the analogous charts. Hence on the ‘new’ P2 we have thatdiv(ω 1/2 ) = − 23 C1 , where we denote the birational transform of C1 , i.e. {Y 0 Z 0 −X 02 = 0},
by the same symbol. The formula of Definition 2.3 yields
Z¡
L−1(L − 1)2 ¢PVω 1/2 = L−2 L2 − L + 2L −1/2
+ −1/2=0L− 1 (L− 1)2
P2(X:Y :Z)andZ¡
ω 1/2 = L−2 L2 + (L + 1)

PV
P2(X 0 :Y 0 :Z 0 )L−1 ¢= −L−3/2 (L + L1/2 + 1) 6= 0.
L−1/2 − 1

ON MOTIVIC PRINCIPAL VALUE INTEGRALS

S4

C4

C3

...
..

C

...........
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............
.
.
.
.
.
.
.
.
.
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............

C

C5

C

.
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....

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....

S5......
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.
.
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.
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.
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1
.
.
.
.
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......

C5

...
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..............
....

S3

C3

•

C

C4
•
•

C1

C

•

S1

C3

•
C2

C1

...
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.

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5 ..........
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11

...
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.

2
P(X
0 :Y 0 :Z 0 )

2
P(X:Y
:Z)

•
C2

π
−→
C1

C1

Figure 2
It is useful to indicate the ‘geometric reason’ why this happens. We decompose in
Figure 2 the map π in a composition of blowing-ups and blowing-downs, where the fat
points indicate the centers of blowing-up and C3 , C4 and C5 are exceptional curves. The
surfaces on the middle row are ruled surfaces. One easily verifies that the multiplicitiesof C3 , C4 and C5 in div ω 1/2 are − 12 , −1 and 0, respectively. This means in particularthat C5 does not occur in the support of div ω 1/2 on S4 , S5 and P2(X 0 :Y 0 :Z 0 ) (where C5 isa line),div ω 1/2Rhas logarithmic poles
So we cannot identify
3 and S4 . R
R and that
R on S2 , S1/2
1/2
1/2
P V P2
ω= P V S1 ω
with P V P2
ω= P V S5 ω 1/2 via principal(X:Y :Z)(X 0 :Y 0 :Z 0 )value integrals on the intermediate surfaces S2 , S3 and S4 since they are not defined
there.
Note. (i) We found this example several years ago; it was briefly mentioned by Jacobs[Ja1, §8] in the context of real principal value integrals.
(ii) Actually, it is also valid when k is not algebraically closed. (And if we would haveintroduced principal value integrals in arbitrary characteristic by the same formula, it
would still work.)

12

WILLEM VEYS

3.5. Example 3.4 can be adapted to the birational equivalence class of any non-rational
ruled surface; then we use onlyR the middle and top row
R of Figure 2. It is possible to givea similar form ω 1/2 with P V S1 ω 1/2 = 0 and P V S5 ω 1/2 6= 0 (maybe div(ω 1/2 ) willcontain more fibres in its support). So for surfaces such examples exist in every birational
equivalence class of Kodaira dimension −∞.
Moreover, by taking Cartesian products with arbitrary complete smooth varieties,
Example 3.4 can be extended to arbitrary dimension.
We now turn to the other case, i.e. when the Kodaira dimension is nonnegative.
3.6. Theorem. Fix the birational equivalence class of a surface of nonnegative Kodairadimension, and a multivalued differential form ω 1/d on it. Suppose that this class contains
a smooth complete model X such that div(ω 1/dR) is a normal crossings divisor on X andω 1/d has no logarithmic poles on X. Then P V X ω 1/d is independent of the chosen suchmodel, and is thus a birational invariant.
Proof. (Recall that for smooth surfaces complete is equivalent to projective.) Let Xm
be the unique (smooth, projective) minimal model in the class, and let h : Y → Xm be
the composition of the minimal set of blowing-ups, needed to make div ω 1/d a normal
crossings divisor on Y . More precisely, if div ω 1/d is a normal crossings divisor on Xm ,put Y := Xm . Otherwise, let h1 : Y1 → Xm be obtained from Xm by blowing up the finite
number of points where div ω 1/d has no normal crossings. If div ω 1/d is a normal crossings
divisor on Y1 , put Y := Y1 . Otherwise, continuing this way abuts in the unique smoothprojective Y such that div ω 1/d is a normal crossings divisor on Y and h : Y → Xm is
minimal with respect to this property.
Suppose now that X is any model as in the énoncé of the theorem. Since div ω 1/d isa normal crossings divisor on X, there is a morphism π : X → Y . Also, sinceR ω 1/d has
no logarithmic poles on X, the same is certainly true on Y . Hence also P V Y ω 1/d isRdefined, and is equal to P V X ω 1/d by Remark 3.2. ¤3.7. In higher dimensions we face the non-existence of a unique minimal model, andthe fact that in general a (smooth, complete) variety does not map to a minimal modelby a morphism. A reasonable idea is to try to adapt Definition 2.7 as follows, assuming
the Minimal Model Program. Take a birational equivalence class of nonnegative Kodairadimension and a multi-valued differential form ω 1/d on it.
Suppose that there exists a minimal model X in the given class, and a log resolution
h : Y → X of div ω 1/d on X, for which ω 1/d has no logarithmic
R poles on Y . Define thenthe (candidate) birational invariant associated to ω 1/d as P V Y ω 1/d . Now independenceof both X and Y has to be checked.
Fixing such a minimal model X, theof Y is proven by the same argument
R independence
1/d
as for Proposition 2.6. Indeed, P V Y ω
is just the evaluation of ZX (div ω 1/d ; s) in
s = 1. (Recall that this zeta function was defined more generally on canonical varieties,
so certainly on minimal models.) We now verify that the zeta function ZX (div ω 1/d ; s)itself in fact does not depend on the chosen X; then a fortiori the same is true for its
evaluation in s = 1.

ON MOTIVIC PRINCIPAL VALUE INTEGRALS

13

3.8. Proposition. Let ω 1/d be a multi-valued differential form on a birational equivalence class of nonnegative Kodaira dimension. Let X1 and X2 be minimal models in thisclass; then ZX1 (div ω 1/d ; s) = ZX2 (div ω 1/d ; s).
Proof. Since X1 and X2 are isomorphic in codimension one, div ω 1/d on X1 and X2 areeach others birational transform. Take a common log resolution hi : Y → Xi of div ω 1/d
on X1 and X2 . Remembering that these two (Q-)divisors are representatives of KX1and KX2 , we have h∗1 (div ω 1/d ) = h∗2 (div ω 1/d ) and KY |X1 = KY |X2 , see [KM, Proof of
Theorem 3.52] or [Wa, Corollary 1.10].
So, computing both zeta functions by their defining expressions on Y , see (1.6), yields
exactly the same formula. ¤Note. When our birational equivalence class is of general type, we can use its uniquecanonical model Xc and work with ZXc (div ω 1/d ; s), which shortens the argument.
Summarizing, we obtained the following well defined invariant.
3.9. Definition. Let ω 1/d be a multi-valued differential form on a birational equivalenceclass of nonnegative Kodaira dimension. Assume the Minimal Model Program. Supposethat there exists a minimal model X and a log resolution hR : Y → X of div ω 1/d on X,
for which ω 1/d has no logarithmic poles on Y . Then P V Y ω 1/d is independent of all
choices and is thus a (partial) birational invariant of ω 1/d .
3.10. We mention ‘partial’ because we are confronted with similar subtle problems as in§2. Suppose that there exists a smooth complete model Z in the given class, for which
div ω 1/d is a normal crossings divisor on Z and ω 1/d has no logarithmic poles on Z.
I) If there exists a minimal model X andlog resolution Y → X as in Definition 3.9,
R a1/dis the invariant above then equal to P V Z ω?
RII) If on the other hand no such X and Y exist, are the expressions P V Z ω 1/d thesame for different such models Z ?
3.11. Remark. An alternative point of view for ‘birational invariance’ is ‘independenceof chosen completion’ for principal value integrals on non-complete smooth varieties. For
real principal value integrals Jacobs [Ja1, §8] mentioned Example 3.4 in this context.
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WILLEM VEYS

[Cr]
[De1]
[De2]
[DJ]
[DL1]
[DL2]
[DL3]

[DM]
[Ha1]
[Ha2]
[Hi]
[Ig1]
[Ig2]
[Ja1]
[Ja2]
[KM]
[Ko]
[Lae]
[Lanl][Lan2]
[Lo]
[LS1]
[LS2]
[Ma]
[Po]
[Ve1]
[Ve2]
[Ve3]
[Ve4]

[Wa]
[WÃl]

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K.U.Leuven, Departement Wiskunde, Celestijnenlaan 200B, B–3001 Leuven, Belgium

ON MOTIVIC PRINCIPAL VALUE INTEGRALS
E-mail address: wim.veys@wis.kuleuven.be
http://www.wis.kuleuven.be/algebra/veys.htm

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