Heights and metrics with logarithmic
singularities
Gerard Freixas i Montplet
Abstract
We prove lower bound and finiteness properties for arakelovian heights
with respect to pre-log-log hermitian ample line bundles. These heights
were introduced by Burgos, Kramer and Kühn in [2], in their extension
of the arithmetic intersection theory of Gillet and Soulé [8], aimed to
deal with hermitian vector bundles equipped with metrics admitting suit-
able logarithmic singularities. Our results generalize the corresponding
properties for the heights of cycles in Bost-Gillet-Soulé [1], as well as the
properties established by Faltings [7] for heights of points attached to her-
mitian line bundles whose metrics have logarithmic singularities. We also
discuss various geometric constructions where such pre-log-log hermitian
ample line bundles naturally arise.
MSC: 14G40 (Primary) 11J25, 11J97 (Secondary).
Contents
1 Introduction 2
2 Differential forms with logarithmic singularities 5
3 Logarithmically singular hermitian vector bundles 13
4 Global bounds for real log-log growth (1,1)-forms 17
5 Bounding height integrals 25
6 Arakelovian heights 39
7 Examples 46
8 Appendix 52
Conventions
We fix some conventions and notations to be followed throughout this paper.
The open disc of C centered at 0 and of radius ε > 0 will be denoted by ∆ε.
If f, g : E → R are two functions on a set E 6= ∅, we write f ≺ g to mean
that there exists a constant C such that f(x) ≤ Cg(x) for all x ∈ E. If the
involved constant depends on some data D that we want to specify, we write
f ≺D g.
http://arxiv.org/abs/0704.1046v1
If X is a complex analytic manifold we decompose the exterior differential
operator as d = ∂+ ∂̄ and we define dc = (4πi)−1(∂− ∂̄), so that ddc = i ∂ ∂̄ /2π.
Let k be a field. By an algebraic variety X over k we mean a separated and
reduced scheme of finite type over k. In particular, X is noetherian and it has
a finite number of irreducible components. When k = C, for every separated
scheme of finite type X over C there is an associated complex analytic space
Xan, whose underlying topological space equals the set of complex points X(C).
If F is a closed subscheme of X , then F an is an analytic subspace of Xan. The
scheme X is proper over C if, and only if, Xan is compact. Also, X is a
nonsingular variety over C if, and only if, Xan is a complex analytic manifold.
To simplify notations we will write X instead of Xan or X(C) (it will be clear
from the context the category we are working on).
Let K be a number field. Its ring of integers is denoted by OK . Let S =
SpecOK . An arithmetic variety over S will be a flat and projective scheme
π : X → S , with regular generic fiber XK = X ×S SpecK of pure dimension
n. The set of complex points X (C) has a natural structure of complex analytic
manifold, and it can be partitioned as
X (C) =
σ:K →֒C
Xσ(C).
The complex conjugation induces an antiholomorphic involution F∞ : X (C) →
X (C).
1 Introduction
In this paper we establish a common generalization of the following two state-
ments.
Theorem 1.1 (Faltings [7], Lemma 3). Let X be a projective arithmetic variety
and Y ⊆ X a Zariski closed subset. Let L = (L , ‖ ·‖) be an ample line bundle
on X endowed with a smooth hermitian metric on L |X (C)\Y (C). Suppose that
‖ · ‖ has logarithmic singularities along Y (C). Fix a number field K. For any
real constant C, there are only finitely many points P ∈ X (K) \ Y (K) with
(P ) ≤ C.
The notions of function and metric with logarithmic singularities are recalled
in §2 and §3 below.
Theorem 1.2 (Bost-Gillet-Soulé [1], Proposition 3.2.4 and Theorem 3.2.5).
Let X be a projective arithmetic variety and L a smooth hermitian ample line
bundle on X .
1. For any real constant A, there are only finitely many effective cycles z ∈
Zp(X ) such that deg
z ≤ A and h
(z) ≤ A.
2. There exists a positive constant κ such that h
(z) ≥ −κ deg
z for every
effective cycle z ∈ Zp(X ).
Let us place under the hypothesis of Theorem 1.1. Let P ∈ X (K) \ Y (K),
extended to P : SpecOK → X by properness of X . Since L is ample, there
exists some positive power L ⊗N admitting a global section s non-vanishing at
P . Then
(P ) = log ♯
(L ⊗N )
(sOK)
σ:K →֒C
log ‖s‖σ(Pσ),
where Pσ is the point in Xσ(C) induced by P . Therefore, in Faltings’ result,
only the metric as a function is required. The definition of the height of a
cycle of positive relative dimension, with respect to a smooth hermitian line
bundle, involves the derivatives of the metric up to second order (see [1] §3 and
also §6 below). Therefore, to extend both theorems, we need to describe the
kind of logarithmic singularities allowed to the derivatives of the metric, up to
second order. The arithmetic intersection theory of Gillet and Soulé needs to
be reinterpreted so we can deal with such metrics. This has been done in [2],
where Burgos, Kramer and Kühn develop a theory of abstract arithmetic Chow
groups, and apply it to the case of logarithmic singularities.
Before the statement of our main theorem we introduce some notations. Let
K be a number field, OK its ring of integers and X a projective arithmetic
variety over SpecOK . Suppose that D ⊆ XK is a divisor such that D(C) has
normal crossings. Write U = X (C) \D(C). By Z
U (X ) we denote the group of
codimension p cycles z on X , such that z(C) intersects D(C) properly. A cycle
on X is called horizontal if it is the Zariski closure of a cycle on XK . We write
U (XK) for the subgroup of Z
U (X ) consisting of horizontal cycles. Now let L
be a line bundle on X , endowed with a pre-log-log hermitian metric ‖ · ‖, with
singularities along D (see §3 below for the definition). According to Burgos,
Kramer and Kühn, this is the notion of metric with logarithmic singularities
well suited to define heights on Z
U (X ) (see [2] §7). If LK is ample there is a
well defined normalized height h̃
U (XK) with respect to L = (L , ‖ · ‖).
We refer to §6 for a summary of the theory of heights in Arakelov theory.
Theorem 1.3 (Main Theorem). Let X be a projective arithmetic variety over
SpecOK , D ⊆ XK a reduced divisor such that D(C) ⊆ X (C) has simple
normal crossings and U = X (C) \ D(C). Let L be a line bundle on X with
LK ample. Let ‖ · ‖ be a pre-log-log hermitian metric on L , with singularities
along D, and ‖ · ‖0 a smooth metric on L . Then there exist constants α, β,
γ > 0 and R ∈ Z≥0 such that for every effective cycle z ∈ Z
U (XK) we have
(z) + γ ≥ 1 and
(1.1)
∣∣∣h̃
(z)− h̃
∣∣∣ ≤ α+ β logR
(z) + γ
If moreover ‖ · ‖ is good along D, then we can take R = 1.
In conjunction with Theorem 1.2, Theorem 1.3 immediately yields the de-
sired finiteness property as well as the existence of a universal lower bound.
Corollary 1.4. Let X be a projective arithmetic variety over SpecOK , D ⊆
XK a reduced divisor such that D(C) has simple normal crossings and U =
X (C) \D(C). Let L be a pre-log-log hermitian ample line bundle on X , with
singularities along D.
1. For any real constant A, there are only finitely many effective cycles z ∈
U (X ) such that degLK z ≤ A and hL (z) ≤ A.
2. There exists a positive constant κ such that h
(z) ≥ −κ deg
z for every
effective cycle z ∈ Z
U (X ).
The techniques employed for the proof of the main theorem were initially in-
spired by the work of Carlson and Griffiths on the defect relation for equidimen-
sional holomorphic maps, in higher dimensional Nevanlinna theory [4]. Moreover
(1.1) may be interpreted as a vast generalization of the naive Liouville’s inequal-
ity for the distance between an algebraic number and a rational number. This
point of view is conceptually interesting, since it opens the natural question of
finding an analogue to Roth’s theorem for the distance(1) between a divisor with
simple normal crossings and effective cycles of arbitrary dimension (all defined
over a number field)(2).
This paper is organized as follows.
In Section 2 we review the theory of differential forms with logarithmic sin-
gularities necessary in the rest of the paper. In Section 3 we study in detail
several notions of logarithmically singular hermitian vector bundles. The re-
sults we recall provide a wealth of constructions to which Theorem 1.1 and
Theorem 1.3 apply. Both sections are complemented with examples for a bet-
ter understanding of the theory. In Section 4 we establish global bounds for
real log-log growth (1,1)-forms. An outstanding consequence is a decomposition
theorem (Theorem 4.3) for pre-log-log functions, which plays a crucial role in
the proof of the main theorem. In Section 5 we prove estimates for integrals of
pre-log-log forms appearing in the archimedian part of the definition of height.
This leads to the proof of the main theorem in Section 6, where the reader will
find an overview of the theory of heights in Arakelov geometry. In Section 7
we present some examples of good hermitian line bundles interesting for arith-
metic purposes (for instance in relation with Theorem 1.3). We treat the case
of fully decomposed automorphic vector bundles on locally symmetric varieties,
the relative dualizing sheaf of the universal curve over the moduli space of stable
curves, equipped with the family hyperbolic metric, the Weil-Petersson metric
on the moduli space of curves and the Kähler-Einstein metric on quasi-projective
varieties. Finally, in the Appendix we prove a Bertini’s type theorem needed
for the preliminary reductions in the proof of the main theorem.
Acknowledgements. I am deeply indebted to J.-B. Bost and J. I. Burgos Gil
for proposing me to work on this subject as the starting point of my PhD. thesis,
and for their guidance and constant encouragement. During the preparation of
this paper I benefited from stimulating discussions with several people: I warmly
thank S. Fischler, U. Kühn, V. Maillot, C. Mourougane and D. Roessler for the
time they devoted to me. Their useful advice is reflected throughout the paper.
This work was presented, in a preliminary form, at the Internationales
Graduiertenkolleg Arithmetic and Geometry of the Humboldt-Universität zu
Berlin and the ETH of Zürich, in May 2005. I am grateful to J. Kramer for
inviting me to talk to their summer school.
(1)Suitable candidates for the logarithm of the distance are provided by the height integrals
introduced in §5 below.
(2)The height morphism h
is defined for any pre-log-log hermitian line bundle L , with
singularities along a divisor in X (C). However, for the main theorem to hold, the divisor
needs to be defined over a number field. This essentially goes back to the construction of
transcendental numbers due to Liouville.
2 Differential forms with logarithmic singulari-
Let X be a complex analytic manifold and F a closed analytic subspace. In this
section we introduce several notions of differential forms on X with logarithmic
singularities along F , relevant to our work. We distinguish the case of functions
from the case of differential forms, since the former can be presented in a more
general geometric frame. Indeed, while we can define functions with logarithmic
singularities along an arbitrary closed analytic subspace F , the appropriate
analogues for differential forms of any order require F to be a divisor with
normal crossings.
2.1 Functions with logarithmic singularities
Let X be a complex analytic manifold and F a closed analytic subspace of X .
We denote by IF ⊆ OX the ideal sheaf defining F and supp(F ) for the support
of F . For every x ∈ X , there exist an open neighborhood V and holomorphic
functions s1, . . . , sm ∈ OX(V ) such that
i. the germ of ideal sheaf IF,x is generated by s1, . . . , sm;
ii. the trace of the support of F on V is supp(F ) ∩ V = {z ∈ V | s1(z) =
. . . = sm(z) = 0}.
Since OX is a coherent sheaf, so is IF . Then, given s1, . . . , sm as above, after
possibly shrinking V , s1, . . . , sm generate all the germs IF,z for z ∈ V . In this
case we say that s1, . . . , sm generate IF |V and we write IF |V = (s1, . . . , sm) as
a shortcut. The reader is referred to [6] for further details on analytic spaces.
Definition 2.1. Let X be a complex analytic manifold and F ⊆ X a closed
analytic subspace. A smooth function f : X \ supp(F ) → C has logarithmic sin-
gularities along F if for every open subset V of X such that IF |V = (s1, . . . , sm)
and every relatively compact open subset V ′ ⊂⊂ V , there exists an integer
N ≥ 0 such that
(2.1) |f|V ′\supp(F ) | ≺
∣∣∣∣log( maxi=1,...,m |si|)
For a function f : X \ supp(F ) → C to be with logarithmic singularities
along F it is enough that (2.1) be satisfied for a given open covering of X and
given local generators of IF . This is the content of the next lemma.
Lemma 2.2. Let X be a complex analytic manifold and F a closed analytic
subspace. Fix an open covering {Vα}α of X such that IF |Vα is generated by
sα1 , . . . , s
. Suppose given V ′α ⊂⊂ Vα still forming an open covering of X.
Then a smooth function f : X \ supp(F ) → C has logarithmic singularities
along F if, and only if, for every α there exists an integer N ≥ 0 such that
|f|V ′α\supp(F )
∣∣∣∣log( maxi=1,...,mα
|sαi |)
Proof. Left as an easy exercise.
Lemma 2.3. Let X be a complex analytic manifold and F , G closed analytic
subspaces with supp(F ) = supp(G). A smooth function f : X\supp(F ) → C has
logarithmic singularities along F if, and only if, it has logarithmic singularities
along G.
Proof. Write I, J for the ideal sheaves of F , G, respectively. Let V be an open
neighborhood of x ∈ X such that I|V = (s1, . . . , sl) and J|V = (t1, . . . , tm).
By Hilbert’s Nullstellensatz (see (4.22) in [6]), after possibly shrinking V , there
exists an integer N ≥ 0 such that
sNi =
i tj , t
µijsi.
for some holomorphic functions λ
i , µ
j . Hence, if V
′ ⊆ V is a relatively compact
open subset, there exists a constant C > 0 such that
N ≤ C max
j=1,...,m
|tj |, |tj |
N ≤ C max
i=1,...,l
|si|.
The lemma follows.
The meaning of the lemma is that the notion of function with logarithmic sin-
gularities along F depends only on the support of F .
Proposition 2.4. i. Let X be a complex analytic manifold and F,G closed an-
alytic subspaces with supp(F ) ⊆ supp(G). A smooth function f : X\supp(F ) →
C with logarithmic singularities along F has logarithmic singularities along G.
ii. Let ϕ : X → Y be a morphism of complex analytic manifolds and F ⊆ Y a
closed analytic subspace. If f : Y \ supp(F ) → C has logarithmic singularities
along F , then ϕ∗f = f ◦ ϕ has logarithmic singularities along ϕ−1(F ). If ϕ is
surjective and proper, the converse holds.
Proof. The first item i is straightforward. We shall prove the second part ii.
Let V be an open subset of Y such that IF |V = (s1, . . . , sm). The ideal sheaf
of ϕ−1(F ∩ V ) is (ϕ∗s1, . . . , ϕ
∗sm). Let {Vα}α be an open covering of V by
relatively compact subsets. For every open U ⊂⊂ ϕ−1(V ) define Uα = U ∩
ϕ−1(Vα). Then Uα is relatively compact in ϕ
−1(V ) and ϕ(Uα) ⊆ Vα. The
estimate
|f|Vα\supp(F )| ≺
∣∣∣∣log( maxi=1,...,m |si|)
implies the corresponding inequality for ϕ∗(f) on Uα.
We now prove the converse under the surjectivity and properness assumption.
Let V be as above and V ′ ⊂⊂ V an open subset. Since ϕ is proper, ϕ−1(V ′)
is relatively compact in ϕ−1(V ). The hypothesis of logarithmic singularities of
ϕ∗f asserts the existence of an integer N ≥ 0 such that
|ϕ∗f|ϕ−1(V ′\supp(F ))| ≺
∣∣∣∣log( maxi=1...,m |ϕ
∗si|)
We come up with the conclusion by the surjectivity of ϕ.
2.2 Differential forms with logarithmic singularities
Definition 2.5 (Divisor with normal crossings). Let X be a complex analytic
manifold of dimension n. A reduced analytic subspace D of X is a divisor
with normal crossings if X can be covered by open subsets V with coordinates
z1, . . . , zn such that D ∩ V is defined by z1 · . . . · zm = 0, for some 0 ≤ m ≤ n.
We say that D has simple normal crossings if it can be written as a finite union
of smooth analytic hypersurfaces of X .
Definition 2.6 (Adapted analytic chart [2]). Let X be a complex analytic
manifold of dimension n and D a divisor with normal crossings in X . An
analytic chart (V ; {zi}
i=1) is said to be adapted to D if |zi| < 1/e, i = 1, . . . , n
and D ∩ V is defined by z1 · . . . · zm = 0, for some 0 ≤ m ≤ n. The integer m
will be understood when no confusion can arise.
Notation 2.7. Let X be a complex analytic manifold and D ⊆ X a divisor
with normal crossings. Let (V ; {zi}i) be an analytic chart such that D ∩ V is
defined by z1 · . . . · zm = 0. We define
dζk =
zk log |zk|−1
, if 1 ≤ k ≤ m
dzk, if k > m
and similarly for dζ̄k. Given I, J ordered subsets of {1, . . . , n}, we abbreviate
dζI ∧ dζ̄J =
dζi ∧
dζ̄j .
In the following definitions we write X for a complex analytic manifold of
dimension n, D a divisor with normal crossings in X , U = X \D and ι : U →֒ X
for the natural open immersion. The sheaf of C∞ complex differential forms on
U is denoted by E∗U .
Definition 2.8 (Poincaré growth forms [14]). The sheaf of Poincaré growth
forms on X , with singularities along D, is the subalgebra PD of ι∗E
U gener-
ated, on every open analytic chart (V ; {zi}i) adapted to D, by C
∞(V \D,C) ∩
L∞loc(V,C) and the differential forms dζk, dζ̄k, k = 1, . . . , n. Namely, for every
analytic chart (V ; {zi}i) adapted to D, ι∗E
U (V ) is the C-vector space generated
by differential forms ∑
αI,JdζI ∧ dζ̄J
where αI,J ∈ C
∞(V \D) are locally bounded on V .
Definition 2.9 (Good forms [14]). A good differential form on an open subset
V of X , with singularities along D, is a section ω ∈ Γ(V,PD) such that dω ∈
Γ(V,PD).
Definition 2.10 (log-log growth forms [2]). The sheaf of log-log growth forms
on X , with singularities along D, is the subalgebra LD of ι∗E
U generated, on
every analytic chart (V ; {zi}i) adapted to D, by the functions f ∈ C
∞(V \D,C)
such that on every open V ′ ⊂⊂ V
(2.2) |f(z1, . . . , zn)| ≺
(log log |zk|
for some integer N ≥ 0, together with the differential forms dζk, dζ̄k, k =
1, . . . , n.
Remark 2.11. Observe that the following inequalities hold:
(log log |zk|
−1)N ≤
(log log |zk|
−1)Nm ≤ 2m−1
(log log |zk|
−1)Nm.
Therefore, a smooth function f on V \ D has log-log growth along D if, and
only if, for every open V ′ ⊂⊂ V there is an estimate
(2.3) |f|V ′\D| ≺
(log log |zk|
−1)N .
In concrete computations involving functions with log-log growth, we may use
either formulations (2.2) or (2.3).
Proposition 2.12. Let f : X \D → C be a smooth function. Suppose that df
has log-log growth with singularities along D. Then f has log-log growth with
singularities along D. More precisely, for every analytic chart (V ; {zi}i) adapted
to D and every open V ′ ⊂⊂ V , if df =
j gjdζj +
j hjdζ̄j with
|gj|V ′\D |, |hj|V ′\D | ≺
(log log |zk|
−1)N ,
then we have
|f|V ′\D| ≺
(log log |zi|
−1)N+1
i<j≤m
(log log |zj |
−1)N .
Proof. After localizing to an analytic chart adapted to D we reduce to V =
with V \D = ∆∗r
. Let 0 < ε < 1 and Uε = ∆
be contained in V . Let (w1, . . . , wn) ∈ Uε \D. Fix 1 ≤ i ≤ r. Define the curve
γ : [0, 1] −→ V
t 7−→ (tw1 + (1− t)
e|w1|
, w2, . . . , wn).
Then we have
(f ◦ γ)(1)− (f ◦ γ)(0) =
γ∗(df).
Now we write
γ∗(df|Uε\D) = γ
z1 log |z1|−1
z1 log |z1|−1
where g, h ∈ C∞(Uε \D,C) satisfy
|g|, |h| ≺
(log log |zk|
−1)N on Uε \D.
A straightforward computation yields
df|Uε\D
) ∣∣∣∣
≺f,Uε
1<j≤r
(log log |wj |
−1)N ·
(log log γ∗|z1|
log γ∗|z1|
log γ∗|z1|−1
≤ (log log |w1|
−1)N+1
1<j≤r
(log log |wj |
−1)N .
It follows that
|f(w1, . . . , wn)| ≺f,Uε
∣∣∣∣f
e|w1|
, w2, . . . , wn
)∣∣∣∣
+(log log |w1|
−1)N+1
1<j≤r
(log log |wj |
−1)N .
By induction we find
|f(w1, . . . , wn)| ≺f,Uε sup
|f(z1, . . . zn)|
(log log |wi|
−1)N+1
i<j≤r
(log log |wj |
Observe that the sup is finite because f is smooth on X \ D. The proof is
complete.
Definition 2.13 (pre-log-log forms [2]). The sheaf of pre-log-log forms on X ,
with singularities along D, is the subalgebra E∗X〈〈D〉〉pre of ι∗E
U generated by
log-log growth forms ω in LD such that ∂ ω, ∂̄ ω and ∂ ∂̄ ω are also log-log growth
forms in LD.
A particular case of pre-log-log forms that deserves to be distinguished is
that of P-singular functions.
Definition 2.14 (P-singular function). Let f : X\D → C be a smooth function.
We say that f is a P-singular function, with singularities along D, if, and only
if, df and ddcf have Poincaré growth along D.
Corollary 2.15. Let f : X \D → C be a P-singular function. Then, for every
adapted analytic chart (V ; {zi}i) and every open V
′ ⊂⊂ V , we have
|fV ′\D| ≺
(log log |zk|
Consequently, f is a pre-log-log function.
Proof. This is a straightforward application of Proposition 2.12.
For later computations it will be worth having at our disposal the following
basic properties of log-log growth forms.
Proposition 2.16. i. Any log-log growth form is locally integrable. Moreover,
log-log growth functions and log-log growth 1-forms are locally L2.
ii. (Stokes’ theorem for pre-log-log forms.) If ω ∈ Γ(X, E∗X〈〈D〉〉pre) and [ω]
denotes its associated current, then
d[ω] = [dω]
and similarly for ∂, ∂̄ and ∂ ∂̄.
iii. If f : X → Y is a morphism of complex analytic manifolds and DX ⊆ X,
DY ⊆ Y normal crossing divisors with f
−1(DY ) ⊆ DX , then f
∗PDY ⊆ PDX
and f∗LDY ⊆ LDX . Therefore f
∗E∗Y 〈〈DY 〉〉pre ⊆ E
X〈〈DX〉〉pre. In particular,
this is true for f being the natural inclusion of a complex analytic submanifold
X ⊆ Y intersecting DY transversally and DX = DY ∩X.
Proof. The proposition quotes [2], Proposition 7.5 and Proposition 7.6. How-
ever, for later use, we may comment on the proof of i. After changing to polar
coordinates, it is enough to observe that for every 0 < δ < 1 we have an estimate
∫ ε/e
(log log t−1)N
t(log t)2
∫ ε/e
t(log t−1)1+δ
< +∞.
We finally give an example showing that the notion of pre-log-log function
depends on the compactification X of U .
Example 2.17. Let X = P2
with projective coordinates (w0 : w1 : w2). As
divisor with normal crossings set D = (w0 = 0) ∪ (w1 = 0). Define the smooth
function on U = X \D
g(w0 : w1 : w2) =
|w0|2 + |w1|2
Denote by X̃ the blowing-up of X at (0 : 0 : 1). X̃ admits the following
description:
((w0 : w1 : w2), (z0 : z1)) ∈ X × P
| w0z1 = w1z0
The map realizing the blowing-up is the projection onto the first factor π : X̃ →
X . In particular, since (0 : 0 : 1) ∈ D, we have an isomorphism π−1(U)
→ U ,
and π−1(D) is a divisor with normal crossings. Observe that the pullback of g
by π is
f((w0 : w1 : w2), (z0 : z1)) =
|z0|2 + |z1|2
The function f extends to a smooth function on the whole X̃, in particular pre-
log-log along π−1(D). However, we claim that g is not a pre-log-log function
alongD. To see this we compute ∂ g. We may localize at the affine neighborhood
w2 6= 0 of (0 : 0 : 1) and write u = w0/w2, v = w1/w2. In coordinates u, v we
∂ g =
|u|2|v|2
(|u|2 + |v|2)2
But the function |u|2|v|2 log |u|−1/(|u|2 + |v|2)2 does not have log-log growth
along D, as we see after restriction to |u| = |v|. This proves the claim.
2.3 Variants: log-log forms
Following [3] we introduce a variant of the sheaf of pre-log-log forms, by imposing
bounds on all the derivatives of the component functions of the differential forms.
There are also corresponding variants for Poincaré growth forms and good forms,
for which we refer to loc. cit.
We fix a complex analytic manifold X and D ⊆ X a divisor with normal
crossings. Write U = X \D and ι : U →֒ X for the natural open immersion.
Definition 2.18 (log-log functions of infinite order [3]). A smooth function
f : X \D → C is said to be a log-log function of infinite order, with singularities
along D, if for every analytic chart (V ; {zi}i) adapted toD, every open V
′ ⊂⊂ V
and multi-indices α = (α1, . . . , αn), β = (β1, . . . , βn), there is a bound on V
∣∣∣∣∣
f(z1, . . . , zn)
∣∣∣∣∣ ≺
k=1(log log |zk|
where zα
= zα11 . . . z
m (similarly for z
β≤m) and N depends on V ′, α, β.
Definition 2.19 (log-log growth forms of infinite order [3]). The sheaf of log-log
growth forms of infinite order onX , with singularities along D, is the subalgebra
of ι∗E
U generated, on every analytic chart V adapted to D, by log-log growth
functions of infinite order and the differential forms dζk, dζ̄k, k = 1, . . . , n (see
Notation 2.7).
Remark 2.20. Let ω be a log-log growth form of infinite order alongD, defined
on some analytic open subset of X . Then the complex conjugate ω̄ is also a
log-log growth form of infinite order along D.
Definition 2.21 (log-log forms [3]). The sheaf of log-log forms on X , with
singularities along D, is the subalgebra E∗X〈〈D〉〉 of ι∗E
U generated by log-log
growth forms ω of infinite order, such that ∂ ω, ∂̄ ω and ∂ ∂̄ ω are also log-log
growth forms of infinite order.
Remark 2.22. There is an obvious inclusion E∗X〈〈D〉〉 ⊆ E
X〈〈D〉〉pre.
Log-log growth forms of infinite order enjoy of analogous properties to the
log-log growth forms introduced before. We refer to [3] for details. An advan-
tage of the sheaf of log-log forms over the sheaf of pre-log-log forms is that a
Poincaré’s type lemma holds for the former. The next essential property follows.
Theorem 2.23 ([3]). The natural inclusion
Ω∗X −→ E
X〈〈D〉〉
is a filtered quasi-isomorphism with respect to the Hodge filtration.
Proposition 2.24. Let f : X \ D → C be a smooth function. Then f is a
log-log form, with singularities along D, if, and only if, df is locally L2 on X
and ddcf is a log-log growth form of infinite order along D.
Proof. The direct implication is an easy exercise. Let us see the converse. By
hypothesis, ∂̄ ∂ f is a log-log form, with singularities along D. Let x ∈ X . By
Theorem 2.23, there exists an open neighborhood V of x and a log-log form ω
on V such that
∂̄ ∂ f = ∂̄ ω.
Therefore, we can write
∂ f = ω + θ.
for some holomorphic form θ on V \D. Observe that θ is locally L2, because
∂ f is locally L2 by hypothesis and ω is a log-log 1-form (see Proposition 2.16).
By Lemma 2.25 below, θ must be holomorphic on V . This proves that ∂ f has
log-log growth of infinite order along D. The same reasoning applied to ∂̄ ∂ f̄
(the complex conjugate of ∂ ∂̄ f) proves that ∂̄ f has log-log growth of infinite
order along D. Therefore df has log-log growth of infinite order.
Again by Theorem 2.23, after possibly shrinking V , there exists a log-log func-
tion of infinite order g and a holomorphic function h on V \D such that f = g+h.
We claim that h is locally L2. Since this is true for g, we are reduced to prove
it for f . But we have already shown that df has log-log growth of infinite order
along D, so that Proposition 2.12 implies that f has log-log growth along D. In
particular, f is locally L2 (Proposition 2.16). By Lemma 2.25, g is holomorphic
on V and hence f has log-log growth of infinite order along D. This finishes the
proof.
Lemma 2.25. Let X be a complex analytic manifold and D ⊆ X a divisor with
normal crossings. Let θ be a holomorphic function on X \D. If θ is locally L2
on X, then θ extends to a holomorphic function on X.
Proof. The lemma is well-known, but we include the proof for lack of reference.
It is enough to treat the case whenX = ∆nε ⊆ C
n andD is defined by z1·. . .·zr =
0, so that X \D = ∆∗rε ×∆
ε. We write δ = (δ1, . . . , δr) ∈ R
>0. Since θ is locally
(2.4) ‖θ‖2ε := lim
Iδ < +∞
where
∆ε/2\∆δk )×∆
∣∣∣∣∣
dzk ∧ dz̄k
∣∣∣∣∣ .
The Laurent series development
(2.5) θ(z1, . . . , zn) =
is absolutely and uniformly convergent on any (
k=1 ∆ε/2\∆δk)×∆
. There-
fore, the integral Iδ can be computed term by term:
ν,µ∈Zn
∆ε/2\∆δk
k |dzk ∧ dz̄k|
zνkk z
k |dzk ∧ dz̄k|
Recall that given integers a, b we have
eaiθebiθdθ = 2πδa,b,
so that
|aν |
∆ε/2\∆δk
2νk |dzk ∧ dz̄k|
2νk |dzk ∧ dz̄k|
(2.6)
We reason by contradiction and assume that θ does not extend to a holomorphic
function on ∆n
. We can suppose that in (2.5) there appears a term aνz
ν 6= 0,
with ν = (ν1, . . . , νn) ∈ Z
<0 × Z
≥0 , 1 ≤ l ≤ r. From (2.6) and by direct
computation we find
(2.7) Iδ ≥ (4π)
n|aν |
k=l+1
(ε/2)2νk+2
2νk + 2
δ2νk+2k
2νk + 2
(ε/2)2νk+2
2νk + 2
where
Jδk =
if νk = −1,
(ε/2)2νk+2
2νk+2
2νk+2
2νk+2
if νk < −1.
Since aν 6= 0 and Jδk → +∞ as δ → 0, we see from (2.7) that Iδ → +∞ as
δ → 0. This contradicts (2.4). The proof is complete.
3 Logarithmically singular hermitian vector bun-
Let X be a complex analytic manifold and D ⊆ X a divisor with normal cross-
ings. Write U = X \D and ι : U →֒ X for the natural open immersion. In this
section we study vector bundles endowed with hermitian metrics with singular-
ities of logarithmic type along D. The reader is referred to §2 for the several
definitions and properties of differential forms with singularities of logarithmic
type along D.
Definition 3.1 ([3] and [14]). Let E be a vector bundle of rank r on X . A
smooth hermitian metric h on E|U is said to have logarithmic singularities along
D if, for every trivializing open subset V and holomorphic frame e1, . . . , er of
E|V , putting hij = h(ei, ej) and H = (hij) on V \D, the following condition is
fulfilled:
(L(E, h)) the functions |hij |, detH
−1 have logarithmic singularities along
D ∩ V (see Definition 2.1).
We say that h is (pre-)log-log (resp. good) along D if moreover, for every such
data V and e1, . . . , er, the following property holds:
(G(E, h)) the entries of the matrix (∂ H)H−1 are (pre-)log-log (resp.
good) forms on V , with singularities along D (see Definition 2.9).
We will usually write E = (E, h) when no confusion on the metric can arise.
Sometimes we use some variants of the definition, and we say for instance “E
has logarithmic singularities along D” or “E is (pre-)log-log (resp. good) along
In the case of line bundles, the notions of (pre-)log-log and good hermitian
metrics can be characterized by slightly simpler properties.
Proposition 3.2. Let L be a hermitian line bundle on X and h a smooth
hermitian metric on L|U . Write ‖ · ‖ for the norm associated to h.
i. The metric h is (pre-)log-log (resp. good) with singularities along D if, and
only if, for every trivializing open subset V and holomorphic frame e of L|V , the
function log h(e, e) is (pre-)log-log (resp. P-singular) on V , with singularities
along D.
ii. The metric h is log-log with singularities along D if, and only if, for every
trivializing open subset V and holomorphic frame e of L|V , the form ∂ log h(e, e)
is locally L2 on V and ∂̄ ∂ log h(e, e) has log-log growth of infinite order, with
singularities along D.
Proof. This follows from the definitions and Proposition 2.12, Proposition 2.15
and Proposition 2.24 applied to the smooth real function log h(e, e) on V \D.
An essential extension property of hermitian vector bundles with logarithmic
singularities is the following observation due to Mumford.
Proposition 3.3. Let (E◦, h) be a smooth hermitian vector bundle on U . Then
there exists at most one extension of (E◦, h) to a hermitian vector bundle (E, h)
on X, with logarithmic singularities along D. More precisely, if (E, h) is such
an extension, then for every open subset V in X
Γ(V,E) =
s ∈ Γ(V, ι∗E
◦) | h(s, s) has log. sing. along D ∩ V
Proof. This is Proposition 1.3 in [14].
Hermitian vector bundles with logarithmic singularities along D admit the
following characterization.
Proposition 3.4. Let E be a vector bundle on X and hE a smooth hermitian
metric on E|U . Denote by hE∨ the dual metric. Then hE has logarithmic
singularities along D if, and only if, the following condition is satisfied with
F = E and F = E
(L̃(F, hF )) for every open subset V and any holomorphic section s of F|V ,
the function hF (s, s) has logarithmic singularities along D ∩ V .
Proof. For the direct implication, first take a holomorphic section s of E over
an open subset V . We may assume that s does not vanish on V . After pos-
sibly shrinking V , we can complete s to a holomorphic frame e1 = s, . . . , er
of E|V . By the definition of metric with logarithmic singularities, the function
hE(s, s) = hE(e1, e1) has logarithmic singularities along D.
Secondly, let V be a trivializing open subset of E and e1, . . . , er a holomor-
phic frame of E|V . Write H = (hij) for the matrix of hE in base {ei}i and
H−1 = (gij) for the inverse matrix. From the very construction of H
−1 and
the logarithmic singularities of the functions hij and detH
−1, it is immedi-
ate to check that the functions gij have logarithmic singularities along D. If
B is the matrix of hE∨ in any holomorphic frame of E
, then there exists
A ∈ GLr(Γ(V,OX)) such that
B = At ·H−1 · A.
Since the entries of A are holomorphic, the entries of B inherit from H−1 the
logarithmic singularities along D ∩ V .
Let now s be a holomorphic section of E∨ over an open subset V . Replacing
V by a smaller open subset, we can complete s to a holomorphic frame of E∨
v1 = s, . . . , vr. As we have just proven the functions hE∨(vi, vj) have logarithmic
singularities along D, in particular so does hE∨(s, s) = h(v1, v1).
Now for the converse. Let V be a trivializing open subset, adapted to D. Let
e1, . . . , er be a frame for E|V . Write H = (hij) for the matrix of the hermitian
metric hE in base {ei}i. By hypothesis, for every open subset V
′ ⊂⊂ V , there
exists an integer N ≥ 0 such that on V ′
hE(ei, ei) ≺ (log |z1 · . . . · zm|
−1)N .
Applying Schwarz’s inequality we get
|hij |
2 ≺ (log |z1 · . . . · zm|
−1)2N .
The same argument provides similar bounds for the entries of the matrix of hE∨
in the dual basis, namely H−1. Since the determinant of H−1 is a polynomial
in the entries of this matrix, we derive a bound
detH−1 ≺ (log |z1 · . . . · zm|
for some integer M . This concludes the proof.
As an immediate consequence of the proposition we establish the following
corollary.
Corollary 3.5. Let E = (E, hE) be a hermitian vector bundle with logarithmic
singularities along D. For every exact sequence of vector bundles
0 −→ F −→ E −→ Q −→ 0,
the induced hermitian vector bundles F = (F, hF ) (restricted metric) and Q =
(Q, hQ) (quotient metric) have logarithmic singularities along D.
Proof. It is enough to prove that for every exact sequence as in the statement,
conditions (L̃(F )) and (L̃(Q)) hold. Indeed, since E
has logarithmic singular-
ities along D, conditions (L̃(F
)) and (L̃(Q
)) automatically follow by duality.
Then we conclude applying Proposition 3.4. The validity of L̃(F ) is clear. For
L̃(Q), we just observe that if s is a holomorphic section of Q|V and s̃ is a
holomorphic section of E|V lifting s, then
hQ(s, s) ≤ hE(s̃, s̃).
Thus we see that L̃(E) implies L̃(Q).
We next state the main formal properties of logarithmically singular (resp.
(pre-)log-log, resp. good) hermitian vector bundles.
Proposition 3.6. Let E, F be two vector bundles on X and hE and hF smooth
hermitian metrics on E|U and F|U , respectively. If hE and hF have logarithmic
(resp. (pre-)log-log, resp. good) singularities along D, then E
, E ⊗ F , SkE
and ∧kE have logarithmic (resp. (pre-)log-log, resp. good) singularities along
Proof. Left as an elementary exercise.
Proposition 3.7. Let X, Y be complex analytic manifolds and DX ⊆ X,
DY ⊆ Y normal crossing divisors. Let f : X → Y be a morphism of com-
plex analytic manifolds. Let E = (E, h) be a hermitian vector bundle on Y
whose metric is defined and smooth on Y \DY .
i. If f−1(DY ) ⊆ DX and h has logarithmic (resp. (pre-)log-log, resp. good)
singularities along DY , then the metric f
∗(h) on f∗(E) has logarithmic (resp.
(pre-)log-log, resp. good) singularities along DX .
ii. Suppose that f is surjective, proper and f−1(DY ) = DX . Then h has
logarithmic singularities along DY if, and only if, f
∗(E) has logarithmic singu-
larities along DX .
Proof. The first item i follows from Proposition 2.4 i and Proposition 2.16 iii.
The second item ii is automatically deduced from Proposition 2.4 ii.
Corollary 3.8. Let (E, h) be a hermitian vector bundle on X, with singularities
along D. Let OE(1) be the dual of the trivial vector bundle of P(E), the projective
space of lines in E∨. Denote by π : P(E) → X the natural projection. Then the
metric on OE(1) induced by π
∗(h) has logarithmic singularities along π−1(D).
Proof. By definition, the line bundle OE(1) is a quotient of π
∗(E). The hermi-
tian metric on OE(1) is the quotient metric from π
∗(E). By Proposition 3.7,
π∗(E) has logarithmic singularities along π−1(D). Then, by Corollary 3.5, so
does the induced metric on OE(1).
The end of this section is devoted to some counter-examples.
Example 3.9. i. Counter-example to Corollary 3.5 and Corollary 3.8 for pre-
log-log hermitian vector bundles. Let X = A1
be the complex line, with analytic
coordinate z. Let D be the divisor with normal crossings z = 0. As vector
bundle we take E = O⊕2X . We consider a hermitian metric h on E such that, in
the standard basis e1, e2 and near the origin, its matrix H looks like
log |z|−1 0
It is easily seen that (E, h) is pre-log-log along D (actually good). However, the
induced metric on the line bundle OE(−1) ⊆ π
∗(E∨) on P(E) = P1
is not
pre-log-log. Indeed, identifiy A1
as an open subset of P1
via t 7→ (1 : t). If e∨1 ,
e∨2 is the dual basis, define the section s of OE(−1)|A1
s = e∨1 + te
Then h∨(s, s) = (log |z|−1)−1 + |t|2 and
log h∨(s, s) =
(log |z|−1)−1 + |t|2
If we restrict ∂ log h∨(s, s)/ ∂ t to the set C : t = (log |z|−1)−1/2, we find
log h∨(s, s)
(log |z|−1)1/2
which does not have log-log growth near z = 0.
ii. The notion of hermitian vector bundle with logarithmic singularities depends
on the compactification. Let Y be a smooth complex projective surface. Let p
be a closed point in Y and π : X → Y the blowing-up of Y at p. Let D be a
divisor with normal crossings in Y with p ∈ D. Then π−1(D) is a divisor with
normal crossings. Define U = X \ π−1(D) and V = Y \D. Then π induces an
isomorphism between U and V . Let h be a smooth hermitian metric on ωY |V ,
and endow ωX |U with the induced metric π
∗(h). Assume that (ωX , π
∗(h)) has
logarithmic singularities along π−1(D). Then we claim that h does not define
a metric on ωY with logarithmic singularities along D. Indeed, suppose that
ωY = (ωY , h) had logarithmic singularities along D. Then, by Proposition 3.7,
π∗(ωY ) would have logarithmic singularities along π
−1(D). Observe that
π∗(ωY )|U = ωX |U .
By Proposition 3.3 we would derive the equality π∗(ωY ) = ωX . However we
know that
π∗(ωY ) = ωX ⊗O(−E)
where E is the exceptional divisor π−1(p). Since the self-intersection (E2) = −1,
O(−E) is not trivial. We thus arrive to a contradiction and the claim is proved.
We remark that we can produce such examples just endowing ωX with a smooth
hermitian metric and then restricting it to U .
4 Global bounds for real log-log growth (1,1)-
forms
4.1 Statement of the theorem and consequences
Let X be a complex analytic manifold and D ⊆ X a divisor with simple normal
crossings. DecomposeD into smooth irreducible components,D = D1∪. . .∪Dm.
For every Dk we fix a global section sk of O(Dk) with divisor div sk = Dk. We
endow O(Dk) with a smooth hermitian metric ‖ · ‖k such that ‖sk‖
k ≤ e
Therefore 1 ≤ log log ‖sk‖
−2 ≤ +∞ on X .
Notation 4.1. For every integer N ≥ 0 we define the real positive smooth
function on X \D
(log log ‖sk‖
The purpose of this section is the proof of the following global bounds for
real log-log growth (1,1)-forms on a compact complex analytic manifold.
Theorem 4.2. Suppose that X is compact and let ω be a smooth positive (1,1)-
form on X. Let η be a real log-log growth (1,1)-form on X, with singularities
along D. Then there exist constants A,B > 0 and an integer N ≥ 0 such that
on X \D
(4.1) η +BΘN (dd
c(−Θ1) +Aω) ≥ 0.
If moreover η has Poincaré growth along D, then N can be chosen to be 0.
The proof of the theorem is postponed until §4.3. Now we may discuss a
result appearing as a particular instance of Theorem 4.2.
Theorem 4.3. Suppose that X is compact and let ω be a smooth positive (1,1)-
form on X. Let f : X \D → R be a pre-log-log function, with singularities along
D. Then there exist positive pre-log-log functions, with singularities along D,
ϕ, ψ : X \D −→ R≥0
and constants A,B ≥ 0, N ∈ Z≥0 with the properties
i. f is the difference of ϕ and ψ: f = ϕ− ψ;
ii. the following inequalities hold on X \D:
ωϕ :=dd
c(−ϕ) +BΘN (dd
c(−Θ1) +Aω) ≥ 0,
ωψ :=dd
c(−ψ) +BΘN (dd
c(−Θ1) +Aω) ≥ 0.
If f is P-singular, then N can be chosen to be 0;
iii. if f is P-singular, one can take ϕ, ψ to be P-singular with
ddc(−ϕ) +Aω ≥ 0,
ddc(−ψ) +Aω ≥ 0
on X \D.
Proof. Since X is compact, from the log-log growth of f it is easily seen that
for some constant C > 0 and integer M ≥ 0,
f + CΘM ≥ 0 on X \D.
If f is P-singular, then by Corollary 2.15 we can take (as we do) M ≤ 1. We
define ϕ̃ = f + CΘM and ψ̃ = CΘM . These are positive pre-log-log functions,
with singularities along D (see Lemma 4.7 below). If f is P-singular (hence
M ≤ 1), then ϕ̃, ψ̃ are P-singular (again by Lemma 4.7). By Theorem 4.2 there
exist constants A,B ≥ 0 and N ∈ Z≥0 such that
eϕ := dd
c(−ϕ̃) +BΘN (dd
c(−Θ1) +Aω) ≥ 0
:= ddc(−ψ̃) +BΘN (dd
c(−Θ1) +Aω) ≥ 0
hold on X \D. Hence ϕ = ϕ̃ and ψ = ψ̃ satisfy the requirements of i and ii. If
f is P-singular, then ddc(−ϕ̃), ddc(−ψ̃) have Poincaré growth along D and we
may take N = 0. In this case we have
eϕ =dd
c(−ϕ̃) +mBddc(−Θ1) +mABω
=ddc(−(ϕ̃+mBΘ1)) +mABω
and similarly
= ddc(−(ψ̃ +mBΘ1)) +mABω
In view of these equalities, ϕ = ϕ̃ + mBΘ1 and ψ = ψ̃ + mBΘ1 fulfill the
requirement of iii.
We include the next corollary for its own interest, but we will not need it in
the sequel.
Corollary 4.4. Suppose that X is compact and Kähler. Let ω be a Kähler
form on X. Let f : X \ D → R be a P-singular function and f = ϕ − ψ a
decomposition as in Theorem 4.3 iii. Then the functions
−ϕ,−ψ : X \D −→ R≤0
uniquely extend to quasiplurisubharmonic functions(3) on X.
Proof. First of all, since −ϕ and −ψ are pre-log-log alongD, by Proposition 2.16
we have the equality of currents ddc[−ϕ] = [ddc(−ϕ)] and ddc[−ψ] = [ddc(−ψ)]
on X . The inequalities
ddc[−ϕ] +Aω ≥ 0,
ddc[−ψ] +Aω ≥ 0
(4.2)
then hold on X in the sense of currents. Let U ⊂ X be an open subset diffeo-
morphic to a complex euclidian ball. Because ω is d-closed (Kähler assumption),
by Poincaré’s lemma ω|U is d-exact. Since U itself is Kähler, ω|U is in fact dd
exact. Write ω|U = dd
ch for some smooth function h on U . Then the currents
ddc[−ϕ|U +Ah] and dd
c[−ψ|U +Ah] are positive on U , by (4.2). Since −ϕ and
−ψ are bounded above and D is polar, ϕ̃ := −ϕ|U + Ah, ψ̃ := −ψ|U + Ah
uniquely extend to plurisubharmonic functions on U (see [6], Theorem 5.24).
Since h is smooth, these extensions determine extensions of −ϕ|U and −ψ|U
to quasiplurisubharmonic functions on U , clearly unique. The corollary fol-
lows.
4.2 Construction of pre-log-log functions
4.2.1 Preliminaries
Lemma 4.5. Let M be a complex analytic manifold and α, β, C∞ differential
forms of type (1,0) on M . For every function K > 0 on M , the following
inequality holds:
2Re(iα ∧ β) ≥ −
α ∧ α− iKβ ∧ β.
(3)A quasiplurisubharmonic function on a complex analytic manifold M is an upper semi-
continuous function h : M → [−∞,+∞[ which is locally the sum of a smooth function and a
plurisubharmonic function.
Proof. On one hand, the (1,1)-form
µ = i(α/K1/2 +K1/2β) ∧ (α/K1/2 +K1/2β)
is semi-positive. On the other hand, there is an equality
α ∧ α+ 2Re(iα ∧ β) + iKβ ∧ β.
The lemma follows.
Lemma 4.6. Let L be a line bundle on X. Suppose that s ∈ Γ(X,L) is a global
section such that div s is a divisor with normal crossings. Let ‖ · ‖ be a smooth
hermitian metric on L with ‖s‖2 ≤ e−e. For every integer N ≥ 0, the smooth
function
θN = (log log ‖s‖
−2)N : X \ div s −→ R≥1
is pre-log-log, with singularities along div s. On X \div s the following identities
hold:
∂ θN = NθN−1
∂ log ‖s‖−2
log ‖s‖−2
ddc(−θN ) =
θ1 −N + 1
∂ θN ∧ ∂̄ θN −NθN−1
c1(L)
log ‖s‖−2
provided N ≥ 1. If N = 1, θ1 is P-singular along div s.
Proof. The lemma is trivial for N = 0. We suppose N ≥ 1. First of all we
remark that θN has log-log growth along div s. Next we compute ∂ θN and
ddc(−θN ) on X \ div s. We find
(4.3) ∂ θN = NθN−1
∂ log ‖s‖−2
log ‖s‖−2
ddc(−θN ) = −
∂ ∂̄ θN = −
N ∂ θN−1 ∧
∂̄ log ‖s‖−2
log ‖s‖−2
NθN−1
∂ log ‖s‖−2 ∧ ∂̄ log ‖s‖−2
(log ‖s‖−2)2
NθN−1
∂ ∂̄ log ‖s‖−2
log ‖s‖−2
To simplify the last equality, rearrange terms, use (4.3) and the trivial fact
θN+M = θNθM , and recall that on X \ div s we have c1(L) = dd
c log ‖s‖−2. We
(4.4) ddc(−θN ) =
θ1 −N + 1
∂ θN ∧ ∂̄ θN −NθN−1
c1(L)
log ‖s‖−2
Observe that the quotient (θ1−N+1)/NθN is bounded, so it has log-log growth
along div s. Also the function 1/ log ‖s‖−2 is bounded and c1(L) is smooth, so
that c1(L)/ log ‖s‖
−2 has log-log growth along div s. Hence from (4.3) and (4.4)
we see that it is enough to prove that ∂ θN has log-log growth along div s or,
still, that ∂ log ‖s‖−2/ log ‖s‖−2 has log-log growth along div s.
Let V be an open analytic chart adapted to div s such that L|V can be trivialized
and s = z1 . . . zme, where e is a holomorphic frame of L|V . We can write
∂ log ‖s‖−2
log ‖s‖−2
∂ log ‖e‖−2
log ‖s‖−2
log |zk|
log ‖s‖−2
zk log |zk|−1
The differential form ∂ log ‖e‖−2 is smooth on V and (log ‖s‖−2)−1 is bounded
on V , so that the first term is bounded on any small enough open V ′ ⊂⊂ V . As
for the sum, we observe that log |zk|/ log ‖s‖
−2 is bounded on any small enough
open V ′ ⊂⊂ V , because log ‖s‖−2 = log ‖e‖−2 +
j=1 log |zj|
−2 and log ‖e‖−2
is smooth. Hence ∂ log ‖s‖−2/ log ‖s‖−2 has log-log growth along div s. The
proof is complete.
Lemma 4.7. Under the hypothesis of §4.1 and with the notations therein, the
functions ΘN are pre-log-log, with singularities along D. If N = 1, then Θ1 is
P-singular.
Proof. Write θ
N = (log log ‖sk‖
N , ΘN =
k=1 θ
N and apply Lemma 4.6.
4.2.2 Local results
Let L be a line bundle on X admitting a global section s ∈ Γ(X,L) whose
associated divisor div s is smooth and irreducible. Let ‖·‖ be a smooth hermitian
metric on L with ‖s‖2 ≤ e−e on X . Define, as before, the function
θ1 = log log ‖s‖
−2 : X \ div s −→ R≥1.
By Lemma 4.6 we can write
(4.5) ddc(−θ1) =
∂ log ‖s‖−2 ∧ ∂̄ log ‖s‖−2
(log ‖s‖−2)2
c1(L)
log ‖s‖−2
Let (V ; z1, . . . , zn) be an analytic chart adapted to div s with V ∩div s 6= ∅. We
suppose that L|V can be trivialized and we denote u for a holomorphic frame
such that s = z1u. In local coordinates equality (4.5) becomes dd
c(−θ1) =
α+ β + γ, where
∂ log |z1|
−2 ∧ ∂̄ log |z1|
(log ‖s‖−2)2
dz1 ∧ dz̄1
|z1|2(log |z1|−2)2
β :=− 2Re
∂ log |z1|
−2 ∧ ∂̄ log ‖u‖−2
(log ‖s‖−2)2
=2aRe
dz1 ∧ ∂̄ log ‖u‖
z1(log |z1|−2)2
∂ log ‖u‖2 ∧ ∂̄ log ‖u‖2
(log ‖s‖−2)2
c1(L)
log ‖s‖−2
and a = (log |z1|/ log ‖s‖)
2. Decompose ∂̄ log ‖u‖2 =
j=1 qjdz̄j , where the
functions qj are smooth on V . Then β can be expanded as a sum β =
j=1 βj ,
βj := 2aRe
dz1 ∧ dz̄j
z1(log |z1|−2)2
Proposition 4.8. Let L be a line bundle on X admitting a global section s ∈
Γ(X,L) such that div s is smooth and irreducible. Let ‖·‖ be a smooth hermitian
metric on L with ‖s‖2 ≤ e−e and define θ1 = log log ‖s‖
−2. Let ω be a positive
(1,1)-form on X. For every analytic chart (V ; z1, . . . , zn) adapted to div s and
any open V ′ ⊂⊂ V intersecting div s, there exists A > 0 such that on V ′ \ div s
ddc(−θ1) +Aω ≥
dz1 ∧ dz̄1
|z1|2(log |z1|−2)2
Proof. Without loss of generality, suppose supV | log ‖u‖| and supV |qj | finite for
all j (otherwise, replace V by a relatively compact open subset containing V ′).
We divide the proof into three steps.
Step 1. Observe that because log ‖s‖ = log |z1|+log ‖u‖ and log ‖u‖ is bounded
on V , the function a uniformly tends to 1 as z1 tends to 0. Therefore there
exists an open V ′′ ⊂ V ′ such that 1/2 ≤ a ≤ 2 on V ′′ \ div s. For later need,
we take V ′′ so that V ′ \ V ′′ does not intersect div s. On V ′′ \div s the following
inequality holds:
(4.6) α ≥
dz1 ∧ dz̄1
|z1|2(log |z1|−2)2
Step 2. Define C = maxj supV |qj |. Shrinking V
′′ if necessary, we assume that
|z1| ≤ 1/16n(C + 1) on V
′′. Since a ≤ 2 on V ′′, we have
β1 = 2aRe
dz1 ∧ dz̄1
|z1|2(log |z1|−2)2
≥ −4C|z1|
dz1 ∧ dz̄1
|z1|2(log |z1|−2)2
dz1 ∧ dz̄1
|z1|2(log |z1|−2)2
(4.7)
on V ′′ \ div s. Fix a constant K ≥ 8nC. Applying Lemma 4.5, for every j > 1
we find
(4.8) βj ≥ −2
dz1 ∧ dz̄1
|z1|2(log |z1|−2)2
− 2CK
dzj ∧ dz̄j
(log |z1|−2)2
also on V ′′ \ div s. From inequalities (4.7) and (4.8) we derive that
(4.9) β ≥ −
dz1 ∧ dz̄1
|z1|2(log |z1|−2)2
− 2CK
dzj ∧ dz̄j
(log |z1|−2)2
holds on V ′′ \ div s.
Step 3. To conclude, add up (4.6) and (4.9) to find
(4.10) ddc(−θ1) ≥
dz1 ∧ dz̄1
|z1|2(log |z1|−2)2
− 2CK
dzj ∧ dz̄j
(log |z1|−2)2
on V ′′ \ div s. The last two terms in (4.10) define a smooth differential form on
V \div s, bounded on V ′′ \div s. Hence, there exists a constant A > 0 such that
(4.11) ddc(−θ1) +Aω ≥
dz1 ∧ dz̄1
|z1|2(log |z1|−2)2
holds on V ′′\div s. Because θ1 is smooth away from div s and V ′ \ V ′′ is compact
and disjoint from div s, after possibly increasing A inequality (4.11) holds on
V ′ \ div s as well, as was to be shown.
4.2.3 Global results
We keep the hypothesis and notations of §4.1.
Proposition 4.9. Suppose that X is compact and let ω be a smooth positive
(1,1)-form on X. Let L be a line bundle on X admitting a global section s ∈
Γ(X,L) such that div s is a divisor with normal crossings. Let ‖ · ‖ be a smooth
hermitian metric on L such that ‖s‖2 ≤ e−e. Define θN = (log log ‖s‖
for any N ∈ Z≥0. Then there exists a constant A = A(N) > 0 such that
ddc(−θN ) +Aω ≥ 0 on X \ div s.
Proof. The case N = 0 is trivial. We treat the case N ≥ 1. By Lemma 4.6 the
following identity holds:
ddc(−θN ) =
θ1 −N + 1
∂ θN ∧ ∂̄ θN −NθN−1
c1(L)
log ‖s‖−2
First of all, the function θN−1/ log ‖s‖
−2 is bounded and the differential form
c1(L) is smooth on X . Since X is compact, there exists a constant A > 0 such
(4.12) −NθN−1
c1(L)
log ‖s‖−2
ω ≥ 0 on X \ div s
Still by the compactness hypothesis and by the very definition of θ1, there
exists an open neighborhood V of div s such that θ1|V ≥ N − 1. Moreover
i ∂ θN ∧ ∂̄ θN ≥ 0, so that
(4.13)
θ1 −N + 1
∂ θN ∧ ∂̄ θN ≥ 0 on V \ div s.
Finally, since θM ≥ 1 is smoooth on X \div s for every integerM ≥ 0 and X \V
is compact, after possibly increasing A we have
(4.14)
θ1 −N + 1
∂ θN ∧ ∂̄ θN +
ω ≥ 0 on X \ V.
Equations (4.12), (4.13) and (4.14) together give the desired positivity property.
Corollary 4.10. Suppose that X is compact and let ω be a smooth positive
(1,1)-form on X. For every integer N ≥ 0 there exists A = A(N) > 0 such that
ddc(−ΘN ) +Aω ≥ 0 holds on X \D.
Proof. This follows from Proposition 4.9 by writing ΘN =
k=1 θ
N , with the
notation θ
N = (log log ‖sk‖
The last proposition of this subsection provides a first approach to Theorem
4.2. We may thus place under the hypothesis and notations therein.
Proposition 4.11. Suppose that X is compact and let ω be a smooth positive
(1,1)-form on X. For every finite covering {(Vα; z
1 , . . . , z
n )}α of X by analytic
charts adapted to D, together with relatively compact open subsets V ′α ⊂⊂ Vα
still forming a covering, there exists a constant A > 0 such that
ddc(−Θ1) +Aω ≥
dzαk ∧ dz̄
|zαk |
2(log |zαk |
holds on V ′α \D for every α.
Proof. This follows immediately from Proposition 4.8.
4.3 Proof of Theorem 4.2
We now complete the proof of Theorem 4.2.
Since X is compact, we can choose a finite open covering Vα of X as in
Proposition 4.11. It is enough to prove the existence of constants A,B,N ful-
filling (4.1) on a single V ′α ⊂⊂ Vα. We write {zi}i for the coordinates on Vα,
instead of {zαi }i. Following Notation 2.7, we develop
dζj ∧ dζ̄j +
2Re(hjk
dζj ∧ dζ̄k),
where the functions hjk, j ≤ k, have log-log growth along D ∩ Vα. There exist
a constant C > 0 and an integer N ≥ 0 such that
|hjk|V ′α\D| ≤ CΘN .
Therefore, by Lemma 4.5, on V ′α \D there is a lower bound
η ≥ −CΘN
dζj ∧ dζ̄j − CΘN
dζj ∧ dζ̄j +
dζk ∧ dζ̄k
From this inequality and Proposition 4.11 we see that there exist B ≥ 1 and a
smooth differential form σ on Vα such that
η +BΘN (dd
c(−Θ1) +Aω) ≥ ΘNσ
holds on V ′α \D. After possibly increasing A, we also have σ +Aω ≥ 0 on V
Since B ≥ 1, we finally find
η +BΘN(dd
c(−Θ1) + 2Aω) ≥ 0
on V ′α \D, as was to be shown. Observe that if η has Poincaré growth, then we
can choose N = 0.
5 Bounding height integrals
5.1 Geometric assumptions and statement of the theorem
Let X be a complex analytic manifold and D ⊂ X a divisor with normal cross-
ings. Let L be a line bundle on X and ‖ · ‖ a pre-log-log hermitian metric on L,
with singularities along D (see Definition 3.1). If ‖ · ‖0 is any smooth hermitian
metric on L, then we can write
‖ · ‖ = e−f/2‖ · ‖0,
where f : X \D → R is a pre-log-log function. If ‖ · ‖ is good along D, then f
is P-singular along D. As usual, abbreviate L = (L, ‖ · ‖) and L0 = (L, ‖ · ‖0).
Suppose now that Y ⊂ X is a compact complex analytic submanifold of pure
dimension d. We assume the following conditions are fulfilled:
i. the submanifold Y meets D in a divisor with normal crossings E in Y ;
ii. the restriction ω := c1(L0)|Y is semi-positive and
degL Y :=
ωd > 0;
iii. there exists a global section s ∈ Γ(Y, L) such that ‖s‖20 ≤ e
−e on Y and
div s is a divisor with normal crossings containing E. In particular div s is
reduced, so that we may indistinctly treat div s as a reduced Weil divisor
or a reduced scheme. For every integer N ≥ 0, we define the pre-log-log
function, with singularities along div s,
ℓN = (log log ‖s‖
N : Y \ div s −→ R≥1;
iv. there exist pre-log-log functions, with singularities along E,
Θ1,ΘN , ϕ, ψ : Y \ E −→ R≥0 (N ∈ Z≥0 depending on f)
with f = ϕ− ψ, and bounds
ϕ ≤ CℓM , ψ ≤ CℓM ,Θ1 ≤ Cℓ1,ΘN ≤ CℓN ,
for some constant C ≥ 0 and integer M ≥ 0. Moreover, if f is P-singular,
we suppose that M = 1 and N = 0;
v. there exists A > 0 such that τ := ddc(−Θ1) + Aω ≥ 0, and for every
integer Q ≥ 0 there exists AQ > 0 such that τQ := dd
c(−ℓQ) + AQω ≥ 0.
For Q = 0, we can choose A0 = 1, so that τ0 = ω;
vi. there exists B > 0 such that
ddc(−ϕ) +BΘNτ ≥ 0,
ddc(−ψ) +BΘNτ ≥ 0
hold on Y \E (and so on Y \div s). Observe that by the bounds in iv, we
then have
ωϕ := dd
c(−ϕ) +BCℓNτ ≥ 0,
ωψ := dd
c(−ψ) +BCℓNτ ≥ 0
on Y \ div s.
The aim of this section is to find bounds for the height integrals
Jp :=
f c1(L)
p c1(L0)
d−p, 0 ≤ p ≤ d.
We observe that f c1(L)
p c1(L0)
is a pre-log-log differential form on Y with
singularities along E, hence locally integrable (see Proposition 2.16). In partic-
ular, since E ⊆ div s and div s is Lebesegue negligible, the integrals Jp can be
computed on Y \ div s.
Theorem 5.1. There exist constants α, β > 0, R ∈ Z≥0, depending only on A,
{AN}N , B, C, M and N , such that, for any p ∈ {0, . . . , d},
|Jp| ≤ α degL Y + β · (degL Y ) · log
log ‖s‖−20
c1(L0)
degL Y
If ‖ · ‖ is good along E, then we can take R = 1, so that
|Jp| ≤ α degL Y + β · (degL Y ) · log
log ‖s‖−20
c1(L0)
degL Y
The theorem will be reduced to the bounds claimed by the following two
propositions.
Proposition 5.2. Let σ be a closed, real and semi-positive pre-log-log (t,t)-form
on Y , with singularities along div s. Let a, b be integers such that a+ b+ t = d.
If a, b ≥ 0, define the integral
I(M,a, b, σ) =
Otherwise set I(M,a, b, σ) = 0. Then, if a > 0, the following bound holds
I(M,a, b, σ) ≤ BCI(M +N, a− 1, b, στ)
+ CI(M,a− 1, b, στM )
+ bBC2I(2M,a− 1, b− 1, σττN )
+ (a− 1)BC2I(2M,a− 2, b, σττN ).
If f is P-singular (in which case N = 0 and M = 1), then
I(1, a, b, σ) ≤ ABCI(1, a− 1, b, στ0)
+ (B + 1)CI(1, a− 1, b, στ1).
Similar bounds are true if b > 0, exchanging the role of a and b.
Proposition 5.3. Let σ be a (d, d)-form which is a product of (1,1)-forms of
the kind τ or τQ, Q ≥ 0. Let W be the set of integers Q ≥ 0 such that τQ
appears in σ. Fix an integer K ≥ 0. Then there exist constants α, β > 0 and
an integer R ≥ 0, depending only on A, {AN}N , C, K and W , such that
ℓKσ ≤ α degL Y + β(degL Y ) log
log ‖s‖−20
c1(L0)
degL Y
If K = 1 and σ is a product of differential forms of the kind τ0, τ1 (i.e. W ⊆
{0, 1}), then we can take R = 1.
Assuming for the moment the propositions, we prove Theorem 5.1.
Proof of Theorem 5.1. We first observe that c1(L) = dd
cf + c1(L0), so that
(5.1) Jp =
f(ddcf)jωd−j.
Next, on Y \ div s we write f = ϕ− ψ and ddcf = ωψ − ωϕ. We get
(−1)j−k
ϕωkψω
d−j −
ψωkψω
The coefficients
can be bounded in terms of dimX (hence independently
of Y and the hermitian line bundles). Therefore we are reduced to bound
integrals of the kind ∫
ϕωaϕω
ψωaϕω
for integers a, b, c ≥ 0 such that a + b + c = d. Since the differential forms
c are semi-positive and 0 ≤ ϕ, ψ ≤ CℓM , we have to find upper bounds
for the integrals
I(M,a, b, ωc) =
Successively applying Proposition 5.2, we reduce our problem to find bounds for
integrals
ℓKσ, where σ is a product of (1,1)-forms of type τ or τQ, for some
integers K, Q ≥ 0. If f is P-singular, then K = 1 and σ is a product of forms
of type τ0 and τ1. We conclude by Proposition 5.3.
5.2 Proofs of Proposition 5.2 and Proposition 5.3
We proceed to prove Proposition 5.2 and Proposition 5.3. The proofs make an
extensive use of Stokes’ theorem for pre-log-log differential forms. We refer to
Proposition 2.16 for the statement and references. To simplify the exposition, it
will be worth having at our disposal the computations summarized in the next
lemma.
Lemma 5.4. Let a, b ≥ 0 be integers. On Y \ E the following equalities hold:
∂(ωaϕω
ψ) = aBC(∂ ℓN)ω
ψτ + bBC(∂ ℓN )ω
ψ τ ;
∂ ∂̄(ωaϕω
ψ) =aBC(∂ ∂̄ ℓN)ω
−aBC(∂̄ ℓN ) ∂(ω
+bBC(∂ ∂̄ ℓN )ω
−bBC(∂̄ ℓN ) ∂(ω
ψ )τ.
Proof. It is enough to apply the definition of ωϕ, ωψ, Leibniz’ rule and the fact
that ddc(−ϕ), ddc(−ψ) and τ are ∂ and ∂̄-closed.
Proof of Proposition 5.2. Write I = I(M,a, b, σ). We suppose that a > 0 and
b ≥ 0. We decompose
ωaϕ = dd
c(−ϕ)ωa−1ϕ +BCℓNτω
Accordingly, the integral I decomposes as I = I1 +BCI2, with
c(−ϕ)ωa−1ϕ ω
ℓM+Nω
Bounding I1. To get bounds on I1 we apply Stokes’ theorem for pre-log-log
forms. For this, we first recall that σ is closed of degree (t, t) and ωϕ, ωψ are of
degree (1, 1). Then, by Leibniz’ rule, we compute
∂̄(−ϕ)ωa−1ϕ ω
=ℓMdd
c(−ϕ)ωa−1ϕ ω
+(∂ ℓM )
∂̄(−ϕ)ωa−1ϕ ω
∂̄(−ϕ) ∂(ωa−1ϕ ω
where we used that ddc = i ∂ ∂̄ /2π. By Stokes’ theorem, we find I1 = I1,1+I1,2,
where
I1,1 = −
(∂ ℓM ) ∂̄(−ϕ)ω
I1,2 =
ℓM ∂̄(−ϕ) ∂(ω
Bounding I1,1. Again we apply Stokes’ theorem. By Lebniz’ rule we have
(−ϕ)(∂ ℓM )ω
(∂ ℓM ) ∂̄(−ϕ)ω
(−ϕ)(∂ ∂̄ ℓM )ω
(−ϕ)(∂ ℓM ) ∂̄(ω
Therefore, by Stokes’ theorem, we get I1,1 = I1,1,1 + I1,1,2, where
I1,1,1 =
ϕddc(−ℓM )ω
I1,1,2 =
(−ϕ)(∂ ℓM ) ∂̄(ω
Bounding I1,1,1. Recall that, by assumption, there exists a constant AM > 0
such that τM = dd
c(−ℓM )+AMω ≥ 0. Then, since ω
ψωσ is a semi-positive
form and 0 ≤ ϕ ≤ CℓM , we have a bound
I1,1,1 ≤
ϕ(ddc(−ℓM ) +AMω)ω
=CI(M,a− 1, b, στM ).
(5.2)
Bounding I1,1,2. To bound the integral I1,1,2 we first appeal to Lemma 5.4 to
develop
∂̄(ωa−1ϕ ω
ψ) = (a− 1)BC(∂̄ ℓN )ω
ψτ + bBC(∂̄ ℓN)ω
Then we write I1,1,2 = (a− 1)BCI
1,1,2 + bBCI
1,1,2, where
1,1,2 =
(−ϕ)(∂ ℓM )(∂̄ ℓN )ω
1,1,2 =
(−ϕ)(∂ ℓM )(∂̄ ℓN )ω
ψ στ.
In these integrals we observe that
(5.3) i ∂ ℓM ∧ ∂̄ ℓN = NMℓN+M−2i ∂ ℓ1 ∧ ∂̄ ℓ1,
which is a semi-positive differential form. Since ϕ ≥ 0 and ωϕ, ωψ, σ and τ are
also semi-positive, we find I
1,1,2 ≤ 0 and I
1,1,2 ≤ 0. This shows
(5.4) I1,1,2 = (a− 1)BCI
1,1,2 + bBCI
1,1,2 ≤ 0.
From (5.2) and (5.4) we conclude with the bound
(5.5) I1,1 = I1,1,1 + I1,1,2 ≤ CI(M,a− 1, b, στM ).
Bounding I1,2. As before we proceed by successive applications of Stokes’ the-
orem. First of all we compute
ℓM (−ϕ) ∂(ω
ℓM ∂̄(−ϕ) ∂(ω
(−ϕ)(∂̄ ℓM ) ∂(ω
(−ϕ)ℓM ∂̄ ∂(ω
By Stokes’ theorem we find I1,2 = I1,2,1 + I1,2,2, with
I1,2,1 =
ϕ(∂̄ ℓM ) ∂(ω
I1,2,2 =
(−ϕ)ℓM ∂ ∂̄(ω
Bounding I1,2,1. By Lemma 5.4 we get the expansion
∂(ωa−1ϕ ω
ψ) = (a− 1)BC(∂ ℓN )ω
ψτ + bBC(∂ ℓN)ω
Accordingly, we decompose I1,2,1 = (a− 1)BCI
1,2,1 + bBCI
1,2,1, where
1,2,1 = −
ϕ(∂ ℓN )(∂̄ ℓM )ω
1,2,1 = −
ϕ(∂ ℓN )(∂̄ ℓM )ω
ψ στ.
By (5.3) the differential form i ∂ ℓN ∧ ∂̄ ℓM is semi-positive. Since ϕ ≥ 0 and
ωϕ, ωψ, σ and τ are also semi-positive, we have I
1,2,1, I
1,2,1 ≤ 0. This proves
(5.6) I1,2,1 = (a− 1)BCI
1,2,1 + bBCI
1,2,1 ≤ 0.
Bounding I1,2,2. To bound the integral I1,2,2 we first recall from Lemma 5.4
∂ ∂̄(ωa−1ϕ ω
ψ) = (a− 1)BC(∂ ∂̄ ℓN)ω
− (a− 1)BC(∂̄ ℓN) ∂(ω
+ bBC(∂ ∂̄ ℓN)ω
− bBC(∂̄ ℓN ) ∂(ω
ψ )τ.
Corresponding to this expansion, we write I1,2,2 = (a − 1)BCI
1,2,2 + (a −
1)BCI
1,2,2 + bBCI
1,2,2 + bBCI
1,2,2, with the obvious notations for the inte-
grals I
1,2,2 (see below).
Bounding I
1,2,2. We have
1,2,2 =
ϕℓMdd
c(−ℓN)ω
Recall that for some constant AN > 0 the differential form τN = dd
c(−ℓN) +
ANω is semi-positive. Moreover 0 ≤ ϕ ≤ CℓM . Thus we find the bound
1,2,2 ≤C
ψσττN
=CI(2M,a− 2, b, σττN ).
(5.7)
Bounding I
1,2,2. We write
1,2,2 =
ϕℓM (∂̄ ℓN) ∂(ω
ψ)στ.
By (5.3) and Lemma 5.4, i ∂(ωa−1ϕ ω
ψ) ∧ ∂̄ ℓN is semi-positive, so that
(5.8) I
1,2,2 ≤ 0.
Bounding I
1,2,2. We have
1,2,2 =
ϕℓMdd
c(−ℓN)ω
Reasoning as for I
1,2,2, we arrive to
1,2,2 ≤C
ψ σττN
=CI(2M,a− 1, b− 1, σττN ).
(5.9)
Bounding I
1,2,2. We finally bound the integral
1,2,2 =
ϕℓM (∂̄ ℓN ) ∂(ω
ψ )στ.
Again by (5.3) and Lemma 5.4, the differential form i ∂(ωa−1ϕ ω
ψ ) ∧ (∂̄ ℓN ) is
semi-positive, so that
(5.10) I
1,2,2 ≤ 0.
We now conclude with a bound for I1,2, since the inequalities (5.6)–(5.10) yield
I1,2 ≤ I1,2,2 ≤(a− 1)BC
2I(2M,a− 2, b, σττN )
+bBC2I(2M,a− 1, b− 1, σττN ).
(5.11)
As for I1, the bounds (5.5) and (5.11) lead to
I1 = I1,1 + I1,2 ≤CI(M,a− 1, b, στM )
+bBC2I(2M,a− 1, b− 1, σττN )
+(a− 1)BC2I(2M,a− 2, b, σττN ).
(5.12)
Bounding I2. We have
(5.13) I2 =
ℓM+Nω
ψστ = I(M +N, a− 1, b, στ)
To conclude we put (5.12) and (5.13) together and we get
I = I1 +BCI2 ≤BCI(M +N, a− 1, b, στ)
+CI(M,a− 1, b, στM )
+bBC2I(2M,a− 1, b− 1, σττN )
+(a− 1)BC2I(2M,a− 2, b, σττN ),
as was to be shown.
Suppose now that f is P-singular (so that N = 0 andM = 1). We can write
ωϕ = dd
c(−ϕ̃) +ABCω,
ωψ = dd
c(−ψ̃) +ABCω,
where ϕ̃ = ϕ + BCΘ1 and ψ̃ = ψ + BCΘ1. The same method followed above
allows to establish the bound
I(1, a, b, σ) ≤ ABCI(1, a− 1, b, στ0)
+ (B + 1)CI(1, a− 1, b, στ1).
The details are left to the reader.
Proof of Proposition 5.3. Let σ be a (d, d)-form which is a product of (1,1)-
forms of type τ or τQ, Q ≥ 0. We write σ = τ
sσ1, for some s ≥ 0 and σ1 a
product of forms of type τQ. Define
(5.14) J(K, s, σ1) =
First of all we show how to reduce s to 0. The argument is by induction. If
s > 0, recalling that τ = ddc(−Θ1) +Aω we write
(5.15) τs = ddc(−Θ1)τ
s−1 +Aωτs−1.
Since τ0 = ω, we get from the definition of J in (5.14) and from (5.15)
(5.16) J(K, s, σ1) = AJ(K, s− 1, σ1τ0) +
c(−Θ1)τ
s−1σ1.
We next bound the integral on the right hand side of (5.16). Since τ and σ1 are
∂ and ∂̄-closed, applying Stokes’ theorem for pre-log-log forms we get
(5.17)
c(−Θ1)τ
s−1σ1 =
c(−ℓK)τ
s−1σ1.
Now ddc(−ℓK) = τK −AKτ0, the forms τ0, τK are semi-positive and 0 < Θ1 ≤
Cℓ1, so that from (5.17) and the definition of J we derive
(5.18)
c(−Θ1)τ
s−1σ1 ≤ CJ(1, s− 1, σ1τK).
Observe that because ℓK = ℓ
1 and ℓ1 ≥ 1, the inequality ℓ1 ≤ ℓK holds.
Therefore
(5.19) J(1, s− 1, σ1τK) ≤ J(K, s− 1, σ1τK).
From (5.16)–(5.19) we arrive to
J(K, s, σ1) ≤ AJ(K, s− 1, σ1τ0) + CJ(K, s− 1, σ1τK).
Hence we may suppose that s = 0, so that σ is a product τQ1 . . . τQd . We have
to deal with
L(K,Q1, . . . , Qd) =
ℓKτQ1 . . . τQd .
Again by an inductive argument we show how to reduce all the integers Qi to
0. Suppose that Q1 > 0. Then we write τQ1 = dd
c(−ℓQ1) +AQ1τ0, so that
(5.20) L(K,Q1, . . . , Qd) = AQ1L(K, 0, Q2, . . . , Qd) +
c(−ℓQ1)σ1
where σ1 = τQ2 . . . τQd . We study the integral on the right hand side of (5.20).
Because σ1 is ∂ and ∂̄-closed, by Stokes’ theorem for pre-log-log forms we find
L2 :=
c(−ℓQ1)σ1 =
(∂ ℓK)(∂̄ ℓQ1)σ1
KQ1ℓK+Q1−2(∂ ℓ1)(∂̄ ℓ1)σ1.
(5.21)
By the very definition of ℓ1, we have
(5.22) ∂ ℓ1 ∧ ∂̄ ℓ1 =
∂ log ‖s‖−20 ∧ ∂̄ log ‖s‖
(log ‖s‖−20 )
On the other hand, since ‖s‖20 ≤ e
−e, there exists a constant D depending only
on K +Q1 − 2 such that
(5.23)
ℓK+Q1−2
(log ‖s‖−20 )
(log log ‖s‖−20 )
K+Q1−2
(log ‖s‖−20 )
(log ‖s‖−20 )
Moreover observe that
(5.24)
(log ‖s‖−20 )
∂ log ‖s‖−20 = −2 ∂(log ‖s‖
−1/2.
Because i ∂ log ‖s‖−20 ∧ ∂̄ log ‖s‖
0 and σ1 are semi-positive, combining (5.21)–
(5.24) we get
(5.25) L2 ≤ −2
KQ1D(∂(log ‖s‖
−1/2)(∂̄ log ‖s‖−20 )σ1.
By Lemma 5.5 below, we can apply Stokes’ theorem to the right hand side of
(5.25) and obtain
(5.26) L2 ≤ 2
KQ1D(log ‖s‖
−1/2ωσ1.
By the hypothesis on ‖ · ‖0, (log ‖s‖
−1/2 ≤ 1. Using the positivity of ωσ1,
from (5.26) we derive
(5.27) L2 ≤ 2KQ1D
Now recall that ω is ∂ and ∂̄-closed, and τQ = dd
c(−ℓQ) +AQω, so that
ωσ1 = AQ2 . . . AQdω
d + ddcσ2
for some pre-log-log form σ2. Applied to (5.27) this provides
(5.28) L2 ≤ 2Q1AQ2 . . . AQdKD
ωd = 2Q1AQ2 · . . . · AQdKD degL Y.
The identity (5.20) together with (5.28) imply the inequality
L(K,Q1, . . . , Qd) ≤ AQ1L(K, 0, Q2, . . . , Qd) + 2Q1AQ2 · . . . · AQdKD degL Y.
Successively repeating this argument, we reduce Q2, . . . , Qd to 0. Hence it
remains to treat the integrals
M(K) =
We want to apply Jensen’s inequality. First of all, rewrite
M(K) = (degL Y )
degL Y
so that ωd/ degL Y defines a probability measure on Y . Secondly, the function
x 7→ (log x)K is concave on ]eK−1,+∞[, because
(log x)K =
(log x)K−2(K − 1− log x).
Since ‖s‖−20 ≥ e
e, in particular eK+1 log ‖s‖−20 > e
K−1. We use that logK is an
increasing function and apply Jensen’s inequality:
M(K) ≤(degL Y )
logK(eK+1 log ‖s‖−20 )
degL Y
≤(degL Y ) log
log ‖s‖−20
degL Y
By the trivial inequality x+ y ≤ 2xy for real x, y ≥ 1, we finally arrive to
M(K) ≤ (degL Y )(2K + 2)
K logK
log ‖s‖−20
degL Y
This concludes the proof of the proposition, except for the fact that we can take
R = 1 when K = 1 and σ is a product of forms τ0 and τ1. This last case is
similarly treated and left to the reader.
Lemma 5.5. Let µ be a closed pre-log-log (d-1,d-1)-form on Y , with singulari-
ties along div s. Then we have
(log ‖s‖−20 )
∧ ∂̄ log ‖s‖−20 µ =
(log ‖s‖−20 )
∧ ωµ.
Proof. The proof of the lemma follows the ideas of Lemma 7.36 in [2].
First of all, as for log-log growth differential forms, the form
(log ‖s‖−20 )
∧ ∂̄ log ‖s‖−20 µ = −
(log ‖s‖−20 )
1/2 ∂ ℓ1 ∧ ∂̄ ℓ1µ
is locally integrable on Y . Indeed, after localizing to an analytic chart adapted
to div s and changing to polar coordinates, we are reduced to point out that,
for every 0 < δ < 1/2, we have an estimate
(5.29)
∫ ε/e
(log log t−1)N (log t−1)1/2
t(log t)2
∫ ε/e
t(log t−1)3/2−δ
< +∞.
Define
I = −
(log ‖s‖−20 )
∧ ∂̄ log ‖s‖−20 µ.
We construct a finite open covering {(V ′α; {z
i }i)}α of div s, by adapted analytic
open charts. Suppose that via the coordinates {zαi }α, V
α is identified with
(r = r(α)), so that V ′α \ D corresponds to ∆
further assume that in these coordinates we can write s = zα1 · . . . · z
r uα, where
uα is a holomorphic unit. After possibly adding a finite number of adapted
analytic charts to {(V ′α; {z
i }i)}α, the open subsets Vα ⊂⊂ V
α identified with
via the coordinates {zαi }i still cover div s. Write Ω = ∪αVα. Take
a finite open covering {Vβ}β of X \ Ω, so that Vβ ∩ (div s) = ∅ for all β. Let
{χα}α ∪ {χβ}β be a partition of unity subordinate to {Vα}α ∪ {Vβ}β, with χγ
vanishing outside Vγ for all γ = α, β. We can expand
where
Iγ := −
(log ‖s‖−20 )
∧ ∂̄ log ‖s‖−20 χγµ.
We first treat the integrals Iβ . Observe that for any C
∞ differential form ν on
Y \ div s, the differential form χβν is C
∞ on Y , because χβ vanishes on Y \ Vβ
and Vβ ∩ (div s) = ∅. Moreover the equality d(χβν) = dχβ ∧ ν + χβ ∧ dν holds
on Y . For ν = i ∂ ∂̄ log ‖s‖−20 /2π we find χβν = χβω on Y . These observations,
together with dµ = 0, yield, by Stokes’ theorem,
(log ‖s‖−20 )
(log ‖s‖−20 )
∂̄ log ‖s‖−20 (dχβ)µ,
(5.30)
On the other hand, for every α define Bαε (div s) to be the ε-neighborhood of
div s given by
Bαε (div s) =
Bαε (Tk),
where Tk is the divisor z
k = 0 in Vα and
Bαε (Tk) = ∆
×∆ε ×∆
×∆s1/2e ⊂ Vα.
Then we write Iα = limε→0 Iα,ε, where Iα,ε = I
α,ε + I
α,ε + I
α,ε and
I(1)α,ε :=−
Y \Bαε (div s)
(log ‖s‖−20 )
∂̄ log ‖s‖−20 χαµ
I(2)α,ε :=
Y \Bαε (div s)
(log ‖s‖−20 )
I(3)α,ε :=−
Y \Bαε (div s)
(log ‖s‖−20 )
∂̄ log ‖s‖−20 (dχα)µ.
The differential form in I
α,ε is integrable on Y , since it has log-log growth along
div s (Proposition 2.16). The differential form in I
α,ε is integrable on Y , too.
Indeed, after localizing to an analytic chart adapted to div s and changing to
polar coordinates, we are reduced to prove the convergence of the integrals
(5.31)
∫ 1/e
(log log t−1)N
(log t−1)1/2
(5.32)∫ 1/e
(log log t−1)N
(log t−1)1/2
t log t−1
∫ 1/e
(log log t−1)N
t(log t−1)3/2
From the boundedness of (log log t−1)N/(log t−1)1/2 the convergence of (5.31)
follows. The second one (5.32) has already been treated (5.29). Therefore
limε→0 I
α,ε = I
α and limε→0 I
α,ε = I
α , where
I(2)α :=
(log ‖s‖−20 )
ωχαµ,
I(3)α :=−
(log ‖s‖−20 )
∂̄ log ‖s‖−20 (dχα)µ.
As for the integral I
α,ε, after applying Stokes’ theorem we find
I(1)α,ε =
∂ Bαε (div s)
(log ‖s‖−20 )
∂̄ log ‖s‖−20 χαµ.
Taking into account that χα vanishes on ∂ Vα, we easily see that
(5.33) |I(1)α,ε| ≤
∂∗ Bαε (Tk)
(log ‖s‖−20 )
∣∣∂̄ log ‖s‖−20 χαµ
with the notation
∂∗Bαε (Tk) = ∆
× ∂∆ε ×∆
Observe that ∂∗Bαε (Tk) is fibered in circles over Tk, via the projection
pαk,ε : ∂
∗Bε(Tk) −→ Tk
(zα1 , . . . , z
d ) 7−→ (z
1 , . . . , z
k−1, 0, z
k+1 . . . , z
we will mean integral along the fibers of pαk,ε. We claim that, for every
k = 1, . . . , r,
(5.34) lim
(log ‖s‖−20 )
∣∣∂̄ log ‖s‖−20 χαµ
∣∣ = 0.
Indeed, write s = zα1 . . . z
r uα on V
α, where uα is a holomorphic unit. Since χαµ
is a (d− 1, d− 1) pre-log-log form, the differential form to integrate under
has the shape
f(zα1 , . . . , z
(log log |zαk |
(log ‖s‖−20 )
∣∣∣∣∣∣
dz̄αk
j 6=k
dzαj ∧ dz̄
∣∣∣∣∣∣
where f behaves as follows:
0 ≤ f(zα1 , . . . , z
d ) ≺
j 6=k
(log log |zαj |
|zαj |
2(log |zαj |
We point out that |dz̄αk /z̄
k | is bounded along the fibers of p
k,ε, that the form
j 6=k dz
j ∧ dz̄
∣∣∣ is integrable and (log log |zαk |−1)M/(log ‖s‖
1/2 vanishes
along Tk. This is enough to prove the claim (5.34). Therefore, from (5.33)
limε→0 I
α,ε = 0 and consequently
Iα =I
α + I
(log ‖s‖−20 )
(log ‖s‖−20 )
∂̄ log ‖s‖−20 (dχα)µ.
(5.35)
Finally, since the open covering {Vα}α ∪ {Vβ}β is finite, from (5.30) and (5.35)
we derive
(log ‖s‖−20 )
χγ)ωµ
(log ‖s‖−20 )
∂̄ log ‖s‖−20 d(
χγ)µ.
(5.36)
The assertion of the lemma follows from (5.36) once we note that
γ χγ = 1
and d(
γ χγ)= 0.
5.3 Application to the projective case
We suppose that X is a nonsingular complex projective variety (non necessarily
connected) and D ⊆ X a divisor with simple normal crossings. Consider the
following elements:
i. as in section 4, for every integer N ≥ 0 introduce a function ΘN (see
Notation 4.1). By Lemma 4.7, the functions ΘN are pre-log-log, with
singularities along D;
ii. an ample line bundle L on X , admitting a global section s ∈ Γ(X,L)
such that div s is a divisor with simple normal crossings containing D. In
particular div s is reduced and may be seen as a reduced Weil divisor or a
reduced scheme;
iii. a pre-log-log metric ‖ · ‖ on L, with singularities along D;
iv. a smooth hermitian metric ‖·‖0 on L with ω := c1(L0) > 0 and ‖s‖
0 ≤ e
v. as in §5.1 we introduce ℓQ = (log log ‖s‖
Q, Q ∈ Z≥0. By Lemma 4.6,
the function ℓQ is pre-log-log, with singularities along div s. Moreover,
Proposition 4.9 asserts the existence of a constant AQ > 0 such that
τQ := dd
c(−ℓQ) +AQω ≥ 0. We can take A0 = 1.
We write ‖ · ‖ = e−f/2‖ · ‖0, where f : X \D → R is a pre-log-log function, with
singularities along D. If ‖ · ‖ is good, then f is P-singular. Since X is compact,
associated to ω = c1(L0) there is a decomposition f = ϕ − ψ as in Theorem
4.3. Moreover, because D ⊆ div s, there exist a constant C ≥ 0 and an integer
M ≥ 0 such that
ϕ ≤ CℓM , ψ ≤ CℓM ,Θ1 ≤ Cℓ1,ΘN ≤ CℓN
hold on X \div s. If ‖ · ‖ is good, then we can take M = 1. Finally, by Theorem
4.3 and Proposition 4.9, there exist constants A,B > 0 and an integer N ≥ 0
such that τ := ddc(−Θ1) +Aω ≥ 0 and
ωϕ :=dd
c(−ϕ) +BΘN(dd
c(−Θ1) +Aω) ≥ 0,
ωψ :=dd
c(−ψ) +BΘN (dd
c(−Θ1) +Aω) ≥ 0.
If ‖ · ‖ is good along D, then we can take N = 0. Therefore the assumptions of
§5.1 are fulfilled for X .
Let now π : X ′ → X be a morphism of complex analytic manifolds such that the
inverse image schemes π−1(D) ⊆ π−1(div s) are divisors with normal crossings.
Let Y ⊆ X ′ be a compact complex analytic submanifold of pure dimension d.
Suppose that Y meets π−1(div s) in a divisor with normal crossings in Y . Then
Y ∩D is a divisor with normal crossings in Y , too. We pull-back by π all the
objects introduced above (π−1(D), π∗ΘN , π
∗L, π∗s, π∗‖ · ‖, etc.) and then
we restrict them to Y . We obtain corresponding objects on Y (π−1(D) ∩ Y ,
(π∗ΘN)|Y , (π
∗L)|Y , (π
∗s)|Y , (π
∗‖ · ‖)|Y , etc.). Provided that degπ∗L Y > 0, the
requirements of §5.1 are fulfilled on Y . It is important to point out that the
involved constants A, {AQ}Q, B, C, M , N don’t depend on the data X
′, π, Y .
For every integer 0 ≤ p ≤ d define
J∗p =
(π∗f) c1(π
∗L)p c1(π
As a consequence of Theorem 5.1 we get the following corollary.
Corollary 5.6. Let π : X ′ → X be a morphism of complex analytic manifolds
such that π−1(D) and π−1(div s) are divisors with normal crossings. Let Y ⊆ X ′
be a compact complex analytic submanifold of pure dimension d, intersecting
div s in a divisor with normal crossings in Y . Suppose that degπ∗L Y > 0.
There exist constants α, β > 0 and an integer R ≥ 0, depending only on A,
{AQ}Q, B, C, M and N such that, for all p ∈ {0, . . . , d},
|J∗p | ≤ α degπ∗L Y + β · (degπ∗L Y ) · log
log π∗‖s‖−20
degπ∗L Y
If ‖ · ‖ is good along D, then we can take R = 1:
|J∗p | ≤ α degπ∗L Y + β · (degπ∗L Y ) · log
log π∗‖s‖−20
degπ∗L Y
Let Z be a reduced closed subscheme of X of pure dimension d, intersecting
div s properly. Denote by Z1, . . . , Zr the irreducible components of Z. For every
i = 1, . . . , r, let πi : Xi → X be an imbedded resolution of singularities of Zi,
such that π−1i (div s) is a divisor with normal crossings intersecting the strict
transform Z̃i of Zi in a divisor with normal crossings in Z̃i (see [2], Theorem
7.27). Then Corollary 5.6 applies to πi, Zi, for every i = 1, . . . , r. If θ is a
smooth differential form of degree 2d on X \div s, locally integrable on X , then
we adopt the convention ∫
π∗i θ.
This definition intrinsically depends on Z, and not on the choice of the resolu-
tions πi. With this convention, define
f c1(L)
p c1(L0)
In this situation Corollary 5.6 reads as follows.
Corollary 5.7. Let Z be a reduced closed subscheme of X of pure dimension d,
intersecting div s properly. There exist positive constants α, β, and an integer
R ≥ 0, depending only on A, {AQ}Q, B, C, M and N such that, for all
p ∈ {0, . . . , d},
|Jp| ≤ α degL Z + β · (degL Z) · log
log ‖s‖−20
c1(L0)
degL Z
with R = 1 whenever ‖ · ‖ is good along D.
Proof. Decompose Z into irreducible components: Z = Z1∪ . . .∪Zr. Following
the convention above, define
J (i)p =
f c1(L)
p c1(L0)
so that
(5.37) Jp =
J (i)p .
By Corollary 5.6, there exist constants α, β > 0 and an integer R ≥ 0, depending
only on A, {AQ}Q, B, C, M and N such that, for all p ∈ {0, . . . , d},
(5.38) |J (i)p | ≤ α degL Zi + β · (degL Zi) · log
log ‖s‖−20
c1(L0)
degL Zi
If ‖ · ‖ is good along D, then we can take R = 1. Now recall that the function
logR is increasing and concave on ]eR−1,+∞[, so that
degL Zi
degL Z
log ‖s‖−20
c1(L0)
degL Zi
≤ logR
degL Zi
degL Z
log ‖s‖−20
c1(L0)
degL Zi
= logR
log ‖s‖−20
c1(L0)
degL Z
≤(2R+ 2)R logR
log ‖s‖−20
c1(L0)
degL Z
(5.39)
For the last inequality we used that x + y ≤ 2xy for real x, y ≥ 1. The lemma
follows combining (5.37)–(5.39).
6 Arakelovian heights
In this section we turn to an arithmetic situation and deal with arakelovian
heights on arithmetic varieties. We prove Theorem 1.3, which can be seen as an
arithmetic counterpart of the bounds in §5. A remarkable and straightforward
outcome is the finiteness property of arakelovian heights with respect to pre-log-
log hermitian ample line bundles, as well as the existence of a universal lower
bound (Corollary 1.4).
6.1 Heights attached to pre-log-log hermitian line bundles
Let K be a number field and OK its ring of integers. Write S = SpecOK .
Throughout this section we work with a fixed arithmetic variety π : X → S
of relative dimension n. We recall this means that X is a flat and projective
scheme over S , with regular generic fiber XK = X ×S SpecK of pure di-
mension n. The set of complex points X (C) of X has a natural structure of
complex analytic manifold, and it can be decomposed as
X (C) =
σ:K →֒C
Xσ(C).
Complex conjugation induces an antiholomorphic involution
F∞ : X (C) −→ X (C).
We fixD ⊂ XK a divisor, such that D(C) ⊂ X (C) has simple normal crossings.
Write U = X (C) \D(C).
Notation 6.1 ([2]). We define Z
U (X ) to be the free abelian group generated
by the irreducible reduced subschemes Z ⊆ X of codimension p, such that Z(C)
intersects D(C) properly. We call Z
U (X ) the group of cycles of codimension
p, intersecting D(C) properly. A cycle z is said to be vertical if its components
are supported on closed fibers X℘, ℘ ∈ S \ {(0)}. A cycle z is said to be
horizontal if its irreducible components are flat over S . We denote by Z
U (XK)
the subgroup of Z
U (X ) of horizontal cycles.
Definition 6.2. A pre-log-log hermitian line bundle on X , with singularities
along D, is a couple L = (L , ‖ · ‖) formed by
i. a line bundle (invertible sheaf) L on X ;
ii. a pre-log-log hermitian metric ‖·‖ on the line LC on X (C), with singular-
ities along D(C), and invariant under the action of complex conjugation
F∞: F
∞‖ · ‖ = ‖ · ‖.
In [2], [3], Burgos, Kramer and Kühn attach a height morphism to a pre-
log-log hermitian line bundle L ,
U (X ) −→ R,
generalizing the height morphism for smooth hermitian line bundles introduced
by Bost, Gillet and Soulé in [1]. We refer the reader to the cited bibliography
for the precise definition and basic properties of h
, both in the smooth and
pre-log-log case. For our purposes, it will be enough to state the following
propositions summarizing the main features of h
Proposition 6.3. Let L be a pre-log-log hermitian line bundle on X , with
singularities along D. The height h
satisfies the following three properties:
H1. if PK : SpecK → X is a K-valued point whose image does not belong to
D, and P : S → X denotes its extension to S , then
(P∗S ) = d̂eg(P
L )(4)
(4)The arithmetic degree ddeg of a hermitian line bundle M = (M, ‖ · ‖) over SpecOK is
defined as follows: if s is a non zero global section of M, then ddeg(M) = log ♯
σ:K →֒C log ‖s‖σ .
H2. if z is a vertical cycle supported on a closed fiber X℘, then
(z) = log(N℘) deg
where N℘ denote the norm of the ideal ℘;
H3. let z ∈ Z
n+1−p
U (XK) be irreducible and reduced. Let s be a rational section
of L ⊗N which does not identically vanish on z. If (div(s).z)(C) intersects
D(C) properly, then
(z) = h
(div(s).z)−
log(‖s‖L⊗N ) c1(L )
Proof. This follows from the definition of h
and the extended arithmetic in-
tersection theory in [2].
Remark 6.4. i. The convergence of the integral in H3 is implicit in the state-
ment.
ii. If LK is an ample line bundle, then an easy inductive argument shows that
the properties H1, H2 and H3 actually characterize h
Proposition 6.5. Let L be a line bundle on X , ‖ · ‖ a pre-log-log hermitian
metric on L , with singularities along D, and ‖ · ‖0 a smooth hermitian metric
on L . Write L = (L , ‖ · ‖), L 0 = (L , ‖ · ‖0) and ‖ · ‖ = e
−f/2‖ · ‖0, where
f : X (C) \D(C) → R is a pre-log-log function, with singularities along D(C).
For any cycle z ∈ Z
n+1−p
U (X ) we have
(z) = h
(z) +
f c1(L )
k c1(L 0)
p−1−k.
Proof. This is contained in [3], Theorem 4.1.
Remark 6.6. Proposition 6.5 allows to recover Proposition 6.3 once it is known
for smooth hermitian line bundles. In this case the properties H1, H2 and H3
are already established in [1].
Let F |K be a finite extension of fields and write T = SpecOF . Base
changing by T → S , we get an arithmetic variety XT → T , together with a
finite flat morphism g : XT → X of degree [F : K]. If D ⊆ XK is an effective
divisor such that D(C) ⊆ X (C) has simple normal crossings, then DF ⊆ XF is
an effective divisor and DF (C) ⊆ XT (C) has simple normal crossings as well.
Let L be a pre-log-log hermitian line bundle on X , with singularities along
D. The pull-back g∗L of L to XT is a pre-log-log hermitian line bundle,
with singularities along DF . Since g is flat, for every cycle z on X there
is a well defined pull-back cycle g∗(z). Observe that if z(C) intersects D(C)
properly, then g∗(z)(C) intersects DF (C) properly. Namely, the correspondence
z 7→ g∗(z) induces a morphism
U (X ) −→ Z
V (XT )
z 7−→ g∗(z),
(6.1)
where V = XT (C) \DF (C). This morphism maps Z
U (XK) into Z
V (XF ).
Lemma 6.7. Let L be a pre-log-log hermitian line bundle on X , with singu-
larities along D. Let F |K be a finite extension of fields and T = SpecOF .
Write g : XT → X for the finite flat projection induced by T → S . Let
w ∈ Z
V (XF ) be an irreducible and reduced cycle and set z = g(w)red. Let δ be
the degree of g |w. Then we have the equality
(w) = δh
Consequently, for the morphism g∗ of (6.1), we have
hg∗L (g
∗(z)) = [F : K]h
for every z ∈ Z
U (X ) and every p.
Proof. This follows for instance from the case of smooth metrics (see [1], §3.1.4
and Proposition 3.2.1) and Proposition 6.5, since g|w(C) : w(C) → z(C) is gener-
ically smooth and finite of degree δ, so that
f c1(L )
k c1(L 0)
p−1−k
f c1(L )
k c1(L 0)
p−1−k.
Notation 6.8. Let L be a pre-log-log hermitian line bundle, with singularities
along D. Let z ∈ Z
U (XK) with degLK z 6= 0. We define its normalized height
to be
(z) =
[K : Q] deg
Lemma 6.9. Let g : XT → X be as before. Let w ∈ Z
V (XF ) be an irreducible
and reduced cycle and set z = g(w)red. For the normalized height we have
(w) = h̃
(z) and h̃
(g∗z) = h̃
Proof. For every w ∈ Z
n+1−p
V (XF ) irreducible and reduced and z = g(w)red, we
have the equalities
deg(g∗L )F w =
[F : Q]
L )p−1
[F : Q]
c1(L )
[F : K]
where δ is the degree of g|w. It follows that deg(g∗L )F (g
∗z) = deg
z. Com-
bined with Lemma 6.7, we get h̃g∗L (w) = h̃L (z) and h̃g∗L (g
∗z) = h̃
Remark 6.10. The normalization h̃
just introduced is not the standard one.
However it appears naturally in the statement and proof of Theorem 1.3.
6.2 Proof of the main theorem
We now proceed to prove Theorem 1.3. The argument mainly relies on the
bounds established in §5 , and more concretely the situation studied in §5.3.
However, in the reduction steps we will need the following Bertini’s type theo-
rem. The proof is essentially well known, but we include it in the Appendix for
lack of reference.
Proposition 6.11. Let X be a nonsingular projective scheme over an alge-
braically closed field k. Let D ⊆ X be a divisor with simple normal crossings.
Let L be an ample line bundle on X. Then there exist an integer N > 0 and
global sections s1, . . . , sr ∈ H
0(X,L⊗N ) such that supp(div s1),. . . , supp(div sr)
are divisors with simple normal crossings and the following equality of schemes
holds:
D = (supp(div s1) ∩ . . . ∩ supp(div sr))red.
Under the hypothesis of Theorem 1.3, since D(C) has simple normal cross-
ings, DK has also simple normal crossings in XK . We will apply Proposition
6.11 through the following straightforward corollary.
Corollary 6.12. There exist a finite extension K ′|K, a positive integer N and
global sections s1, . . . , sr ∈ H
0(XK′ ,L
K′ ) such that
B1. supp(div sj)(C) is a divisor with simple normal crossings in X (C), for
every j = 1, . . . , r;
B2. DK′ = (supp(div s1) ∩ . . . ∩ supp(div sr))red.
The next two lemmas provide the final reductions before the proof of Theo-
rem 1.3.
Lemma 6.13. It is enough to proof Theorem 1.3 in the following situations:
i. after some finite extension K ′ | K;
ii. L is very ample and z ∈ Z
U (XK) is irreducible and reduced.
Proof. The first claim i is clear. For the proof of ii, we first note that the
statement of Theorem 1.3 for L ⊗N already implies the statement of the theorem
for L , since h̃
⊗N (z) = Nh̃
(z) and h̃
(z) = Nh̃
(z). Hence we assume
LK is very ample. We then proceed in two steps.
Step 1. We can suppose that L is very ample. Indeed, there exists some model
(Y ,A ) of (XK ,LK) with A very ample. The metrics ‖ · ‖, ‖ · ‖0 on L induce
metrics on A , and we write A and A 0 for the corresponding hermitian line
bundles. For every effective cycle z ∈ Z
U (XK), we write z̃ for the corresponding
effective and horizontal cycle on Y . By Proposition 3.2.2 in [1], there exists a
positive constant C, independent of z, such that
(6.2)
(z̃)− h
∣∣ ≤ C deg
Since A C and L C are isometric, Proposition 6.5 and (6.2) together give
(z̃)− h
(z̃)− h
∣∣ ≤ C deg
Step 2. We can suppose that z is irreducible and reduced. Indeed, suppose that
we have shown the existence of constants α, β, γ > 0 and R ∈ Z≥0 such that,
for every w ∈ Z
U (XK) irreducible and reduced, we have h̃L 0(w) + γ ≥ 1 and
(6.3)
∣∣∣h̃
(w)− h̃
∣∣∣ ≤ α+ β logR
(w) + γ
After possibly increasing γ we can suppose that h̃
(w) + γ > eR−1, for every
irreducible and reduced w ∈ Z
U (XK). Let us now consider w =
i∈I wi ∈
U (XK) \ {0} where the wi ∈ Z
U (XK) are irreducible and reduced. Then (6.3)
yields
∣∣∣h̃
(w) − h̃
∣∣∣ ≤
∣∣∣h̃
(wi)− h̃L 0(wi)
≤ α+ β
(wi) + γ
Since h̃
(wi) + γ > e
R−1 for all i and logR is concave on ]eR−1,+∞[, we
conclude
(wi) + γ
≤ logR
(wi) + γ)
= logR
(w) + γ
This completes the proof.
By Corollary 6.12 and Lemma 6.13, after possibly extending K and choosing
a suitable model of (XK ,LK), we can suppose that L is very ample and there
exist sections s1, . . . , sr ∈ H
0(XK ,LK) with the properties B1, B2 above (with
K ′ = K). After possibly multiplying the sections sj by a sufficiently divisible
integer, we can even suppose that s1, . . . , sr ∈ H
0(X ,L ). We denote Ej =
supp(div sj)K . We fix these data until the end of the proof.
Lemma 6.14. Let z ∈ Z
U (XK) be irreducible and reduced. Let F be a finite
extension of K over which all the irreducible components of zK are defined.
Let T = SpecOF and g : XT → X be the finite flat projection induced by
T → S . Write g∗(z) =
i∈I zi, zi irreducible, reduced and flat over T . Then
z(C) intersects D(C) properly if, and only if, every zi(C) intersects one of the
Ej,F (C) properly.
Proof. Straightforward.
Now we can complete the proof of Theorem 1.3.
Proof of Theorem 1.3. First of all, for every integer M ≥ 0 we construct a
function ΘM , as in §4, for the complex analytic variety X (C) and the divisor
with simple normal crossings D(C).
Observe that we can suppose that the metric ‖ · ‖0 satisfies c1(L 0) > 0 and
0 ≤ e
−e for every j = 1, . . . , r. Indeed, by [1], Proposition 3.2.2 (or also
Proposition 6.5 for smooth metrics), a change of smooth metric causes only
bounded variations of the normalized height.
We introduce the pre-log-log functions
Q = (log log ‖sj‖
Q : X (C) \ Ej(C) −→ R.
For everyQ we fix a positive constantAQ > 0 such that dd
Q )+AQ c1(L 0) ≥
0, A0 = 1, for all j = 1, . . . , r (see Proposition 4.9).
Write ‖ · ‖ = e−f/2‖ · ‖0, where f : X (C) \ D(C) → R is a pre-log-log
function (resp. P-singular if ‖ · ‖ is good). Attached to c1(L 0) we perform
a decomposition f = ϕ − ψ as in Theorem 4.3. Recall that ϕ, ψ are positive
pre-log-log (resp. P-singular) functions along D(C), with
ωϕ :=dd
c(−ϕ) +BΘN(dd
c(−Θ1) +A c1(L 0)) ≥ 0,
ωψ :=dd
c(−ψ) + BΘN(dd
c(−Θ1) +A c1(L 0)) ≥ 0,
for some A,B > 0 and N ∈ Z≥0. If ‖ · ‖ is good, then we can take N = 1.
For every j = 1, . . . , r, D ⊆ Ej . By compactness of X (C) there exist constants
C > 0, M ∈ Z≥0 such that
ϕ ≤ Cℓ
M , ψ ≤ Cℓ
M ,Θ1 ≤ Cℓ
1 ,ΘN ≤ Cℓ
for all j ∈ {1, . . . , r}.
Let z ∈ Z
n+1−p
U (XK) be irreducible and reduced. Denote by F an extension
of K such that all the irreducible components of zK are defined over F . Let
T = SpecOF , g : XT → X be as before. Decompose g
∗(z) =
i∈I zi, with
the zi irreducible and flat over T . By Lemma 6.14, for every zi there exists
j = j(i) ∈ {0, . . . , r} such that zi(C) intersects Ej,F (C) properly. By Corollary
5.7 and Proposition 6.5 we have
(6.4)
∣∣∣h̃g∗L (zi)− h̃g∗L 0(zi)
∣∣∣ ≤ α+β logR
zi(C)
log g∗‖sj‖
[K : Q] deg(g∗L )F zi
for α, β > 0, R ∈ Z≥0 depending only on A, {AQ}Q, B, C,M and N . Moreover,
if ‖ · ‖ is good, then we can take R = 1. Applying the property H3 of heights
(see Proposition 6.3), we rewrite (6.4) as
(6.5)∣∣∣h̃g∗L (zi)− h̃g∗L 0(zi)
∣∣∣ ≤ α+ β logR
g∗L 0
(zi)− 2h̃g∗L 0(div(g
∗sj).zi)
To derive this inequality we point out that
deg(g∗L )F (div(g
∗sj).zi) = deg(g∗L )F zi,
so that
hg∗L 0(div(g
∗sj).zi)
[F : Q] deg(g∗L )F zi
= h̃g∗L 0(div(g
∗sj).zi).
By Theorem 1.2 there exists a positive constant κ > 0 such that, for every effec-
tive cycle w 6= 0 on X , we have h̃
(w) > −κ. Combined with Lemma 6.9 this
yields h̃g∗L 0(div(g
∗sj).zi) > −κ, because div(g
∗sj).zi is effective. Therefore,
from (6.5) we deduce
∣∣∣h̃g∗L (zi)− h̃g∗L 0(zi)
∣∣∣ ≤α+ β logR
2h̃g∗L 0(zi) + 2κ+ 2e
≤α+ 2Rβ logR
h̃g∗L 0(zi) + κ+ e
(6.6)
where we applied the trivial inequalities log 2 ≤ 1 and x + y ≤ 2xy for real
x, y ≥ 1. Now h̃g∗L 0(zi)+κ+ e
R+1 > eR−1 and logR is concave on ]eR−1,+∞[.
From (6.6) we derive
∣∣∣h̃g∗L (g
∗z)−h̃g∗L 0(g
deg(g∗L )F (zi)
deg(g∗L )F (g
∣∣∣h̃g∗L (zi)− h̃g∗L 0(zi)
≤α+ 2Rβ logR
deg(g∗L )F (zi)
deg(g∗L )F (g
h̃g∗L 0(zi) + κ+ e
=α+ 2Rβ logR
h̃g∗L 0(g
∗z) + κ+ eR+1
(6.7)
By Lemma 6.9, h̃
(g∗z) = h̃
(z) and h̃
g∗L 0
(g∗z) = h̃
(z). Hence (6.7) is
equivalent to
(6.8)
∣∣∣h̃
(z)− h̃
∣∣∣ ≤ α+ 2Rβ logR
(z) + κ+ eR+1
The constants α, 2Rβ, γ := κ + eR+1 > 0, R ∈ Z≥0 (R = 1 if ‖ · ‖ is good) in
(6.8) depend only on L and L 0, and not on z. This concludes the proof of the
theorem.
7 Examples
7.1 Automorphic vector bundles on toroidal compactifi-
cations
The first natural examples of good hermitian vector bundles are provided by the
theory of (fully decomposed) automorphic vector bundles on locally symmetric
varieties, and their extensions to smooth toroidal compactifications. These have
been firstly worked by Mumford in his proof of Hirzebruch’s proportionality
principle in the non-compact case [14]. In this section we quote from loc. cit.
the main construction and Mumford’s theorem. As an application, we briefly
discuss the example of modular forms.
Let B be a bounded symmetric domain. We can write B = G/K, where
G is a semi-simple adjoint group and K is a maximal compact subgroup. De-
note KC, GC the complexifications of K and G. Inside GC there is a parabolic
subgroup of the form P+ · KC (P+ being its unipotent radical), such that
K = G∩ (P+ ·KC) and G · (P+ ·KC) is open in GC. Then B̌ := GC/G · (P+ ·KC)
is a rational projective variety and there is a G-equivariant immersion B →֒ B̌
compatible with the complex structure of B. Let E0 be a G-equivariant vector
bundle on B attached to a representation σ : K → GLr(C). We complexify
σ and extend it to P+ · KC, by letting it act trivially on P+. This extension
induces a GC equivariant analytic vector bundle Ě0 on B̌, with ι
∗(Ě0) = E0.
This way we get a holomorphic structure on E0 (which depends on the chosen
extension of σ to P+ ·KC).
Let Γ be a neat arithmetic subgroup of G acting on B. Then X = Γ\B is a
smooth quasi-projective complex variety. The vector bundle E0 descends to a
holomorphic vector bundle E on X . Such a vector bundle is called fully decom-
posed automorphic vector bundle. Since K is compact, there is a G-invariant
hermitian metric h0 on E0, thus inducing a hermitian metric h on E.
Theorem 7.1. Let X be a smooth toroidal compactification of X with D =
X \X a divisor with normal crossings. Then the automorphic vector bundle E
extends to a vector bundle E1 over X, such that h is good along D.
The following proposition may be interesting for some arithmetic purposes.
Proposition 7.2. Suppose that E0 = ωB is the canonical bundle of B. Then
E1 = ωX(D) and coincides with the pull-back of an ample line bundle O(1) on
the Baily-Borel compactification X∗ of X. The global sections of O(n) naturally
correspond to the modular forms whose automorphy factor is the nth power of
the jacobian.
Remark 7.3. Under the hypothesis of the proposition, the line bundle ωX(D)
is not ample in general.
Equip the line bundle E0 = ωB with the hermitian metric h0 induced by the
Kähler-Einstein metric on B, say with Einstein constant −1. The existence and
uniqueness is guaranteed by a result of Mok and Yau [13]. Since the Kähler-
Einstein metric is invariant under automorphisms, h0 is G-equivariant. By The-
orem 7.1 h0 induces a good hermitian metric h on E1 = ωX(D), with singulari-
ties along D. Observe that this metric is induced by the Kähler-Einstein metric
on X with Einstein constant −1, by uniqueness. Since the Kähler-Einstein met-
ric has negative Ricci curvature, c1(ωX(D), h) ≥ 0 on X . Together with the
fact that ωX(D) is the pull-back of an ample line bundle O(1) on X
∗, this can
be shown to be enough for the main theorem to hold, as soon as X and X
are defined over a number field K and we have chosen suitable models X of
X, X ∗ of X∗, etc. over SpecOK . Suppose that O(1) extends to an ample
line bundle A on X ∗, that there is a morphism π : X → X ∗ extending the
natural projection X → X∗ and put L = π∗(A). The line bundle L is a model
of ωX(D) and it can be endowed with the good hermitian metric induced by
h. Then Corollary 1.4 hold for L , provided we restrict to effective horizontal
cycles. The proof follows the same lines as for pre-log-log hermitian ample line
bundles, and it will be detailed elsewhere.
7.2 Some natural hermitian vector bundles on the moduli
space of curves
Let g ≥ 2 be an integer and Mg the moduli space of complex stable curves of
genus g. We denote by π : Cg → Mg the universal curve. By Mg we mean the
open subset of Mg parametrizing smooth curves, and we write Cg = π
−1(Mg).
The boundary ∂Mg = Mg \ Mg, which classifies singular stable curves, is a
divisor with normal crossings. We write ∂ Cg = π
−1(∂Mg), which is a divisor
with normal crossings, too. For the sake of simplicity we neglect that Mg and
Cg are actually V -manifolds, and we work as if they were complex analytic
manifolds (see [17] for the definition of V -manifold and the description of the
moduli space of curves as a V -manifold).
The first example concerns ωCg/Mg , the relative dualizing sheaf of π. Every
fiber of π|Cg admits a unique complete hyperbolic metric of constant negative
curvature −1. By Teichmüller’s theory these metrics glue together and define
a smooth hermitian metric on ω∨
Cg/Mg
. We get a smooth hermitian metric on
ωCg/Mg . By a theorem of Wolpert [18] this metric extends to a good hermitian
metric on ωCg/Mg , with singularities along ∂ Cg. It is well known that ωCg/Mg
is relatively ample ([5], Corollary to Theorem 1.2).
Let us now consider (ΩMg , hWP ) the cotangent bundle of Mg with the Weil-
Petersson metric. Recall that if p is a point of Mg representing a Riemann
surface R, then ΩMg ,p is isomorphic to the space of holomorphic quadratic
differentials on R. If R is written as H/Γ (H Poincaré’s upper half plane and
Γ ⊆ PSL2(R) a discrete subgroup), then the metric hWP on the stalk ΩMg ,p is
the usual scalar product
〈ϕ, ψ〉 =
ϕ(z)ψ(z)y2dxdy
on automorphic forms of weight (2,0) for the group Γ. By a result of Trapani [16],
(detΩMg , dethWP ) extends to a good hermitian line bundle ωMg (log ∂Mg),
with singularities along ∂Mg. Moreover Trapani shows that ωMg (log ∂Mg)
admits a smooth hermitian metric with positive curvature form. Therefore, its
pull-back to the moduli space of curves of genus g with level n structure (n ≥ 3)
is ample.
In an ambitious program pioneered by [11], [12], Liu, Sun and Yau study
the goodness and bounded geometry of several natural Kähler metrics on the
moduli space of curves. The interested reader is referred to loc. cit. for precise
statements.
7.3 Kähler-Einstein metrics on quasi-projective varieties
7.3.1 Complex theory
The main references we follow are [10], [15], and [19].
Let M be a complex analytic manifold of dimension n and Ω a smooth
positive (n, n)-form on M , namely a volume form. For every analytic chart
(V ; z1, . . . , zn) of M write
Ω|V = ξV
dzk ∧ dz̄k
The functions {ξV }V define a smooth hermitian metric ‖ · ‖Ω on the canonical
line bundle ωM . By definition, the Ricci form of Ω is the real (1,1)-form
RicΩ = c1(ωM , ‖ · ‖Ω)
which is locally given by
RicΩ |V= dd
c log ξV .
The generalized Fefferman operator J acting on volume forms is defined as
J : Ω 7−→
(RicΩ)n
Theorem 7.4 (Kobayashi [10]). Let X be a compact complex analytic manifold
and D ⊆ X a divisor with simple normal crossings. Suppose that the line bundle
ωX(D) is ample on X. Then there exists a unique complete Kähler-Einstein
metric gKE on X \D with constant negative Ricci curvature -1. If ΩKE is the
volume form of gKE, then gKE is characterized by being complete on X \D and
by the equation
J(ΩKE) = 1.
Proposition 7.5. Let X be a compact complex analytic manifold and D ⊆
X a divisor with simple normal crossings. Suppose that ωX(D) is ample on
X. Let U = X \ D and endow ωU with the smooth hermitian metric ‖ · ‖KE
induced by ΩKE. Then (ωU , ‖ · ‖KE) extends to a good hermitian line bundle
(ωX(D), ‖ · ‖KE), with singularities along D.
The proof of Proposition 7.5 follows easily from the more precise growth
properties established in the proof of Theorem 7.4. For the sake of completeness
we now deepen some of the details involved. In the sequel we fix X and D as
in the proposition.
Definition 7.6. Let M be a complex analytic manifold of dimension n. Let
V ⊆ Cn be an open subset. A holomorphic map φ : V → M is called a
quasicoordinate if rank(dvφ) = n for every v ∈ V . Then (V, φ) is called a local
quasicoordinate of M .
Let x ∈ D and (V = V (x); z1, . . . , zn) be an analytic chart of X centered at
x, by means of which V gets identified with ∆r1 ×∆
1 and V \D with ∆
(r = r(x)). For every η ∈ (0, 1)r, define the quasicoordinate
φη : Vη = (∆3/4)
r ×∆s1 −→ V \D
v = (v1, . . . , vn) 7−→ (φη,1(v), . . . , φη,n(v))
where
φη,k(v) =
if 1 ≤ k ≤ r,
vk if k > r.
Observe that
η∈(0,1)r
φη(Vη).
We now construct a quasicoordinate covering V = {(Vβ , φβ)}β , containing ex-
actly:
• all the quasicoordinates {(Vη, φη)}η∈(0,1)r , for (V = V (x); z1, . . . , zn),
x ∈ D, as above. Denote by W the union of the images of all these
quasicoordinates. This is an open neighborhood of D;
• a finite coordinate covering of the compact subset X \W .
We introduce the Hölder space of Ck,α-functions on U = X \D, with respect to
the quasicoordinate covering V .
Definition 7.7 (Hölder spaces). Let k ≥ 0 be an integer and α ∈ (0, 1). The
Ck,α-norm with respect to V of a function u ∈ Ck(U) is
|u‖k,α,V = sup
(V,φ)∈V
‖φ∗(u)‖k,α
= sup
(V,φ)∈V
|p|+|q|≤k
∣∣∣∣∣
∂|p|+|q|
∂ vp ∂ vq
φ∗(u)(v)
∣∣∣∣∣
+ sup
v,v′∈V
|p|+|q|=k
|v − v′|−α
∣∣∣∣∣
∂|p|+|q|
∂ vp ∂ vq
φ∗(u)(v)
|p|+|q|
∂ vp ∂ vq
φ∗(u)(v′)
∣∣∣∣∣
We define the space of Ck,α-functions on U as
Ck,α(U) = {u ∈ Ck(U) | ‖u‖k,α,V < +∞},
which is seen to be a Banach space with respect to the norm ‖ · ‖k,α,V .
Definition 7.8. We define Rr,s(U) as the space of (r, s)-differential forms ω on
U such that, for all quasicoordinate (V, φ) ∈ V ,
φ∗(ω) =
|p|=r
|q|=s
(apdv
p + bqdv
‖ap‖k,α, ‖bq‖k,α < +∞
for all multi-indices p, q with |p| = r, |q| = s and all k ≥ 0, α ∈ (0, 1).
If (v1, . . . , vn) are the standard coordinates on V ⊂ C
n and p = (i1, . . . , ir),
q = (j1, . . . , js) are ordered multi-indices, we wrote
dvp = dvi1 ∧ . . . ∧ dvir , dv
q = dvj1 ∧ . . . ∧ dvjs .
Lemma 7.9. i. Rr,s(U) is a C-vector space;
ii. Rr,s(U) ∧Rr
′,s′(U) ⊆ Rr+r
′,s+s′(U);
iii. ∂ Rr,s(U) ⊆ Rr+1,s(U) and ∂̄ Rr,s(U) ⊆ Rr,s+1(U).
Proof. Immediate to check from the definition of Rr,s(U).
Recall that PD denotes the sheaf of Poincaré forms on X with singularities
along D (Definition 2.8).
Lemma 7.10. We have an inclusion Rr,s(U) ⊆ Γ(X,PD)
(r,s), where the su-
perscript stands for the (r, s) part with respect to the usual bigrading of complex
differential forms.
Proof. We localize near the divisor D and consider a quasicoordinate φη : Vη →
V \ D in V . Hence V has coordinates z1, . . . , zn and V \ D is the divisor
z1 . . . zr = 0. For simplicity we consider the differential form
ω = h
z1 log(|z1|−1)
∧ . . . ∧
zr log(|zr|−1)
∧ dzr+1 ∧ . . . dzn
and suppose that φ∗η(ω) has finite C
k,α-norm for all k ≥ 0 and α ∈ (0, 1). We
have to prove that h is bounded on the image of φη. From the definition of φη,
a straightforward computation shows that
φ∗η(ω) = φ
1− |vi|2
|vi − 1|
(vi − 1)2
dv1 ∧ . . . ∧ dvn.
The hypothesis implies the coefficient of dv1 ∧ . . .∧ dvn has bounded sup-norm.
Since |vi| ≤ 3/4 for i = 1, . . . , r, this immediately yields the boundedness of
φ∗(h).
Corollary 7.11. Let u : U = X \D → C be a smooth function. If u ∈ R0,0(U),
then u is a P-singular function with singularities along D.
Proof. By Lemma 7.9, du ∈ R1,0(U)⊕R0,1(U) and ddcu ∈ R1,1(U). By Lemma
7.10, du and ddcu have Poincaré growth with singularities along D. Hence u is
a P-singular function with singularities along D.
Theorem 7.12 (Kobayashi [10]). Let X be a compact complex analytic manifold
and D ⊆ X a divisor with simple normal crossings. Suppose that ωX(D) is
ample on X. Let D1, . . . , Dm be the irreducible components of D and si ∈
Γ(X,O(Di)) sections with div si = Di, for all i = 1, . . . ,m. Let Ω be a volume
form on X. There exist suitable smooth hermitian metrics on the line bundles
O(Di), that we write ‖ · ‖ for simplicity, with ‖si‖ small enough, and a function
u ∈ R0,0(U) such that the volume form ΩKE of the Kähler-Einstein metric on
ΩKE = e
u Ω∏m
i=1 ‖si‖
2 log(‖si‖2)2
With Theorem 7.12 at hand, we can prove Proposition 7.5.
Proof of Proposition 7.5. We may localize at an analytic chart (V ; z1, . . . , zn)
of X by means of which V gets identified with ∆r1 ×∆
1 and V \D = ∆
For simplicity suppose that Di ∩ V gets identified with zi = 0, for i = 1, . . . , r
and Di ∩ V = ∅ for i > r. A local analytic frame of ωX(D)|V is
, . . . ,
, dzr+1, . . . , dzn.
Write ‖si‖
2 = |zi|
2hi for i = 1, . . . , r. Then by Theorem 7.12 we can write
ΩKE|V = e
hk log(‖sk‖2)2
‖sk‖2 log(‖sk‖2)2
dzk ∧ dz̄k
where γ is a smooth positive function. Observe also that the functions hk as
well as the second product are smooth positive functions. We are thus reduced
to prove that u and log(log(‖si‖
2)2) are P-singular functions with singularities
along D. On one hand, Lemma 7.11 proves that u is P-singular. On the other
hand, log(log(‖si‖
2)2) is P-singular by Lemma 4.6.
7.3.2 Arithmetic theory
Let K be a number field and X a nonsingular projective variety over K. Let
D ⊆ X be a reduced effective divisor such that D(C) ⊆ X(C) has simple
normal crossings. Suppose that ωX(D) is an ample divisor on X . Then, for
every σ : K →֒ C, ωXσ,C(Dσ,C) is ample and there exists a unique Kähler-
Einstein metric on Xσ(C) \ Dσ(C) with constant negative Ricci curvature -1
(see Theorem 7.4). By Proposition 7.5, these metrics induce good hermitian
metrics on the lines ωXσ,C(Dσ,C), with singularities along Dσ(C), respectively.
The collection of these metrics, for varying σ : K →֒ C, is invariant under the
action of complex conjugation F∞. Indeed, let gKE,σ be the Kähler-Einstein
metric on Xσ(C) \Dσ(C). Then F
∞(gKE,σ) defines a complete Kähler metric
on Xσ(C) \Dσ(C). Let ΩKE,σ be the volume form of gKE,σ, so that F
∞ΩKE,σ
is the volume form of F ∗∞(gKE,σ). Since RicF
∞ΩKE,σ = F
∞ RicΩKE,σ, we find
J(F ∗∞ΩKEσ) = F
∞J(ΩKE,σ) = 1.
By uniqueness we derive F ∗∞(gKE,σ) = gKE,σ. We write ωX(D)KE for the
resulting good hermitian line bundle, with singularities along D.
Let now (X ,L ) be any model of (X,ωX(D)KE) over S = SpecOK . Then
Theorem 1.3 can be applied for any choice of smooth metric ‖ · ‖0 on L . If L
is ample over X , then Corollary 1.4 applies to (X ,L ). In this situation L
verifies the finiteness and the universal lower bound properties.
8 Appendix
The appendix is aimed to prove Proposition 6.11.
Proof of Proposition 6.11. Decompose D into smooth irreducible components,
D = D1 ∪ . . . ∪ Dm. Let D
∗ be the Weil divisor D1 + . . . + Dm. Denote by
F the family of nonsingular subschemes of X consisting of X itself and all the
irreducible components of the intersections
where I runs over all the subsets of {1, . . . ,m}. Since L is ample, there ex-
ists some positive integer N such that L⊗N and L⊗N(−D∗) are very ample.
Consider the exact sequence
0 → L⊗N (−D∗) → L⊗N → L⊗ND → 0.
Taking global sections, we find the exact sequence
(8.1) 0 → Γ(X,L⊗N(−D∗))
→ Γ(X,L⊗N )
→ Γ(X,L⊗ND ).
For every Y ∈ F , Bertini’s theorem ([9], Chapter II, Theorem 8.18) provides us
with a dense open subset UY of the projective space P = P(Γ(X,L
⊗N(−D∗)))
such that, for any t ∈ UY , supp(div t) intersects Y transversally. Since F is
finite and the open subsets UY are dense, the intersection
is a non-empty open subset of P. Therefore we can take t1, . . . , tr ∈ U such
that supp(div t1) ∩ . . . ∩ supp(div tr)=∅. Let t1, . . . , tr ∈ Γ(X,L
⊗N(−D∗)) be
representatives of t1, . . . , tr, respectively. Let s1 = j(t1), . . . , sr = j(tr) be their
images by the morphism j of (8.1). Since p(si) = 0, D ⊆ supp(div si) for all
i = 1, . . . , r. Actually, for every i = 1, . . . , r, we have
div si = div ti +D
Therefore, for the support of div si we find
supp(div si) = supp(div ti) ∪D.
By the choice of the sections ti (ti ∈ U), supp(div si) is a divisor with simple
normal crossings. Finally, since supp(div t1) ∩ . . . ∩ supp(div tr) = ∅, we have
an equality of reduced closed subschemes of X
D = (supp(div s1) ∩ . . . ∩ supp(div sr))red.
The proof of the proposition is complete.
References
[1] J.-B. Bost, H. Gillet and C. Soulé, Heights of projective varieties and pos-
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G. Freixas i Montplet, Département de Mathématiques, Univer-
sité Paris-Sud, Bâtiment 425, 91405 Orsay cedex, France
E-mail address : gerard.freixas@math.u-psud.fr
	Introduction
	Differential forms with logarithmic singularities
	Logarithmically singular hermitian vector bundles
	Global bounds for real log-log growth (1,1)-forms
	Bounding height integrals
	Arakelovian heights
	Examples
	Appendix

  We prove lower bound and finiteness properties for arakelovian heights with
respect to pre-log-log hermitian ample line bundles. These heights were
introduced by Burgos, Kramer and K\"uhn, in their extension of the arithmetic
intersection theory of Gillet and Soul\'e, aimed to deal with hermitian vector
bundles equipped with metrics admitting suitable logarithmic singularities. Our
results generalize the corresponding properties for the heights of
Bost-Gillet-Soul\'e, as well as the properties established by Faltings for
heights of points attached to hermitian line bundles whose metrics have
logarithmic singularities. We also discuss various geometric constructions
where such pre-log-log hermitian ample line bundles naturally arise.

Introduction 2
2 Differential forms with logarithmic singularities 5
3 Logarithmically singular hermitian vector bundles 13
4 Global bounds for real log-log growth (1,1)-forms 17
5 Bounding height integrals 25
6 Arakelovian heights 39
7 Examples 46
8 Appendix 52
Conventions
We fix some conventions and notations to be followed throughout this paper.
The open disc of C centered at 0 and of radius ε > 0 will be denoted by ∆ε.
If f, g : E → R are two functions on a set E 6= ∅, we write f ≺ g to mean
that there exists a constant C such that f(x) ≤ Cg(x) for all x ∈ E. If the
involved constant depends on some data D that we want to specify, we write
f ≺D g.
http://arxiv.org/abs/0704.1046v1
If X is a complex analytic manifold we decompose the exterior differential
operator as d = ∂+ ∂̄ and we define dc = (4πi)−1(∂− ∂̄), so that ddc = i ∂ ∂̄ /2π.
Let k be a field. By an algebraic variety X over k we mean a separated and
reduced scheme of finite type over k. In particular, X is noetherian and it has
a finite number of irreducible components. When k = C, for every separated
scheme of finite type X over C there is an associated complex analytic space
Xan, whose underlying topological space equals the set of complex points X(C).
If F is a closed subscheme of X , then F an is an analytic subspace of Xan. The
scheme X is proper over C if, and only if, Xan is compact. Also, X is a
nonsingular variety over C if, and only if, Xan is a complex analytic manifold.
To simplify notations we will write X instead of Xan or X(C) (it will be clear
from the context the category we are working on).
Let K be a number field. Its ring of integers is denoted by OK . Let S =
SpecOK . An arithmetic variety over S will be a flat and projective scheme
π : X → S , with regular generic fiber XK = X ×S SpecK of pure dimension
n. The set of complex points X (C) has a natural structure of complex analytic
manifold, and it can be partitioned as
X (C) =
σ:K →֒C
Xσ(C).
The complex conjugation induces an antiholomorphic involution F∞ : X (C) →
X (C).
1 Introduction
In this paper we establish a common generalization of the following two state-
ments.
Theorem 1.1 (Faltings [7], Lemma 3). Let X be a projective arithmetic variety
and Y ⊆ X a Zariski closed subset. Let L = (L , ‖ ·‖) be an ample line bundle
on X endowed with a smooth hermitian metric on L |X (C)\Y (C). Suppose that
‖ · ‖ has logarithmic singularities along Y (C). Fix a number field K. For any
real constant C, there are only finitely many points P ∈ X (K) \ Y (K) with
(P ) ≤ C.
The notions of function and metric with logarithmic singularities are recalled
in §2 and §3 below.
Theorem 1.2 (Bost-Gillet-Soulé [1], Proposition 3.2.4 and Theorem 3.2.5).
Let X be a projective arithmetic variety and L a smooth hermitian ample line
bundle on X .
1. For any real constant A, there are only finitely many effective cycles z ∈
Zp(X ) such that deg
z ≤ A and h
(z) ≤ A.
2. There exists a positive constant κ such that h
(z) ≥ −κ deg
z for every
effective cycle z ∈ Zp(X ).
Let us place under the hypothesis of Theorem 1.1. Let P ∈ X (K) \ Y (K),
extended to P : SpecOK → X by properness of X . Since L is ample, there
exists some positive power L ⊗N admitting a global section s non-vanishing at
P . Then
(P ) = log ♯
(L ⊗N )
(sOK)
σ:K →֒C
log ‖s‖σ(Pσ),
where Pσ is the point in Xσ(C) induced by P . Therefore, in Faltings’ result,
only the metric as a function is required. The definition of the height of a
cycle of positive relative dimension, with respect to a smooth hermitian line
bundle, involves the derivatives of the metric up to second order (see [1] §3 and
also §6 below). Therefore, to extend both theorems, we need to describe the
kind of logarithmic singularities allowed to the derivatives of the metric, up to
second order. The arithmetic intersection theory of Gillet and Soulé needs to
be reinterpreted so we can deal with such metrics. This has been done in [2],
where Burgos, Kramer and Kühn develop a theory of abstract arithmetic Chow
groups, and apply it to the case of logarithmic singularities.
Before the statement of our main theorem we introduce some notations. Let
K be a number field, OK its ring of integers and X a projective arithmetic
variety over SpecOK . Suppose that D ⊆ XK is a divisor such that D(C) has
normal crossings. Write U = X (C) \D(C). By Z
U (X ) we denote the group of
codimension p cycles z on X , such that z(C) intersects D(C) properly. A cycle
on X is called horizontal if it is the Zariski closure of a cycle on XK . We write
U (XK) for the subgroup of Z
U (X ) consisting of horizontal cycles. Now let L
be a line bundle on X , endowed with a pre-log-log hermitian metric ‖ · ‖, with
singularities along D (see §3 below for the definition). According to Burgos,
Kramer and Kühn, this is the notion of metric with logarithmic singularities
well suited to define heights on Z
U (X ) (see [2] §7). If LK is ample there is a
well defined normalized height h̃
U (XK) with respect to L = (L , ‖ · ‖).
We refer to §6 for a summary of the theory of heights in Arakelov theory.
Theorem 1.3 (Main Theorem). Let X be a projective arithmetic variety over
SpecOK , D ⊆ XK a reduced divisor such that D(C) ⊆ X (C) has simple
normal crossings and U = X (C) \ D(C). Let L be a line bundle on X with
LK ample. Let ‖ · ‖ be a pre-log-log hermitian metric on L , with singularities
along D, and ‖ · ‖0 a smooth metric on L . Then there exist constants α, β,
γ > 0 and R ∈ Z≥0 such that for every effective cycle z ∈ Z
U (XK) we have
(z) + γ ≥ 1 and
(1.1)
∣∣∣h̃
(z)− h̃
∣∣∣ ≤ α+ β logR
(z) + γ
If moreover ‖ · ‖ is good along D, then we can take R = 1.
In conjunction with Theorem 1.2, Theorem 1.3 immediately yields the de-
sired finiteness property as well as the existence of a universal lower bound.
Corollary 1.4. Let X be a projective arithmetic variety over SpecOK , D ⊆
XK a reduced divisor such that D(C) has simple normal crossings and U =
X (C) \D(C). Let L be a pre-log-log hermitian ample line bundle on X , with
singularities along D.
1. For any real constant A, there are only finitely many effective cycles z ∈
U (X ) such that degLK z ≤ A and hL (z) ≤ A.
2. There exists a positive constant κ such that h
(z) ≥ −κ deg
z for every
effective cycle z ∈ Z
U (X ).
The techniques employed for the proof of the main theorem were initially in-
spired by the work of Carlson and Griffiths on the defect relation for equidimen-
sional holomorphic maps, in higher dimensional Nevanlinna theory [4]. Moreover
(1.1) may be interpreted as a vast generalization of the naive Liouville’s inequal-
ity for the distance between an algebraic number and a rational number. This
point of view is conceptually interesting, since it opens the natural question of
finding an analogue to Roth’s theorem for the distance(1) between a divisor with
simple normal crossings and effective cycles of arbitrary dimension (all defined
over a number field)(2).
This paper is organized as follows.
In Section 2 we review the theory of differential forms with logarithmic sin-
gularities necessary in the rest of the paper. In Section 3 we study in detail
several notions of logarithmically singular hermitian vector bundles. The re-
sults we recall provide a wealth of constructions to which Theorem 1.1 and
Theorem 1.3 apply. Both sections are complemented with examples for a bet-
ter understanding of the theory. In Section 4 we establish global bounds for
real log-log growth (1,1)-forms. An outstanding consequence is a decomposition
theorem (Theorem 4.3) for pre-log-log functions, which plays a crucial role in
the proof of the main theorem. In Section 5 we prove estimates for integrals of
pre-log-log forms appearing in the archimedian part of the definition of height.
This leads to the proof of the main theorem in Section 6, where the reader will
find an overview of the theory of heights in Arakelov geometry. In Section 7
we present some examples of good hermitian line bundles interesting for arith-
metic purposes (for instance in relation with Theorem 1.3). We treat the case
of fully decomposed automorphic vector bundles on locally symmetric varieties,
the relative dualizing sheaf of the universal curve over the moduli space of stable
curves, equipped with the family hyperbolic metric, the Weil-Petersson metric
on the moduli space of curves and the Kähler-Einstein metric on quasi-projective
varieties. Finally, in the Appendix we prove a Bertini’s type theorem needed
for the preliminary reductions in the proof of the main theorem.
Acknowledgements. I am deeply indebted to J.-B. Bost and J. I. Burgos Gil
for proposing me to work on this subject as the starting point of my PhD. thesis,
and for their guidance and constant encouragement. During the preparation of
this paper I benefited from stimulating discussions with several people: I warmly
thank S. Fischler, U. Kühn, V. Maillot, C. Mourougane and D. Roessler for the
time they devoted to me. Their useful advice is reflected throughout the paper.
This work was presented, in a preliminary form, at the Internationales
Graduiertenkolleg Arithmetic and Geometry of the Humboldt-Universität zu
Berlin and the ETH of Zürich, in May 2005. I am grateful to J. Kramer for
inviting me to talk to their summer school.
(1)Suitable candidates for the logarithm of the distance are provided by the height integrals
introduced in §5 below.
(2)The height morphism h
is defined for any pre-log-log hermitian line bundle L , with
singularities along a divisor in X (C). However, for the main theorem to hold, the divisor
needs to be defined over a number field. This essentially goes back to the construction of
transcendental numbers due to Liouville.
2 Differential forms with logarithmic singulari-
Let X be a complex analytic manifold and F a closed analytic subspace. In this
section we introduce several notions of differential forms on X with logarithmic
singularities along F , relevant to our work. We distinguish the case of functions
from the case of differential forms, since the former can be presented in a more
general geometric frame. Indeed, while we can define functions with logarithmic
singularities along an arbitrary closed analytic subspace F , the appropriate
analogues for differential forms of any order require F to be a divisor with
normal crossings.
2.1 Functions with logarithmic singularities
Let X be a complex analytic manifold and F a closed analytic subspace of X .
We denote by IF ⊆ OX the ideal sheaf defining F and supp(F ) for the support
of F . For every x ∈ X , there exist an open neighborhood V and holomorphic
functions s1, . . . , sm ∈ OX(V ) such that
i. the germ of ideal sheaf IF,x is generated by s1, . . . , sm;
ii. the trace of the support of F on V is supp(F ) ∩ V = {z ∈ V | s1(z) =
. . . = sm(z) = 0}.
Since OX is a coherent sheaf, so is IF . Then, given s1, . . . , sm as above, after
possibly shrinking V , s1, . . . , sm generate all the germs IF,z for z ∈ V . In this
case we say that s1, . . . , sm generate IF |V and we write IF |V = (s1, . . . , sm) as
a shortcut. The reader is referred to [6] for further details on analytic spaces.
Definition 2.1. Let X be a complex analytic manifold and F ⊆ X a closed
analytic subspace. A smooth function f : X \ supp(F ) → C has logarithmic sin-
gularities along F if for every open subset V of X such that IF |V = (s1, . . . , sm)
and every relatively compact open subset V ′ ⊂⊂ V , there exists an integer
N ≥ 0 such that
(2.1) |f|V ′\supp(F ) | ≺
∣∣∣∣log( maxi=1,...,m |si|)
For a function f : X \ supp(F ) → C to be with logarithmic singularities
along F it is enough that (2.1) be satisfied for a given open covering of X and
given local generators of IF . This is the content of the next lemma.
Lemma 2.2. Let X be a complex analytic manifold and F a closed analytic
subspace. Fix an open covering {Vα}α of X such that IF |Vα is generated by
sα1 , . . . , s
. Suppose given V ′α ⊂⊂ Vα still forming an open covering of X.
Then a smooth function f : X \ supp(F ) → C has logarithmic singularities
along F if, and only if, for every α there exists an integer N ≥ 0 such that
|f|V ′α\supp(F )
∣∣∣∣log( maxi=1,...,mα
|sαi |)
Proof. Left as an easy exercise.
Lemma 2.3. Let X be a complex analytic manifold and F , G closed analytic
subspaces with supp(F ) = supp(G). A smooth function f : X\supp(F ) → C has
logarithmic singularities along F if, and only if, it has logarithmic singularities
along G.
Proof. Write I, J for the ideal sheaves of F , G, respectively. Let V be an open
neighborhood of x ∈ X such that I|V = (s1, . . . , sl) and J|V = (t1, . . . , tm).
By Hilbert’s Nullstellensatz (see (4.22) in [6]), after possibly shrinking V , there
exists an integer N ≥ 0 such that
sNi =
i tj , t
µijsi.
for some holomorphic functions λ
i , µ
j . Hence, if V
′ ⊆ V is a relatively compact
open subset, there exists a constant C > 0 such that
N ≤ C max
j=1,...,m
|tj |, |tj |
N ≤ C max
i=1,...,l
|si|.
The lemma follows.
The meaning of the lemma is that the notion of function with logarithmic sin-
gularities along F depends only on the support of F .
Proposition 2.4. i. Let X be a complex analytic manifold and F,G closed an-
alytic subspaces with supp(F ) ⊆ supp(G). A smooth function f : X\supp(F ) →
C with logarithmic singularities along F has logarithmic singularities along G.
ii. Let ϕ : X → Y be a morphism of complex analytic manifolds and F ⊆ Y a
closed analytic subspace. If f : Y \ supp(F ) → C has logarithmic singularities
along F , then ϕ∗f = f ◦ ϕ has logarithmic singularities along ϕ−1(F ). If ϕ is
surjective and proper, the converse holds.
Proof. The first item i is straightforward. We shall prove the second part ii.
Let V be an open subset of Y such that IF |V = (s1, . . . , sm). The ideal sheaf
of ϕ−1(F ∩ V ) is (ϕ∗s1, . . . , ϕ
∗sm). Let {Vα}α be an open covering of V by
relatively compact subsets. For every open U ⊂⊂ ϕ−1(V ) define Uα = U ∩
ϕ−1(Vα). Then Uα is relatively compact in ϕ
−1(V ) and ϕ(Uα) ⊆ Vα. The
estimate
|f|Vα\supp(F )| ≺
∣∣∣∣log( maxi=1,...,m |si|)
implies the corresponding inequality for ϕ∗(f) on Uα.
We now prove the converse under the surjectivity and properness assumption.
Let V be as above and V ′ ⊂⊂ V an open subset. Since ϕ is proper, ϕ−1(V ′)
is relatively compact in ϕ−1(V ). The hypothesis of logarithmic singularities of
ϕ∗f asserts the existence of an integer N ≥ 0 such that
|ϕ∗f|ϕ−1(V ′\supp(F ))| ≺
∣∣∣∣log( maxi=1...,m |ϕ
∗si|)
We come up with the conclusion by the surjectivity of ϕ.
2.2 Differential forms with logarithmic singularities
Definition 2.5 (Divisor with normal crossings). Let X be a complex analytic
manifold of dimension n. A reduced analytic subspace D of X is a divisor
with normal crossings if X can be covered by open subsets V with coordinates
z1, . . . , zn such that D ∩ V is defined by z1 · . . . · zm = 0, for some 0 ≤ m ≤ n.
We say that D has simple normal crossings if it can be written as a finite union
of smooth analytic hypersurfaces of X .
Definition 2.6 (Adapted analytic chart [2]). Let X be a complex analytic
manifold of dimension n and D a divisor with normal crossings in X . An
analytic chart (V ; {zi}
i=1) is said to be adapted to D if |zi| < 1/e, i = 1, . . . , n
and D ∩ V is defined by z1 · . . . · zm = 0, for some 0 ≤ m ≤ n. The integer m
will be understood when no confusion can arise.
Notation 2.7. Let X be a complex analytic manifold and D ⊆ X a divisor
with normal crossings. Let (V ; {zi}i) be an analytic chart such that D ∩ V is
defined by z1 · . . . · zm = 0. We define
dζk =
zk log |zk|−1
, if 1 ≤ k ≤ m
dzk, if k > m
and similarly for dζ̄k. Given I, J ordered subsets of {1, . . . , n}, we abbreviate
dζI ∧ dζ̄J =
dζi ∧
dζ̄j .
In the following definitions we write X for a complex analytic manifold of
dimension n, D a divisor with normal crossings in X , U = X \D and ι : U →֒ X
for the natural open immersion. The sheaf of C∞ complex differential forms on
U is denoted by E∗U .
Definition 2.8 (Poincaré growth forms [14]). The sheaf of Poincaré growth
forms on X , with singularities along D, is the subalgebra PD of ι∗E
U gener-
ated, on every open analytic chart (V ; {zi}i) adapted to D, by C
∞(V \D,C) ∩
L∞loc(V,C) and the differential forms dζk, dζ̄k, k = 1, . . . , n. Namely, for every
analytic chart (V ; {zi}i) adapted to D, ι∗E
U (V ) is the C-vector space generated
by differential forms ∑
αI,JdζI ∧ dζ̄J
where αI,J ∈ C
∞(V \D) are locally bounded on V .
Definition 2.9 (Good forms [14]). A good differential form on an open subset
V of X , with singularities along D, is a section ω ∈ Γ(V,PD) such that dω ∈
Γ(V,PD).
Definition 2.10 (log-log growth forms [2]). The sheaf of log-log growth forms
on X , with singularities along D, is the subalgebra LD of ι∗E
U generated, on
every analytic chart (V ; {zi}i) adapted to D, by the functions f ∈ C
∞(V \D,C)
such that on every open V ′ ⊂⊂ V
(2.2) |f(z1, . . . , zn)| ≺
(log log |zk|
for some integer N ≥ 0, together with the differential forms dζk, dζ̄k, k =
1, . . . , n.
Remark 2.11. Observe that the following inequalities hold:
(log log |zk|
−1)N ≤
(log log |zk|
−1)Nm ≤ 2m−1
(log log |zk|
−1)Nm.
Therefore, a smooth function f on V \ D has log-log growth along D if, and
only if, for every open V ′ ⊂⊂ V there is an estimate
(2.3) |f|V ′\D| ≺
(log log |zk|
−1)N .
In concrete computations involving functions with log-log growth, we may use
either formulations (2.2) or (2.3).
Proposition 2.12. Let f : X \D → C be a smooth function. Suppose that df
has log-log growth with singularities along D. Then f has log-log growth with
singularities along D. More precisely, for every analytic chart (V ; {zi}i) adapted
to D and every open V ′ ⊂⊂ V , if df =
j gjdζj +
j hjdζ̄j with
|gj|V ′\D |, |hj|V ′\D | ≺
(log log |zk|
−1)N ,
then we have
|f|V ′\D| ≺
(log log |zi|
−1)N+1
i<j≤m
(log log |zj |
−1)N .
Proof. After localizing to an analytic chart adapted to D we reduce to V =
with V \D = ∆∗r
. Let 0 < ε < 1 and Uε = ∆
be contained in V . Let (w1, . . . , wn) ∈ Uε \D. Fix 1 ≤ i ≤ r. Define the curve
γ : [0, 1] −→ V
t 7−→ (tw1 + (1− t)
e|w1|
, w2, . . . , wn).
Then we have
(f ◦ γ)(1)− (f ◦ γ)(0) =
γ∗(df).
Now we write
γ∗(df|Uε\D) = γ
z1 log |z1|−1
z1 log |z1|−1
where g, h ∈ C∞(Uε \D,C) satisfy
|g|, |h| ≺
(log log |zk|
−1)N on Uε \D.
A straightforward computation yields
df|Uε\D
) ∣∣∣∣
≺f,Uε
1<j≤r
(log log |wj |
−1)N ·
(log log γ∗|z1|
log γ∗|z1|
log γ∗|z1|−1
≤ (log log |w1|
−1)N+1
1<j≤r
(log log |wj |
−1)N .
It follows that
|f(w1, . . . , wn)| ≺f,Uε
∣∣∣∣f
e|w1|
, w2, . . . , wn
)∣∣∣∣
+(log log |w1|
−1)N+1
1<j≤r
(log log |wj |
−1)N .
By induction we find
|f(w1, . . . , wn)| ≺f,Uε sup
|f(z1, . . . zn)|
(log log |wi|
−1)N+1
i<j≤r
(log log |wj |
Observe that the sup is finite because f is smooth on X \ D. The proof is
complete.
Definition 2.13 (pre-log-log forms [2]). The sheaf of pre-log-log forms on X ,
with singularities along D, is the subalgebra E∗X〈〈D〉〉pre of ι∗E
U generated by
log-log growth forms ω in LD such that ∂ ω, ∂̄ ω and ∂ ∂̄ ω are also log-log growth
forms in LD.
A particular case of pre-log-log forms that deserves to be distinguished is
that of P-singular functions.
Definition 2.14 (P-singular function). Let f : X\D → C be a smooth function.
We say that f is a P-singular function, with singularities along D, if, and only
if, df and ddcf have Poincaré growth along D.
Corollary 2.15. Let f : X \D → C be a P-singular function. Then, for every
adapted analytic chart (V ; {zi}i) and every open V
′ ⊂⊂ V , we have
|fV ′\D| ≺
(log log |zk|
Consequently, f is a pre-log-log function.
Proof. This is a straightforward application of Proposition 2.12.
For later computations it will be worth having at our disposal the following
basic properties of log-log growth forms.
Proposition 2.16. i. Any log-log growth form is locally integrable. Moreover,
log-log growth functions and log-log growth 1-forms are locally L2.
ii. (Stokes’ theorem for pre-log-log forms.) If ω ∈ Γ(X, E∗X〈〈D〉〉pre) and [ω]
denotes its associated current, then
d[ω] = [dω]
and similarly for ∂, ∂̄ and ∂ ∂̄.
iii. If f : X → Y is a morphism of complex analytic manifolds and DX ⊆ X,
DY ⊆ Y normal crossing divisors with f
−1(DY ) ⊆ DX , then f
∗PDY ⊆ PDX
and f∗LDY ⊆ LDX . Therefore f
∗E∗Y 〈〈DY 〉〉pre ⊆ E
X〈〈DX〉〉pre. In particular,
this is true for f being the natural inclusion of a complex analytic submanifold
X ⊆ Y intersecting DY transversally and DX = DY ∩X.
Proof. The proposition quotes [2], Proposition 7.5 and Proposition 7.6. How-
ever, for later use, we may comment on the proof of i. After changing to polar
coordinates, it is enough to observe that for every 0 < δ < 1 we have an estimate
∫ ε/e
(log log t−1)N
t(log t)2
∫ ε/e
t(log t−1)1+δ
< +∞.
We finally give an example showing that the notion of pre-log-log function
depends on the compactification X of U .
Example 2.17. Let X = P2
with projective coordinates (w0 : w1 : w2). As
divisor with normal crossings set D = (w0 = 0) ∪ (w1 = 0). Define the smooth
function on U = X \D
g(w0 : w1 : w2) =
|w0|2 + |w1|2
Denote by X̃ the blowing-up of X at (0 : 0 : 1). X̃ admits the following
description:
((w0 : w1 : w2), (z0 : z1)) ∈ X × P
| w0z1 = w1z0
The map realizing the blowing-up is the projection onto the first factor π : X̃ →
X . In particular, since (0 : 0 : 1) ∈ D, we have an isomorphism π−1(U)
→ U ,
and π−1(D) is a divisor with normal crossings. Observe that the pullback of g
by π is
f((w0 : w1 : w2), (z0 : z1)) =
|z0|2 + |z1|2
The function f extends to a smooth function on the whole X̃, in particular pre-
log-log along π−1(D). However, we claim that g is not a pre-log-log function
alongD. To see this we compute ∂ g. We may localize at the affine neighborhood
w2 6= 0 of (0 : 0 : 1) and write u = w0/w2, v = w1/w2. In coordinates u, v we
∂ g =
|u|2|v|2
(|u|2 + |v|2)2
But the function |u|2|v|2 log |u|−1/(|u|2 + |v|2)2 does not have log-log growth
along D, as we see after restriction to |u| = |v|. This proves the claim.
2.3 Variants: log-log forms
Following [3] we introduce a variant of the sheaf of pre-log-log forms, by imposing
bounds on all the derivatives of the component functions of the differential forms.
There are also corresponding variants for Poincaré growth forms and good forms,
for which we refer to loc. cit.
We fix a complex analytic manifold X and D ⊆ X a divisor with normal
crossings. Write U = X \D and ι : U →֒ X for the natural open immersion.
Definition 2.18 (log-log functions of infinite order [3]). A smooth function
f : X \D → C is said to be a log-log function of infinite order, with singularities
along D, if for every analytic chart (V ; {zi}i) adapted toD, every open V
′ ⊂⊂ V
and multi-indices α = (α1, . . . , αn), β = (β1, . . . , βn), there is a bound on V
∣∣∣∣∣
f(z1, . . . , zn)
∣∣∣∣∣ ≺
k=1(log log |zk|
where zα
= zα11 . . . z
m (similarly for z
β≤m) and N depends on V ′, α, β.
Definition 2.19 (log-log growth forms of infinite order [3]). The sheaf of log-log
growth forms of infinite order onX , with singularities along D, is the subalgebra
of ι∗E
U generated, on every analytic chart V adapted to D, by log-log growth
functions of infinite order and the differential forms dζk, dζ̄k, k = 1, . . . , n (see
Notation 2.7).
Remark 2.20. Let ω be a log-log growth form of infinite order alongD, defined
on some analytic open subset of X . Then the complex conjugate ω̄ is also a
log-log growth form of infinite order along D.
Definition 2.21 (log-log forms [3]). The sheaf of log-log forms on X , with
singularities along D, is the subalgebra E∗X〈〈D〉〉 of ι∗E
U generated by log-log
growth forms ω of infinite order, such that ∂ ω, ∂̄ ω and ∂ ∂̄ ω are also log-log
growth forms of infinite order.
Remark 2.22. There is an obvious inclusion E∗X〈〈D〉〉 ⊆ E
X〈〈D〉〉pre.
Log-log growth forms of infinite order enjoy of analogous properties to the
log-log growth forms introduced before. We refer to [3] for details. An advan-
tage of the sheaf of log-log forms over the sheaf of pre-log-log forms is that a
Poincaré’s type lemma holds for the former. The next essential property follows.
Theorem 2.23 ([3]). The natural inclusion
Ω∗X −→ E
X〈〈D〉〉
is a filtered quasi-isomorphism with respect to the Hodge filtration.
Proposition 2.24. Let f : X \ D → C be a smooth function. Then f is a
log-log form, with singularities along D, if, and only if, df is locally L2 on X
and ddcf is a log-log growth form of infinite order along D.
Proof. The direct implication is an easy exercise. Let us see the converse. By
hypothesis, ∂̄ ∂ f is a log-log form, with singularities along D. Let x ∈ X . By
Theorem 2.23, there exists an open neighborhood V of x and a log-log form ω
on V such that
∂̄ ∂ f = ∂̄ ω.
Therefore, we can write
∂ f = ω + θ.
for some holomorphic form θ on V \D. Observe that θ is locally L2, because
∂ f is locally L2 by hypothesis and ω is a log-log 1-form (see Proposition 2.16).
By Lemma 2.25 below, θ must be holomorphic on V . This proves that ∂ f has
log-log growth of infinite order along D. The same reasoning applied to ∂̄ ∂ f̄
(the complex conjugate of ∂ ∂̄ f) proves that ∂̄ f has log-log growth of infinite
order along D. Therefore df has log-log growth of infinite order.
Again by Theorem 2.23, after possibly shrinking V , there exists a log-log func-
tion of infinite order g and a holomorphic function h on V \D such that f = g+h.
We claim that h is locally L2. Since this is true for g, we are reduced to prove
it for f . But we have already shown that df has log-log growth of infinite order
along D, so that Proposition 2.12 implies that f has log-log growth along D. In
particular, f is locally L2 (Proposition 2.16). By Lemma 2.25, g is holomorphic
on V and hence f has log-log growth of infinite order along D. This finishes the
proof.
Lemma 2.25. Let X be a complex analytic manifold and D ⊆ X a divisor with
normal crossings. Let θ be a holomorphic function on X \D. If θ is locally L2
on X, then θ extends to a holomorphic function on X.
Proof. The lemma is well-known, but we include the proof for lack of reference.
It is enough to treat the case whenX = ∆nε ⊆ C
n andD is defined by z1·. . .·zr =
0, so that X \D = ∆∗rε ×∆
ε. We write δ = (δ1, . . . , δr) ∈ R
>0. Since θ is locally
(2.4) ‖θ‖2ε := lim
Iδ < +∞
where
∆ε/2\∆δk )×∆
∣∣∣∣∣
dzk ∧ dz̄k
∣∣∣∣∣ .
The Laurent series development
(2.5) θ(z1, . . . , zn) =
is absolutely and uniformly convergent on any (
k=1 ∆ε/2\∆δk)×∆
. There-
fore, the integral Iδ can be computed term by term:
ν,µ∈Zn
∆ε/2\∆δk
k |dzk ∧ dz̄k|
zνkk z
k |dzk ∧ dz̄k|
Recall that given integers a, b we have
eaiθebiθdθ = 2πδa,b,
so that
|aν |
∆ε/2\∆δk
2νk |dzk ∧ dz̄k|
2νk |dzk ∧ dz̄k|
(2.6)
We reason by contradiction and assume that θ does not extend to a holomorphic
function on ∆n
. We can suppose that in (2.5) there appears a term aνz
ν 6= 0,
with ν = (ν1, . . . , νn) ∈ Z
<0 × Z
≥0 , 1 ≤ l ≤ r. From (2.6) and by direct
computation we find
(2.7) Iδ ≥ (4π)
n|aν |
k=l+1
(ε/2)2νk+2
2νk + 2
δ2νk+2k
2νk + 2
(ε/2)2νk+2
2νk + 2
where
Jδk =
if νk = −1,
(ε/2)2νk+2
2νk+2
2νk+2
2νk+2
if νk < −1.
Since aν 6= 0 and Jδk → +∞ as δ → 0, we see from (2.7) that Iδ → +∞ as
δ → 0. This contradicts (2.4). The proof is complete.
3 Logarithmically singular hermitian vector bun-
Let X be a complex analytic manifold and D ⊆ X a divisor with normal cross-
ings. Write U = X \D and ι : U →֒ X for the natural open immersion. In this
section we study vector bundles endowed with hermitian metrics with singular-
ities of logarithmic type along D. The reader is referred to §2 for the several
definitions and properties of differential forms with singularities of logarithmic
type along D.
Definition 3.1 ([3] and [14]). Let E be a vector bundle of rank r on X . A
smooth hermitian metric h on E|U is said to have logarithmic singularities along
D if, for every trivializing open subset V and holomorphic frame e1, . . . , er of
E|V , putting hij = h(ei, ej) and H = (hij) on V \D, the following condition is
fulfilled:
(L(E, h)) the functions |hij |, detH
−1 have logarithmic singularities along
D ∩ V (see Definition 2.1).
We say that h is (pre-)log-log (resp. good) along D if moreover, for every such
data V and e1, . . . , er, the following property holds:
(G(E, h)) the entries of the matrix (∂ H)H−1 are (pre-)log-log (resp.
good) forms on V , with singularities along D (see Definition 2.9).
We will usually write E = (E, h) when no confusion on the metric can arise.
Sometimes we use some variants of the definition, and we say for instance “E
has logarithmic singularities along D” or “E is (pre-)log-log (resp. good) along
In the case of line bundles, the notions of (pre-)log-log and good hermitian
metrics can be characterized by slightly simpler properties.
Proposition 3.2. Let L be a hermitian line bundle on X and h a smooth
hermitian metric on L|U . Write ‖ · ‖ for the norm associated to h.
i. The metric h is (pre-)log-log (resp. good) with singularities along D if, and
only if, for every trivializing open subset V and holomorphic frame e of L|V , the
function log h(e, e) is (pre-)log-log (resp. P-singular) on V , with singularities
along D.
ii. The metric h is log-log with singularities along D if, and only if, for every
trivializing open subset V and holomorphic frame e of L|V , the form ∂ log h(e, e)
is locally L2 on V and ∂̄ ∂ log h(e, e) has log-log growth of infinite order, with
singularities along D.
Proof. This follows from the definitions and Proposition 2.12, Proposition 2.15
and Proposition 2.24 applied to the smooth real function log h(e, e) on V \D.
An essential extension property of hermitian vector bundles with logarithmic
singularities is the following observation due to Mumford.
Proposition 3.3. Let (E◦, h) be a smooth hermitian vector bundle on U . Then
there exists at most one extension of (E◦, h) to a hermitian vector bundle (E, h)
on X, with logarithmic singularities along D. More precisely, if (E, h) is such
an extension, then for every open subset V in X
Γ(V,E) =
s ∈ Γ(V, ι∗E
◦) | h(s, s) has log. sing. along D ∩ V
Proof. This is Proposition 1.3 in [14].
Hermitian vector bundles with logarithmic singularities along D admit the
following characterization.
Proposition 3.4. Let E be a vector bundle on X and hE a smooth hermitian
metric on E|U . Denote by hE∨ the dual metric. Then hE has logarithmic
singularities along D if, and only if, the following condition is satisfied with
F = E and F = E
(L̃(F, hF )) for every open subset V and any holomorphic section s of F|V ,
the function hF (s, s) has logarithmic singularities along D ∩ V .
Proof. For the direct implication, first take a holomorphic section s of E over
an open subset V . We may assume that s does not vanish on V . After pos-
sibly shrinking V , we can complete s to a holomorphic frame e1 = s, . . . , er
of E|V . By the definition of metric with logarithmic singularities, the function
hE(s, s) = hE(e1, e1) has logarithmic singularities along D.
Secondly, let V be a trivializing open subset of E and e1, . . . , er a holomor-
phic frame of E|V . Write H = (hij) for the matrix of hE in base {ei}i and
H−1 = (gij) for the inverse matrix. From the very construction of H
−1 and
the logarithmic singularities of the functions hij and detH
−1, it is immedi-
ate to check that the functions gij have logarithmic singularities along D. If
B is the matrix of hE∨ in any holomorphic frame of E
, then there exists
A ∈ GLr(Γ(V,OX)) such that
B = At ·H−1 · A.
Since the entries of A are holomorphic, the entries of B inherit from H−1 the
logarithmic singularities along D ∩ V .
Let now s be a holomorphic section of E∨ over an open subset V . Replacing
V by a smaller open subset, we can complete s to a holomorphic frame of E∨
v1 = s, . . . , vr. As we have just proven the functions hE∨(vi, vj) have logarithmic
singularities along D, in particular so does hE∨(s, s) = h(v1, v1).
Now for the converse. Let V be a trivializing open subset, adapted to D. Let
e1, . . . , er be a frame for E|V . Write H = (hij) for the matrix of the hermitian
metric hE in base {ei}i. By hypothesis, for every open subset V
′ ⊂⊂ V , there
exists an integer N ≥ 0 such that on V ′
hE(ei, ei) ≺ (log |z1 · . . . · zm|
−1)N .
Applying Schwarz’s inequality we get
|hij |
2 ≺ (log |z1 · . . . · zm|
−1)2N .
The same argument provides similar bounds for the entries of the matrix of hE∨
in the dual basis, namely H−1. Since the determinant of H−1 is a polynomial
in the entries of this matrix, we derive a bound
detH−1 ≺ (log |z1 · . . . · zm|
for some integer M . This concludes the proof.
As an immediate consequence of the proposition we establish the following
corollary.
Corollary 3.5. Let E = (E, hE) be a hermitian vector bundle with logarithmic
singularities along D. For every exact sequence of vector bundles
0 −→ F −→ E −→ Q −→ 0,
the induced hermitian vector bundles F = (F, hF ) (restricted metric) and Q =
(Q, hQ) (quotient metric) have logarithmic singularities along D.
Proof. It is enough to prove that for every exact sequence as in the statement,
conditions (L̃(F )) and (L̃(Q)) hold. Indeed, since E
has logarithmic singular-
ities along D, conditions (L̃(F
)) and (L̃(Q
)) automatically follow by duality.
Then we conclude applying Proposition 3.4. The validity of L̃(F ) is clear. For
L̃(Q), we just observe that if s is a holomorphic section of Q|V and s̃ is a
holomorphic section of E|V lifting s, then
hQ(s, s) ≤ hE(s̃, s̃).
Thus we see that L̃(E) implies L̃(Q).
We next state the main formal properties of logarithmically singular (resp.
(pre-)log-log, resp. good) hermitian vector bundles.
Proposition 3.6. Let E, F be two vector bundles on X and hE and hF smooth
hermitian metrics on E|U and F|U , respectively. If hE and hF have logarithmic
(resp. (pre-)log-log, resp. good) singularities along D, then E
, E ⊗ F , SkE
and ∧kE have logarithmic (resp. (pre-)log-log, resp. good) singularities along
Proof. Left as an elementary exercise.
Proposition 3.7. Let X, Y be complex analytic manifolds and DX ⊆ X,
DY ⊆ Y normal crossing divisors. Let f : X → Y be a morphism of com-
plex analytic manifolds. Let E = (E, h) be a hermitian vector bundle on Y
whose metric is defined and smooth on Y \DY .
i. If f−1(DY ) ⊆ DX and h has logarithmic (resp. (pre-)log-log, resp. good)
singularities along DY , then the metric f
∗(h) on f∗(E) has logarithmic (resp.
(pre-)log-log, resp. good) singularities along DX .
ii. Suppose that f is surjective, proper and f−1(DY ) = DX . Then h has
logarithmic singularities along DY if, and only if, f
∗(E) has logarithmic singu-
larities along DX .
Proof. The first item i follows from Proposition 2.4 i and Proposition 2.16 iii.
The second item ii is automatically deduced from Proposition 2.4 ii.
Corollary 3.8. Let (E, h) be a hermitian vector bundle on X, with singularities
along D. Let OE(1) be the dual of the trivial vector bundle of P(E), the projective
space of lines in E∨. Denote by π : P(E) → X the natural projection. Then the
metric on OE(1) induced by π
∗(h) has logarithmic singularities along π−1(D).
Proof. By definition, the line bundle OE(1) is a quotient of π
∗(E). The hermi-
tian metric on OE(1) is the quotient metric from π
∗(E). By Proposition 3.7,
π∗(E) has logarithmic singularities along π−1(D). Then, by Corollary 3.5, so
does the induced metric on OE(1).
The end of this section is devoted to some counter-examples.
Example 3.9. i. Counter-example to Corollary 3.5 and Corollary 3.8 for pre-
log-log hermitian vector bundles. Let X = A1
be the complex line, with analytic
coordinate z. Let D be the divisor with normal crossings z = 0. As vector
bundle we take E = O⊕2X . We consider a hermitian metric h on E such that, in
the standard basis e1, e2 and near the origin, its matrix H looks like
log |z|−1 0
It is easily seen that (E, h) is pre-log-log along D (actually good). However, the
induced metric on the line bundle OE(−1) ⊆ π
∗(E∨) on P(E) = P1
is not
pre-log-log. Indeed, identifiy A1
as an open subset of P1
via t 7→ (1 : t). If e∨1 ,
e∨2 is the dual basis, define the section s of OE(−1)|A1
s = e∨1 + te
Then h∨(s, s) = (log |z|−1)−1 + |t|2 and
log h∨(s, s) =
(log |z|−1)−1 + |t|2
If we restrict ∂ log h∨(s, s)/ ∂ t to the set C : t = (log |z|−1)−1/2, we find
log h∨(s, s)
(log |z|−1)1/2
which does not have log-log growth near z = 0.
ii. The notion of hermitian vector bundle with logarithmic singularities depends
on the compactification. Let Y be a smooth complex projective surface. Let p
be a closed point in Y and π : X → Y the blowing-up of Y at p. Let D be a
divisor with normal crossings in Y with p ∈ D. Then π−1(D) is a divisor with
normal crossings. Define U = X \ π−1(D) and V = Y \D. Then π induces an
isomorphism between U and V . Let h be a smooth hermitian metric on ωY |V ,
and endow ωX |U with the induced metric π
∗(h). Assume that (ωX , π
∗(h)) has
logarithmic singularities along π−1(D). Then we claim that h does not define
a metric on ωY with logarithmic singularities along D. Indeed, suppose that
ωY = (ωY , h) had logarithmic singularities along D. Then, by Proposition 3.7,
π∗(ωY ) would have logarithmic singularities along π
−1(D). Observe that
π∗(ωY )|U = ωX |U .
By Proposition 3.3 we would derive the equality π∗(ωY ) = ωX . However we
know that
π∗(ωY ) = ωX ⊗O(−E)
where E is the exceptional divisor π−1(p). Since the self-intersection (E2) = −1,
O(−E) is not trivial. We thus arrive to a contradiction and the claim is proved.
We remark that we can produce such examples just endowing ωX with a smooth
hermitian metric and then restricting it to U .
4 Global bounds for real log-log growth (1,1)-
forms
4.1 Statement of the theorem and consequences
Let X be a complex analytic manifold and D ⊆ X a divisor with simple normal
crossings. DecomposeD into smooth irreducible components,D = D1∪. . .∪Dm.
For every Dk we fix a global section sk of O(Dk) with divisor div sk = Dk. We
endow O(Dk) with a smooth hermitian metric ‖ · ‖k such that ‖sk‖
k ≤ e
Therefore 1 ≤ log log ‖sk‖
−2 ≤ +∞ on X .
Notation 4.1. For every integer N ≥ 0 we define the real positive smooth
function on X \D
(log log ‖sk‖
The purpose of this section is the proof of the following global bounds for
real log-log growth (1,1)-forms on a compact complex analytic manifold.
Theorem 4.2. Suppose that X is compact and let ω be a smooth positive (1,1)-
form on X. Let η be a real log-log growth (1,1)-form on X, with singularities
along D. Then there exist constants A,B > 0 and an integer N ≥ 0 such that
on X \D
(4.1) η +BΘN (dd
c(−Θ1) +Aω) ≥ 0.
If moreover η has Poincaré growth along D, then N can be chosen to be 0.
The proof of the theorem is postponed until §4.3. Now we may discuss a
result appearing as a particular instance of Theorem 4.2.
Theorem 4.3. Suppose that X is compact and let ω be a smooth positive (1,1)-
form on X. Let f : X \D → R be a pre-log-log function, with singularities along
D. Then there exist positive pre-log-log functions, with singularities along D,
ϕ, ψ : X \D −→ R≥0
and constants A,B ≥ 0, N ∈ Z≥0 with the properties
i. f is the difference of ϕ and ψ: f = ϕ− ψ;
ii. the following inequalities hold on X \D:
ωϕ :=dd
c(−ϕ) +BΘN (dd
c(−Θ1) +Aω) ≥ 0,
ωψ :=dd
c(−ψ) +BΘN (dd
c(−Θ1) +Aω) ≥ 0.
If f is P-singular, then N can be chosen to be 0;
iii. if f is P-singular, one can take ϕ, ψ to be P-singular with
ddc(−ϕ) +Aω ≥ 0,
ddc(−ψ) +Aω ≥ 0
on X \D.
Proof. Since X is compact, from the log-log growth of f it is easily seen that
for some constant C > 0 and integer M ≥ 0,
f + CΘM ≥ 0 on X \D.
If f is P-singular, then by Corollary 2.15 we can take (as we do) M ≤ 1. We
define ϕ̃ = f + CΘM and ψ̃ = CΘM . These are positive pre-log-log functions,
with singularities along D (see Lemma 4.7 below). If f is P-singular (hence
M ≤ 1), then ϕ̃, ψ̃ are P-singular (again by Lemma 4.7). By Theorem 4.2 there
exist constants A,B ≥ 0 and N ∈ Z≥0 such that
eϕ := dd
c(−ϕ̃) +BΘN (dd
c(−Θ1) +Aω) ≥ 0
:= ddc(−ψ̃) +BΘN (dd
c(−Θ1) +Aω) ≥ 0
hold on X \D. Hence ϕ = ϕ̃ and ψ = ψ̃ satisfy the requirements of i and ii. If
f is P-singular, then ddc(−ϕ̃), ddc(−ψ̃) have Poincaré growth along D and we
may take N = 0. In this case we have
eϕ =dd
c(−ϕ̃) +mBddc(−Θ1) +mABω
=ddc(−(ϕ̃+mBΘ1)) +mABω
and similarly
= ddc(−(ψ̃ +mBΘ1)) +mABω
In view of these equalities, ϕ = ϕ̃ + mBΘ1 and ψ = ψ̃ + mBΘ1 fulfill the
requirement of iii.
We include the next corollary for its own interest, but we will not need it in
the sequel.
Corollary 4.4. Suppose that X is compact and Kähler. Let ω be a Kähler
form on X. Let f : X \ D → R be a P-singular function and f = ϕ − ψ a
decomposition as in Theorem 4.3 iii. Then the functions
−ϕ,−ψ : X \D −→ R≤0
uniquely extend to quasiplurisubharmonic functions(3) on X.
Proof. First of all, since −ϕ and −ψ are pre-log-log alongD, by Proposition 2.16
we have the equality of currents ddc[−ϕ] = [ddc(−ϕ)] and ddc[−ψ] = [ddc(−ψ)]
on X . The inequalities
ddc[−ϕ] +Aω ≥ 0,
ddc[−ψ] +Aω ≥ 0
(4.2)
then hold on X in the sense of currents. Let U ⊂ X be an open subset diffeo-
morphic to a complex euclidian ball. Because ω is d-closed (Kähler assumption),
by Poincaré’s lemma ω|U is d-exact. Since U itself is Kähler, ω|U is in fact dd
exact. Write ω|U = dd
ch for some smooth function h on U . Then the currents
ddc[−ϕ|U +Ah] and dd
c[−ψ|U +Ah] are positive on U , by (4.2). Since −ϕ and
−ψ are bounded above and D is polar, ϕ̃ := −ϕ|U + Ah, ψ̃ := −ψ|U + Ah
uniquely extend to plurisubharmonic functions on U (see [6], Theorem 5.24).
Since h is smooth, these extensions determine extensions of −ϕ|U and −ψ|U
to quasiplurisubharmonic functions on U , clearly unique. The corollary fol-
lows.
4.2 Construction of pre-log-log functions
4.2.1 Preliminaries
Lemma 4.5. Let M be a complex analytic manifold and α, β, C∞ differential
forms of type (1,0) on M . For every function K > 0 on M , the following
inequality holds:
2Re(iα ∧ β) ≥ −
α ∧ α− iKβ ∧ β.
(3)A quasiplurisubharmonic function on a complex analytic manifold M is an upper semi-
continuous function h : M → [−∞,+∞[ which is locally the sum of a smooth function and a
plurisubharmonic function.
Proof. On one hand, the (1,1)-form
µ = i(α/K1/2 +K1/2β) ∧ (α/K1/2 +K1/2β)
is semi-positive. On the other hand, there is an equality
α ∧ α+ 2Re(iα ∧ β) + iKβ ∧ β.
The lemma follows.
Lemma 4.6. Let L be a line bundle on X. Suppose that s ∈ Γ(X,L) is a global
section such that div s is a divisor with normal crossings. Let ‖ · ‖ be a smooth
hermitian metric on L with ‖s‖2 ≤ e−e. For every integer N ≥ 0, the smooth
function
θN = (log log ‖s‖
−2)N : X \ div s −→ R≥1
is pre-log-log, with singularities along div s. On X \div s the following identities
hold:
∂ θN = NθN−1
∂ log ‖s‖−2
log ‖s‖−2
ddc(−θN ) =
θ1 −N + 1
∂ θN ∧ ∂̄ θN −NθN−1
c1(L)
log ‖s‖−2
provided N ≥ 1. If N = 1, θ1 is P-singular along div s.
Proof. The lemma is trivial for N = 0. We suppose N ≥ 1. First of all we
remark that θN has log-log growth along div s. Next we compute ∂ θN and
ddc(−θN ) on X \ div s. We find
(4.3) ∂ θN = NθN−1
∂ log ‖s‖−2
log ‖s‖−2
ddc(−θN ) = −
∂ ∂̄ θN = −
N ∂ θN−1 ∧
∂̄ log ‖s‖−2
log ‖s‖−2
NθN−1
∂ log ‖s‖−2 ∧ ∂̄ log ‖s‖−2
(log ‖s‖−2)2
NθN−1
∂ ∂̄ log ‖s‖−2
log ‖s‖−2
To simplify the last equality, rearrange terms, use (4.3) and the trivial fact
θN+M = θNθM , and recall that on X \ div s we have c1(L) = dd
c log ‖s‖−2. We
(4.4) ddc(−θN ) =
θ1 −N + 1
∂ θN ∧ ∂̄ θN −NθN−1
c1(L)
log ‖s‖−2
Observe that the quotient (θ1−N+1)/NθN is bounded, so it has log-log growth
along div s. Also the function 1/ log ‖s‖−2 is bounded and c1(L) is smooth, so
that c1(L)/ log ‖s‖
−2 has log-log growth along div s. Hence from (4.3) and (4.4)
we see that it is enough to prove that ∂ θN has log-log growth along div s or,
still, that ∂ log ‖s‖−2/ log ‖s‖−2 has log-log growth along div s.
Let V be an open analytic chart adapted to div s such that L|V can be trivialized
and s = z1 . . . zme, where e is a holomorphic frame of L|V . We can write
∂ log ‖s‖−2
log ‖s‖−2
∂ log ‖e‖−2
log ‖s‖−2
log |zk|
log ‖s‖−2
zk log |zk|−1
The differential form ∂ log ‖e‖−2 is smooth on V and (log ‖s‖−2)−1 is bounded
on V , so that the first term is bounded on any small enough open V ′ ⊂⊂ V . As
for the sum, we observe that log |zk|/ log ‖s‖
−2 is bounded on any small enough
open V ′ ⊂⊂ V , because log ‖s‖−2 = log ‖e‖−2 +
j=1 log |zj|
−2 and log ‖e‖−2
is smooth. Hence ∂ log ‖s‖−2/ log ‖s‖−2 has log-log growth along div s. The
proof is complete.
Lemma 4.7. Under the hypothesis of §4.1 and with the notations therein, the
functions ΘN are pre-log-log, with singularities along D. If N = 1, then Θ1 is
P-singular.
Proof. Write θ
N = (log log ‖sk‖
N , ΘN =
k=1 θ
N and apply Lemma 4.6.
4.2.2 Local results
Let L be a line bundle on X admitting a global section s ∈ Γ(X,L) whose
associated divisor div s is smooth and irreducible. Let ‖·‖ be a smooth hermitian
metric on L with ‖s‖2 ≤ e−e on X . Define, as before, the function
θ1 = log log ‖s‖
−2 : X \ div s −→ R≥1.
By Lemma 4.6 we can write
(4.5) ddc(−θ1) =
∂ log ‖s‖−2 ∧ ∂̄ log ‖s‖−2
(log ‖s‖−2)2
c1(L)
log ‖s‖−2
Let (V ; z1, . . . , zn) be an analytic chart adapted to div s with V ∩div s 6= ∅. We
suppose that L|V can be trivialized and we denote u for a holomorphic frame
such that s = z1u. In local coordinates equality (4.5) becomes dd
c(−θ1) =
α+ β + γ, where
∂ log |z1|
−2 ∧ ∂̄ log |z1|
(log ‖s‖−2)2
dz1 ∧ dz̄1
|z1|2(log |z1|−2)2
β :=− 2Re
∂ log |z1|
−2 ∧ ∂̄ log ‖u‖−2
(log ‖s‖−2)2
=2aRe
dz1 ∧ ∂̄ log ‖u‖
z1(log |z1|−2)2
∂ log ‖u‖2 ∧ ∂̄ log ‖u‖2
(log ‖s‖−2)2
c1(L)
log ‖s‖−2
and a = (log |z1|/ log ‖s‖)
2. Decompose ∂̄ log ‖u‖2 =
j=1 qjdz̄j , where the
functions qj are smooth on V . Then β can be expanded as a sum β =
j=1 βj ,
βj := 2aRe
dz1 ∧ dz̄j
z1(log |z1|−2)2
Proposition 4.8. Let L be a line bundle on X admitting a global section s ∈
Γ(X,L) such that div s is smooth and irreducible. Let ‖·‖ be a smooth hermitian
metric on L with ‖s‖2 ≤ e−e and define θ1 = log log ‖s‖
−2. Let ω be a positive
(1,1)-form on X. For every analytic chart (V ; z1, . . . , zn) adapted to div s and
any open V ′ ⊂⊂ V intersecting div s, there exists A > 0 such that on V ′ \ div s
ddc(−θ1) +Aω ≥
dz1 ∧ dz̄1
|z1|2(log |z1|−2)2
Proof. Without loss of generality, suppose supV | log ‖u‖| and supV |qj | finite for
all j (otherwise, replace V by a relatively compact open subset containing V ′).
We divide the proof into three steps.
Step 1. Observe that because log ‖s‖ = log |z1|+log ‖u‖ and log ‖u‖ is bounded
on V , the function a uniformly tends to 1 as z1 tends to 0. Therefore there
exists an open V ′′ ⊂ V ′ such that 1/2 ≤ a ≤ 2 on V ′′ \ div s. For later need,
we take V ′′ so that V ′ \ V ′′ does not intersect div s. On V ′′ \div s the following
inequality holds:
(4.6) α ≥
dz1 ∧ dz̄1
|z1|2(log |z1|−2)2
Step 2. Define C = maxj supV |qj |. Shrinking V
′′ if necessary, we assume that
|z1| ≤ 1/16n(C + 1) on V
′′. Since a ≤ 2 on V ′′, we have
β1 = 2aRe
dz1 ∧ dz̄1
|z1|2(log |z1|−2)2
≥ −4C|z1|
dz1 ∧ dz̄1
|z1|2(log |z1|−2)2
dz1 ∧ dz̄1
|z1|2(log |z1|−2)2
(4.7)
on V ′′ \ div s. Fix a constant K ≥ 8nC. Applying Lemma 4.5, for every j > 1
we find
(4.8) βj ≥ −2
dz1 ∧ dz̄1
|z1|2(log |z1|−2)2
− 2CK
dzj ∧ dz̄j
(log |z1|−2)2
also on V ′′ \ div s. From inequalities (4.7) and (4.8) we derive that
(4.9) β ≥ −
dz1 ∧ dz̄1
|z1|2(log |z1|−2)2
− 2CK
dzj ∧ dz̄j
(log |z1|−2)2
holds on V ′′ \ div s.
Step 3. To conclude, add up (4.6) and (4.9) to find
(4.10) ddc(−θ1) ≥
dz1 ∧ dz̄1
|z1|2(log |z1|−2)2
− 2CK
dzj ∧ dz̄j
(log |z1|−2)2
on V ′′ \ div s. The last two terms in (4.10) define a smooth differential form on
V \div s, bounded on V ′′ \div s. Hence, there exists a constant A > 0 such that
(4.11) ddc(−θ1) +Aω ≥
dz1 ∧ dz̄1
|z1|2(log |z1|−2)2
holds on V ′′\div s. Because θ1 is smooth away from div s and V ′ \ V ′′ is compact
and disjoint from div s, after possibly increasing A inequality (4.11) holds on
V ′ \ div s as well, as was to be shown.
4.2.3 Global results
We keep the hypothesis and notations of §4.1.
Proposition 4.9. Suppose that X is compact and let ω be a smooth positive
(1,1)-form on X. Let L be a line bundle on X admitting a global section s ∈
Γ(X,L) such that div s is a divisor with normal crossings. Let ‖ · ‖ be a smooth
hermitian metric on L such that ‖s‖2 ≤ e−e. Define θN = (log log ‖s‖
for any N ∈ Z≥0. Then there exists a constant A = A(N) > 0 such that
ddc(−θN ) +Aω ≥ 0 on X \ div s.
Proof. The case N = 0 is trivial. We treat the case N ≥ 1. By Lemma 4.6 the
following identity holds:
ddc(−θN ) =
θ1 −N + 1
∂ θN ∧ ∂̄ θN −NθN−1
c1(L)
log ‖s‖−2
First of all, the function θN−1/ log ‖s‖
−2 is bounded and the differential form
c1(L) is smooth on X . Since X is compact, there exists a constant A > 0 such
(4.12) −NθN−1
c1(L)
log ‖s‖−2
ω ≥ 0 on X \ div s
Still by the compactness hypothesis and by the very definition of θ1, there
exists an open neighborhood V of div s such that θ1|V ≥ N − 1. Moreover
i ∂ θN ∧ ∂̄ θN ≥ 0, so that
(4.13)
θ1 −N + 1
∂ θN ∧ ∂̄ θN ≥ 0 on V \ div s.
Finally, since θM ≥ 1 is smoooth on X \div s for every integerM ≥ 0 and X \V
is compact, after possibly increasing A we have
(4.14)
θ1 −N + 1
∂ θN ∧ ∂̄ θN +
ω ≥ 0 on X \ V.
Equations (4.12), (4.13) and (4.14) together give the desired positivity property.
Corollary 4.10. Suppose that X is compact and let ω be a smooth positive
(1,1)-form on X. For every integer N ≥ 0 there exists A = A(N) > 0 such that
ddc(−ΘN ) +Aω ≥ 0 holds on X \D.
Proof. This follows from Proposition 4.9 by writing ΘN =
k=1 θ
N , with the
notation θ
N = (log log ‖sk‖
The last proposition of this subsection provides a first approach to Theorem
4.2. We may thus place under the hypothesis and notations therein.
Proposition 4.11. Suppose that X is compact and let ω be a smooth positive
(1,1)-form on X. For every finite covering {(Vα; z
1 , . . . , z
n )}α of X by analytic
charts adapted to D, together with relatively compact open subsets V ′α ⊂⊂ Vα
still forming a covering, there exists a constant A > 0 such that
ddc(−Θ1) +Aω ≥
dzαk ∧ dz̄
|zαk |
2(log |zαk |
holds on V ′α \D for every α.
Proof. This follows immediately from Proposition 4.8.
4.3 Proof of Theorem 4.2
We now complete the proof of Theorem 4.2.
Since X is compact, we can choose a finite open covering Vα of X as in
Proposition 4.11. It is enough to prove the existence of constants A,B,N ful-
filling (4.1) on a single V ′α ⊂⊂ Vα. We write {zi}i for the coordinates on Vα,
instead of {zαi }i. Following Notation 2.7, we develop
dζj ∧ dζ̄j +
2Re(hjk
dζj ∧ dζ̄k),
where the functions hjk, j ≤ k, have log-log growth along D ∩ Vα. There exist
a constant C > 0 and an integer N ≥ 0 such that
|hjk|V ′α\D| ≤ CΘN .
Therefore, by Lemma 4.5, on V ′α \D there is a lower bound
η ≥ −CΘN
dζj ∧ dζ̄j − CΘN
dζj ∧ dζ̄j +
dζk ∧ dζ̄k
From this inequality and Proposition 4.11 we see that there exist B ≥ 1 and a
smooth differential form σ on Vα such that
η +BΘN (dd
c(−Θ1) +Aω) ≥ ΘNσ
holds on V ′α \D. After possibly increasing A, we also have σ +Aω ≥ 0 on V
Since B ≥ 1, we finally find
η +BΘN(dd
c(−Θ1) + 2Aω) ≥ 0
on V ′α \D, as was to be shown. Observe that if η has Poincaré growth, then we
can choose N = 0.
5 Bounding height integrals
5.1 Geometric assumptions and statement of the theorem
Let X be a complex analytic manifold and D ⊂ X a divisor with normal cross-
ings. Let L be a line bundle on X and ‖ · ‖ a pre-log-log hermitian metric on L,
with singularities along D (see Definition 3.1). If ‖ · ‖0 is any smooth hermitian
metric on L, then we can write
‖ · ‖ = e−f/2‖ · ‖0,
where f : X \D → R is a pre-log-log function. If ‖ · ‖ is good along D, then f
is P-singular along D. As usual, abbreviate L = (L, ‖ · ‖) and L0 = (L, ‖ · ‖0).
Suppose now that Y ⊂ X is a compact complex analytic submanifold of pure
dimension d. We assume the following conditions are fulfilled:
i. the submanifold Y meets D in a divisor with normal crossings E in Y ;
ii. the restriction ω := c1(L0)|Y is semi-positive and
degL Y :=
ωd > 0;
iii. there exists a global section s ∈ Γ(Y, L) such that ‖s‖20 ≤ e
−e on Y and
div s is a divisor with normal crossings containing E. In particular div s is
reduced, so that we may indistinctly treat div s as a reduced Weil divisor
or a reduced scheme. For every integer N ≥ 0, we define the pre-log-log
function, with singularities along div s,
ℓN = (log log ‖s‖
N : Y \ div s −→ R≥1;
iv. there exist pre-log-log functions, with singularities along E,
Θ1,ΘN , ϕ, ψ : Y \ E −→ R≥0 (N ∈ Z≥0 depending on f)
with f = ϕ− ψ, and bounds
ϕ ≤ CℓM , ψ ≤ CℓM ,Θ1 ≤ Cℓ1,ΘN ≤ CℓN ,
for some constant C ≥ 0 and integer M ≥ 0. Moreover, if f is P-singular,
we suppose that M = 1 and N = 0;
v. there exists A > 0 such that τ := ddc(−Θ1) + Aω ≥ 0, and for every
integer Q ≥ 0 there exists AQ > 0 such that τQ := dd
c(−ℓQ) + AQω ≥ 0.
For Q = 0, we can choose A0 = 1, so that τ0 = ω;
vi. there exists B > 0 such that
ddc(−ϕ) +BΘNτ ≥ 0,
ddc(−ψ) +BΘNτ ≥ 0
hold on Y \E (and so on Y \div s). Observe that by the bounds in iv, we
then have
ωϕ := dd
c(−ϕ) +BCℓNτ ≥ 0,
ωψ := dd
c(−ψ) +BCℓNτ ≥ 0
on Y \ div s.
The aim of this section is to find bounds for the height integrals
Jp :=
f c1(L)
p c1(L0)
d−p, 0 ≤ p ≤ d.
We observe that f c1(L)
p c1(L0)
is a pre-log-log differential form on Y with
singularities along E, hence locally integrable (see Proposition 2.16). In partic-
ular, since E ⊆ div s and div s is Lebesegue negligible, the integrals Jp can be
computed on Y \ div s.
Theorem 5.1. There exist constants α, β > 0, R ∈ Z≥0, depending only on A,
{AN}N , B, C, M and N , such that, for any p ∈ {0, . . . , d},
|Jp| ≤ α degL Y + β · (degL Y ) · log
log ‖s‖−20
c1(L0)
degL Y
If ‖ · ‖ is good along E, then we can take R = 1, so that
|Jp| ≤ α degL Y + β · (degL Y ) · log
log ‖s‖−20
c1(L0)
degL Y
The theorem will be reduced to the bounds claimed by the following two
propositions.
Proposition 5.2. Let σ be a closed, real and semi-positive pre-log-log (t,t)-form
on Y , with singularities along div s. Let a, b be integers such that a+ b+ t = d.
If a, b ≥ 0, define the integral
I(M,a, b, σ) =
Otherwise set I(M,a, b, σ) = 0. Then, if a > 0, the following bound holds
I(M,a, b, σ) ≤ BCI(M +N, a− 1, b, στ)
+ CI(M,a− 1, b, στM )
+ bBC2I(2M,a− 1, b− 1, σττN )
+ (a− 1)BC2I(2M,a− 2, b, σττN ).
If f is P-singular (in which case N = 0 and M = 1), then
I(1, a, b, σ) ≤ ABCI(1, a− 1, b, στ0)
+ (B + 1)CI(1, a− 1, b, στ1).
Similar bounds are true if b > 0, exchanging the role of a and b.
Proposition 5.3. Let σ be a (d, d)-form which is a product of (1,1)-forms of
the kind τ or τQ, Q ≥ 0. Let W be the set of integers Q ≥ 0 such that τQ
appears in σ. Fix an integer K ≥ 0. Then there exist constants α, β > 0 and
an integer R ≥ 0, depending only on A, {AN}N , C, K and W , such that
ℓKσ ≤ α degL Y + β(degL Y ) log
log ‖s‖−20
c1(L0)
degL Y
If K = 1 and σ is a product of differential forms of the kind τ0, τ1 (i.e. W ⊆
{0, 1}), then we can take R = 1.
Assuming for the moment the propositions, we prove Theorem 5.1.
Proof of Theorem 5.1. We first observe that c1(L) = dd
cf + c1(L0), so that
(5.1) Jp =
f(ddcf)jωd−j.
Next, on Y \ div s we write f = ϕ− ψ and ddcf = ωψ − ωϕ. We get
(−1)j−k
ϕωkψω
d−j −
ψωkψω
The coefficients
can be bounded in terms of dimX (hence independently
of Y and the hermitian line bundles). Therefore we are reduced to bound
integrals of the kind ∫
ϕωaϕω
ψωaϕω
for integers a, b, c ≥ 0 such that a + b + c = d. Since the differential forms
c are semi-positive and 0 ≤ ϕ, ψ ≤ CℓM , we have to find upper bounds
for the integrals
I(M,a, b, ωc) =
Successively applying Proposition 5.2, we reduce our problem to find bounds for
integrals
ℓKσ, where σ is a product of (1,1)-forms of type τ or τQ, for some
integers K, Q ≥ 0. If f is P-singular, then K = 1 and σ is a product of forms
of type τ0 and τ1. We conclude by Proposition 5.3.
5.2 Proofs of Proposition 5.2 and Proposition 5.3
We proceed to prove Proposition 5.2 and Proposition 5.3. The proofs make an
extensive use of Stokes’ theorem for pre-log-log differential forms. We refer to
Proposition 2.16 for the statement and references. To simplify the exposition, it
will be worth having at our disposal the computations summarized in the next
lemma.
Lemma 5.4. Let a, b ≥ 0 be integers. On Y \ E the following equalities hold:
∂(ωaϕω
ψ) = aBC(∂ ℓN)ω
ψτ + bBC(∂ ℓN )ω
ψ τ ;
∂ ∂̄(ωaϕω
ψ) =aBC(∂ ∂̄ ℓN)ω
−aBC(∂̄ ℓN ) ∂(ω
+bBC(∂ ∂̄ ℓN )ω
−bBC(∂̄ ℓN ) ∂(ω
ψ )τ.
Proof. It is enough to apply the definition of ωϕ, ωψ, Leibniz’ rule and the fact
that ddc(−ϕ), ddc(−ψ) and τ are ∂ and ∂̄-closed.
Proof of Proposition 5.2. Write I = I(M,a, b, σ). We suppose that a > 0 and
b ≥ 0. We decompose
ωaϕ = dd
c(−ϕ)ωa−1ϕ +BCℓNτω
Accordingly, the integral I decomposes as I = I1 +BCI2, with
c(−ϕ)ωa−1ϕ ω
ℓM+Nω
Bounding I1. To get bounds on I1 we apply Stokes’ theorem for pre-log-log
forms. For this, we first recall that σ is closed of degree (t, t) and ωϕ, ωψ are of
degree (1, 1). Then, by Leibniz’ rule, we compute
∂̄(−ϕ)ωa−1ϕ ω
=ℓMdd
c(−ϕ)ωa−1ϕ ω
+(∂ ℓM )
∂̄(−ϕ)ωa−1ϕ ω
∂̄(−ϕ) ∂(ωa−1ϕ ω
where we used that ddc = i ∂ ∂̄ /2π. By Stokes’ theorem, we find I1 = I1,1+I1,2,
where
I1,1 = −
(∂ ℓM ) ∂̄(−ϕ)ω
I1,2 =
ℓM ∂̄(−ϕ) ∂(ω
Bounding I1,1. Again we apply Stokes’ theorem. By Lebniz’ rule we have
(−ϕ)(∂ ℓM )ω
(∂ ℓM ) ∂̄(−ϕ)ω
(−ϕ)(∂ ∂̄ ℓM )ω
(−ϕ)(∂ ℓM ) ∂̄(ω
Therefore, by Stokes’ theorem, we get I1,1 = I1,1,1 + I1,1,2, where
I1,1,1 =
ϕddc(−ℓM )ω
I1,1,2 =
(−ϕ)(∂ ℓM ) ∂̄(ω
Bounding I1,1,1. Recall that, by assumption, there exists a constant AM > 0
such that τM = dd
c(−ℓM )+AMω ≥ 0. Then, since ω
ψωσ is a semi-positive
form and 0 ≤ ϕ ≤ CℓM , we have a bound
I1,1,1 ≤
ϕ(ddc(−ℓM ) +AMω)ω
=CI(M,a− 1, b, στM ).
(5.2)
Bounding I1,1,2. To bound the integral I1,1,2 we first appeal to Lemma 5.4 to
develop
∂̄(ωa−1ϕ ω
ψ) = (a− 1)BC(∂̄ ℓN )ω
ψτ + bBC(∂̄ ℓN)ω
Then we write I1,1,2 = (a− 1)BCI
1,1,2 + bBCI
1,1,2, where
1,1,2 =
(−ϕ)(∂ ℓM )(∂̄ ℓN )ω
1,1,2 =
(−ϕ)(∂ ℓM )(∂̄ ℓN )ω
ψ στ.
In these integrals we observe that
(5.3) i ∂ ℓM ∧ ∂̄ ℓN = NMℓN+M−2i ∂ ℓ1 ∧ ∂̄ ℓ1,
which is a semi-positive differential form. Since ϕ ≥ 0 and ωϕ, ωψ, σ and τ are
also semi-positive, we find I
1,1,2 ≤ 0 and I
1,1,2 ≤ 0. This shows
(5.4) I1,1,2 = (a− 1)BCI
1,1,2 + bBCI
1,1,2 ≤ 0.
From (5.2) and (5.4) we conclude with the bound
(5.5) I1,1 = I1,1,1 + I1,1,2 ≤ CI(M,a− 1, b, στM ).
Bounding I1,2. As before we proceed by successive applications of Stokes’ the-
orem. First of all we compute
ℓM (−ϕ) ∂(ω
ℓM ∂̄(−ϕ) ∂(ω
(−ϕ)(∂̄ ℓM ) ∂(ω
(−ϕ)ℓM ∂̄ ∂(ω
By Stokes’ theorem we find I1,2 = I1,2,1 + I1,2,2, with
I1,2,1 =
ϕ(∂̄ ℓM ) ∂(ω
I1,2,2 =
(−ϕ)ℓM ∂ ∂̄(ω
Bounding I1,2,1. By Lemma 5.4 we get the expansion
∂(ωa−1ϕ ω
ψ) = (a− 1)BC(∂ ℓN )ω
ψτ + bBC(∂ ℓN)ω
Accordingly, we decompose I1,2,1 = (a− 1)BCI
1,2,1 + bBCI
1,2,1, where
1,2,1 = −
ϕ(∂ ℓN )(∂̄ ℓM )ω
1,2,1 = −
ϕ(∂ ℓN )(∂̄ ℓM )ω
ψ στ.
By (5.3) the differential form i ∂ ℓN ∧ ∂̄ ℓM is semi-positive. Since ϕ ≥ 0 and
ωϕ, ωψ, σ and τ are also semi-positive, we have I
1,2,1, I
1,2,1 ≤ 0. This proves
(5.6) I1,2,1 = (a− 1)BCI
1,2,1 + bBCI
1,2,1 ≤ 0.
Bounding I1,2,2. To bound the integral I1,2,2 we first recall from Lemma 5.4
∂ ∂̄(ωa−1ϕ ω
ψ) = (a− 1)BC(∂ ∂̄ ℓN)ω
− (a− 1)BC(∂̄ ℓN) ∂(ω
+ bBC(∂ ∂̄ ℓN)ω
− bBC(∂̄ ℓN ) ∂(ω
ψ )τ.
Corresponding to this expansion, we write I1,2,2 = (a − 1)BCI
1,2,2 + (a −
1)BCI
1,2,2 + bBCI
1,2,2 + bBCI
1,2,2, with the obvious notations for the inte-
grals I
1,2,2 (see below).
Bounding I
1,2,2. We have
1,2,2 =
ϕℓMdd
c(−ℓN)ω
Recall that for some constant AN > 0 the differential form τN = dd
c(−ℓN) +
ANω is semi-positive. Moreover 0 ≤ ϕ ≤ CℓM . Thus we find the bound
1,2,2 ≤C
ψσττN
=CI(2M,a− 2, b, σττN ).
(5.7)
Bounding I
1,2,2. We write
1,2,2 =
ϕℓM (∂̄ ℓN) ∂(ω
ψ)στ.
By (5.3) and Lemma 5.4, i ∂(ωa−1ϕ ω
ψ) ∧ ∂̄ ℓN is semi-positive, so that
(5.8) I
1,2,2 ≤ 0.
Bounding I
1,2,2. We have
1,2,2 =
ϕℓMdd
c(−ℓN)ω
Reasoning as for I
1,2,2, we arrive to
1,2,2 ≤C
ψ σττN
=CI(2M,a− 1, b− 1, σττN ).
(5.9)
Bounding I
1,2,2. We finally bound the integral
1,2,2 =
ϕℓM (∂̄ ℓN ) ∂(ω
ψ )στ.
Again by (5.3) and Lemma 5.4, the differential form i ∂(ωa−1ϕ ω
ψ ) ∧ (∂̄ ℓN ) is
semi-positive, so that
(5.10) I
1,2,2 ≤ 0.
We now conclude with a bound for I1,2, since the inequalities (5.6)–(5.10) yield
I1,2 ≤ I1,2,2 ≤(a− 1)BC
2I(2M,a− 2, b, σττN )
+bBC2I(2M,a− 1, b− 1, σττN ).
(5.11)
As for I1, the bounds (5.5) and (5.11) lead to
I1 = I1,1 + I1,2 ≤CI(M,a− 1, b, στM )
+bBC2I(2M,a− 1, b− 1, σττN )
+(a− 1)BC2I(2M,a− 2, b, σττN ).
(5.12)
Bounding I2. We have
(5.13) I2 =
ℓM+Nω
ψστ = I(M +N, a− 1, b, στ)
To conclude we put (5.12) and (5.13) together and we get
I = I1 +BCI2 ≤BCI(M +N, a− 1, b, στ)
+CI(M,a− 1, b, στM )
+bBC2I(2M,a− 1, b− 1, σττN )
+(a− 1)BC2I(2M,a− 2, b, σττN ),
as was to be shown.
Suppose now that f is P-singular (so that N = 0 andM = 1). We can write
ωϕ = dd
c(−ϕ̃) +ABCω,
ωψ = dd
c(−ψ̃) +ABCω,
where ϕ̃ = ϕ + BCΘ1 and ψ̃ = ψ + BCΘ1. The same method followed above
allows to establish the bound
I(1, a, b, σ) ≤ ABCI(1, a− 1, b, στ0)
+ (B + 1)CI(1, a− 1, b, στ1).
The details are left to the reader.
Proof of Proposition 5.3. Let σ be a (d, d)-form which is a product of (1,1)-
forms of type τ or τQ, Q ≥ 0. We write σ = τ
sσ1, for some s ≥ 0 and σ1 a
product of forms of type τQ. Define
(5.14) J(K, s, σ1) =
First of all we show how to reduce s to 0. The argument is by induction. If
s > 0, recalling that τ = ddc(−Θ1) +Aω we write
(5.15) τs = ddc(−Θ1)τ
s−1 +Aωτs−1.
Since τ0 = ω, we get from the definition of J in (5.14) and from (5.15)
(5.16) J(K, s, σ1) = AJ(K, s− 1, σ1τ0) +
c(−Θ1)τ
s−1σ1.
We next bound the integral on the right hand side of (5.16). Since τ and σ1 are
∂ and ∂̄-closed, applying Stokes’ theorem for pre-log-log forms we get
(5.17)
c(−Θ1)τ
s−1σ1 =
c(−ℓK)τ
s−1σ1.
Now ddc(−ℓK) = τK −AKτ0, the forms τ0, τK are semi-positive and 0 < Θ1 ≤
Cℓ1, so that from (5.17) and the definition of J we derive
(5.18)
c(−Θ1)τ
s−1σ1 ≤ CJ(1, s− 1, σ1τK).
Observe that because ℓK = ℓ
1 and ℓ1 ≥ 1, the inequality ℓ1 ≤ ℓK holds.
Therefore
(5.19) J(1, s− 1, σ1τK) ≤ J(K, s− 1, σ1τK).
From (5.16)–(5.19) we arrive to
J(K, s, σ1) ≤ AJ(K, s− 1, σ1τ0) + CJ(K, s− 1, σ1τK).
Hence we may suppose that s = 0, so that σ is a product τQ1 . . . τQd . We have
to deal with
L(K,Q1, . . . , Qd) =
ℓKτQ1 . . . τQd .
Again by an inductive argument we show how to reduce all the integers Qi to
0. Suppose that Q1 > 0. Then we write τQ1 = dd
c(−ℓQ1) +AQ1τ0, so that
(5.20) L(K,Q1, . . . , Qd) = AQ1L(K, 0, Q2, . . . , Qd) +
c(−ℓQ1)σ1
where σ1 = τQ2 . . . τQd . We study the integral on the right hand side of (5.20).
Because σ1 is ∂ and ∂̄-closed, by Stokes’ theorem for pre-log-log forms we find
L2 :=
c(−ℓQ1)σ1 =
(∂ ℓK)(∂̄ ℓQ1)σ1
KQ1ℓK+Q1−2(∂ ℓ1)(∂̄ ℓ1)σ1.
(5.21)
By the very definition of ℓ1, we have
(5.22) ∂ ℓ1 ∧ ∂̄ ℓ1 =
∂ log ‖s‖−20 ∧ ∂̄ log ‖s‖
(log ‖s‖−20 )
On the other hand, since ‖s‖20 ≤ e
−e, there exists a constant D depending only
on K +Q1 − 2 such that
(5.23)
ℓK+Q1−2
(log ‖s‖−20 )
(log log ‖s‖−20 )
K+Q1−2
(log ‖s‖−20 )
(log ‖s‖−20 )
Moreover observe that
(5.24)
(log ‖s‖−20 )
∂ log ‖s‖−20 = −2 ∂(log ‖s‖
−1/2.
Because i ∂ log ‖s‖−20 ∧ ∂̄ log ‖s‖
0 and σ1 are semi-positive, combining (5.21)–
(5.24) we get
(5.25) L2 ≤ −2
KQ1D(∂(log ‖s‖
−1/2)(∂̄ log ‖s‖−20 )σ1.
By Lemma 5.5 below, we can apply Stokes’ theorem to the right hand side of
(5.25) and obtain
(5.26) L2 ≤ 2
KQ1D(log ‖s‖
−1/2ωσ1.
By the hypothesis on ‖ · ‖0, (log ‖s‖
−1/2 ≤ 1. Using the positivity of ωσ1,
from (5.26) we derive
(5.27) L2 ≤ 2KQ1D
Now recall that ω is ∂ and ∂̄-closed, and τQ = dd
c(−ℓQ) +AQω, so that
ωσ1 = AQ2 . . . AQdω
d + ddcσ2
for some pre-log-log form σ2. Applied to (5.27) this provides
(5.28) L2 ≤ 2Q1AQ2 . . . AQdKD
ωd = 2Q1AQ2 · . . . · AQdKD degL Y.
The identity (5.20) together with (5.28) imply the inequality
L(K,Q1, . . . , Qd) ≤ AQ1L(K, 0, Q2, . . . , Qd) + 2Q1AQ2 · . . . · AQdKD degL Y.
Successively repeating this argument, we reduce Q2, . . . , Qd to 0. Hence it
remains to treat the integrals
M(K) =
We want to apply Jensen’s inequality. First of all, rewrite
M(K) = (degL Y )
degL Y
so that ωd/ degL Y defines a probability measure on Y . Secondly, the function
x 7→ (log x)K is concave on ]eK−1,+∞[, because
(log x)K =
(log x)K−2(K − 1− log x).
Since ‖s‖−20 ≥ e
e, in particular eK+1 log ‖s‖−20 > e
K−1. We use that logK is an
increasing function and apply Jensen’s inequality:
M(K) ≤(degL Y )
logK(eK+1 log ‖s‖−20 )
degL Y
≤(degL Y ) log
log ‖s‖−20
degL Y
By the trivial inequality x+ y ≤ 2xy for real x, y ≥ 1, we finally arrive to
M(K) ≤ (degL Y )(2K + 2)
K logK
log ‖s‖−20
degL Y
This concludes the proof of the proposition, except for the fact that we can take
R = 1 when K = 1 and σ is a product of forms τ0 and τ1. This last case is
similarly treated and left to the reader.
Lemma 5.5. Let µ be a closed pre-log-log (d-1,d-1)-form on Y , with singulari-
ties along div s. Then we have
(log ‖s‖−20 )
∧ ∂̄ log ‖s‖−20 µ =
(log ‖s‖−20 )
∧ ωµ.
Proof. The proof of the lemma follows the ideas of Lemma 7.36 in [2].
First of all, as for log-log growth differential forms, the form
(log ‖s‖−20 )
∧ ∂̄ log ‖s‖−20 µ = −
(log ‖s‖−20 )
1/2 ∂ ℓ1 ∧ ∂̄ ℓ1µ
is locally integrable on Y . Indeed, after localizing to an analytic chart adapted
to div s and changing to polar coordinates, we are reduced to point out that,
for every 0 < δ < 1/2, we have an estimate
(5.29)
∫ ε/e
(log log t−1)N (log t−1)1/2
t(log t)2
∫ ε/e
t(log t−1)3/2−δ
< +∞.
Define
I = −
(log ‖s‖−20 )
∧ ∂̄ log ‖s‖−20 µ.
We construct a finite open covering {(V ′α; {z
i }i)}α of div s, by adapted analytic
open charts. Suppose that via the coordinates {zαi }α, V
α is identified with
(r = r(α)), so that V ′α \ D corresponds to ∆
further assume that in these coordinates we can write s = zα1 · . . . · z
r uα, where
uα is a holomorphic unit. After possibly adding a finite number of adapted
analytic charts to {(V ′α; {z
i }i)}α, the open subsets Vα ⊂⊂ V
α identified with
via the coordinates {zαi }i still cover div s. Write Ω = ∪αVα. Take
a finite open covering {Vβ}β of X \ Ω, so that Vβ ∩ (div s) = ∅ for all β. Let
{χα}α ∪ {χβ}β be a partition of unity subordinate to {Vα}α ∪ {Vβ}β, with χγ
vanishing outside Vγ for all γ = α, β. We can expand
where
Iγ := −
(log ‖s‖−20 )
∧ ∂̄ log ‖s‖−20 χγµ.
We first treat the integrals Iβ . Observe that for any C
∞ differential form ν on
Y \ div s, the differential form χβν is C
∞ on Y , because χβ vanishes on Y \ Vβ
and Vβ ∩ (div s) = ∅. Moreover the equality d(χβν) = dχβ ∧ ν + χβ ∧ dν holds
on Y . For ν = i ∂ ∂̄ log ‖s‖−20 /2π we find χβν = χβω on Y . These observations,
together with dµ = 0, yield, by Stokes’ theorem,
(log ‖s‖−20 )
(log ‖s‖−20 )
∂̄ log ‖s‖−20 (dχβ)µ,
(5.30)
On the other hand, for every α define Bαε (div s) to be the ε-neighborhood of
div s given by
Bαε (div s) =
Bαε (Tk),
where Tk is the divisor z
k = 0 in Vα and
Bαε (Tk) = ∆
×∆ε ×∆
×∆s1/2e ⊂ Vα.
Then we write Iα = limε→0 Iα,ε, where Iα,ε = I
α,ε + I
α,ε + I
α,ε and
I(1)α,ε :=−
Y \Bαε (div s)
(log ‖s‖−20 )
∂̄ log ‖s‖−20 χαµ
I(2)α,ε :=
Y \Bαε (div s)
(log ‖s‖−20 )
I(3)α,ε :=−
Y \Bαε (div s)
(log ‖s‖−20 )
∂̄ log ‖s‖−20 (dχα)µ.
The differential form in I
α,ε is integrable on Y , since it has log-log growth along
div s (Proposition 2.16). The differential form in I
α,ε is integrable on Y , too.
Indeed, after localizing to an analytic chart adapted to div s and changing to
polar coordinates, we are reduced to prove the convergence of the integrals
(5.31)
∫ 1/e
(log log t−1)N
(log t−1)1/2
(5.32)∫ 1/e
(log log t−1)N
(log t−1)1/2
t log t−1
∫ 1/e
(log log t−1)N
t(log t−1)3/2
From the boundedness of (log log t−1)N/(log t−1)1/2 the convergence of (5.31)
follows. The second one (5.32) has already been treated (5.29). Therefore
limε→0 I
α,ε = I
α and limε→0 I
α,ε = I
α , where
I(2)α :=
(log ‖s‖−20 )
ωχαµ,
I(3)α :=−
(log ‖s‖−20 )
∂̄ log ‖s‖−20 (dχα)µ.
As for the integral I
α,ε, after applying Stokes’ theorem we find
I(1)α,ε =
∂ Bαε (div s)
(log ‖s‖−20 )
∂̄ log ‖s‖−20 χαµ.
Taking into account that χα vanishes on ∂ Vα, we easily see that
(5.33) |I(1)α,ε| ≤
∂∗ Bαε (Tk)
(log ‖s‖−20 )
∣∣∂̄ log ‖s‖−20 χαµ
with the notation
∂∗Bαε (Tk) = ∆
× ∂∆ε ×∆
Observe that ∂∗Bαε (Tk) is fibered in circles over Tk, via the projection
pαk,ε : ∂
∗Bε(Tk) −→ Tk
(zα1 , . . . , z
d ) 7−→ (z
1 , . . . , z
k−1, 0, z
k+1 . . . , z
we will mean integral along the fibers of pαk,ε. We claim that, for every
k = 1, . . . , r,
(5.34) lim
(log ‖s‖−20 )
∣∣∂̄ log ‖s‖−20 χαµ
∣∣ = 0.
Indeed, write s = zα1 . . . z
r uα on V
α, where uα is a holomorphic unit. Since χαµ
is a (d− 1, d− 1) pre-log-log form, the differential form to integrate under
has the shape
f(zα1 , . . . , z
(log log |zαk |
(log ‖s‖−20 )
∣∣∣∣∣∣
dz̄αk
j 6=k
dzαj ∧ dz̄
∣∣∣∣∣∣
where f behaves as follows:
0 ≤ f(zα1 , . . . , z
d ) ≺
j 6=k
(log log |zαj |
|zαj |
2(log |zαj |
We point out that |dz̄αk /z̄
k | is bounded along the fibers of p
k,ε, that the form
j 6=k dz
j ∧ dz̄
∣∣∣ is integrable and (log log |zαk |−1)M/(log ‖s‖
1/2 vanishes
along Tk. This is enough to prove the claim (5.34). Therefore, from (5.33)
limε→0 I
α,ε = 0 and consequently
Iα =I
α + I
(log ‖s‖−20 )
(log ‖s‖−20 )
∂̄ log ‖s‖−20 (dχα)µ.
(5.35)
Finally, since the open covering {Vα}α ∪ {Vβ}β is finite, from (5.30) and (5.35)
we derive
(log ‖s‖−20 )
χγ)ωµ
(log ‖s‖−20 )
∂̄ log ‖s‖−20 d(
χγ)µ.
(5.36)
The assertion of the lemma follows from (5.36) once we note that
γ χγ = 1
and d(
γ χγ)= 0.
5.3 Application to the projective case
We suppose that X is a nonsingular complex projective variety (non necessarily
connected) and D ⊆ X a divisor with simple normal crossings. Consider the
following elements:
i. as in section 4, for every integer N ≥ 0 introduce a function ΘN (see
Notation 4.1). By Lemma 4.7, the functions ΘN are pre-log-log, with
singularities along D;
ii. an ample line bundle L on X , admitting a global section s ∈ Γ(X,L)
such that div s is a divisor with simple normal crossings containing D. In
particular div s is reduced and may be seen as a reduced Weil divisor or a
reduced scheme;
iii. a pre-log-log metric ‖ · ‖ on L, with singularities along D;
iv. a smooth hermitian metric ‖·‖0 on L with ω := c1(L0) > 0 and ‖s‖
0 ≤ e
v. as in §5.1 we introduce ℓQ = (log log ‖s‖
Q, Q ∈ Z≥0. By Lemma 4.6,
the function ℓQ is pre-log-log, with singularities along div s. Moreover,
Proposition 4.9 asserts the existence of a constant AQ > 0 such that
τQ := dd
c(−ℓQ) +AQω ≥ 0. We can take A0 = 1.
We write ‖ · ‖ = e−f/2‖ · ‖0, where f : X \D → R is a pre-log-log function, with
singularities along D. If ‖ · ‖ is good, then f is P-singular. Since X is compact,
associated to ω = c1(L0) there is a decomposition f = ϕ − ψ as in Theorem
4.3. Moreover, because D ⊆ div s, there exist a constant C ≥ 0 and an integer
M ≥ 0 such that
ϕ ≤ CℓM , ψ ≤ CℓM ,Θ1 ≤ Cℓ1,ΘN ≤ CℓN
hold on X \div s. If ‖ · ‖ is good, then we can take M = 1. Finally, by Theorem
4.3 and Proposition 4.9, there exist constants A,B > 0 and an integer N ≥ 0
such that τ := ddc(−Θ1) +Aω ≥ 0 and
ωϕ :=dd
c(−ϕ) +BΘN(dd
c(−Θ1) +Aω) ≥ 0,
ωψ :=dd
c(−ψ) +BΘN (dd
c(−Θ1) +Aω) ≥ 0.
If ‖ · ‖ is good along D, then we can take N = 0. Therefore the assumptions of
§5.1 are fulfilled for X .
Let now π : X ′ → X be a morphism of complex analytic manifolds such that the
inverse image schemes π−1(D) ⊆ π−1(div s) are divisors with normal crossings.
Let Y ⊆ X ′ be a compact complex analytic submanifold of pure dimension d.
Suppose that Y meets π−1(div s) in a divisor with normal crossings in Y . Then
Y ∩D is a divisor with normal crossings in Y , too. We pull-back by π all the
objects introduced above (π−1(D), π∗ΘN , π
∗L, π∗s, π∗‖ · ‖, etc.) and then
we restrict them to Y . We obtain corresponding objects on Y (π−1(D) ∩ Y ,
(π∗ΘN)|Y , (π
∗L)|Y , (π
∗s)|Y , (π
∗‖ · ‖)|Y , etc.). Provided that degπ∗L Y > 0, the
requirements of §5.1 are fulfilled on Y . It is important to point out that the
involved constants A, {AQ}Q, B, C, M , N don’t depend on the data X
′, π, Y .
For every integer 0 ≤ p ≤ d define
J∗p =
(π∗f) c1(π
∗L)p c1(π
As a consequence of Theorem 5.1 we get the following corollary.
Corollary 5.6. Let π : X ′ → X be a morphism of complex analytic manifolds
such that π−1(D) and π−1(div s) are divisors with normal crossings. Let Y ⊆ X ′
be a compact complex analytic submanifold of pure dimension d, intersecting
div s in a divisor with normal crossings in Y . Suppose that degπ∗L Y > 0.
There exist constants α, β > 0 and an integer R ≥ 0, depending only on A,
{AQ}Q, B, C, M and N such that, for all p ∈ {0, . . . , d},
|J∗p | ≤ α degπ∗L Y + β · (degπ∗L Y ) · log
log π∗‖s‖−20
degπ∗L Y
If ‖ · ‖ is good along D, then we can take R = 1:
|J∗p | ≤ α degπ∗L Y + β · (degπ∗L Y ) · log
log π∗‖s‖−20
degπ∗L Y
Let Z be a reduced closed subscheme of X of pure dimension d, intersecting
div s properly. Denote by Z1, . . . , Zr the irreducible components of Z. For every
i = 1, . . . , r, let πi : Xi → X be an imbedded resolution of singularities of Zi,
such that π−1i (div s) is a divisor with normal crossings intersecting the strict
transform Z̃i of Zi in a divisor with normal crossings in Z̃i (see [2], Theorem
7.27). Then Corollary 5.6 applies to πi, Zi, for every i = 1, . . . , r. If θ is a
smooth differential form of degree 2d on X \div s, locally integrable on X , then
we adopt the convention ∫
π∗i θ.
This definition intrinsically depends on Z, and not on the choice of the resolu-
tions πi. With this convention, define
f c1(L)
p c1(L0)
In this situation Corollary 5.6 reads as follows.
Corollary 5.7. Let Z be a reduced closed subscheme of X of pure dimension d,
intersecting div s properly. There exist positive constants α, β, and an integer
R ≥ 0, depending only on A, {AQ}Q, B, C, M and N such that, for all
p ∈ {0, . . . , d},
|Jp| ≤ α degL Z + β · (degL Z) · log
log ‖s‖−20
c1(L0)
degL Z
with R = 1 whenever ‖ · ‖ is good along D.
Proof. Decompose Z into irreducible components: Z = Z1∪ . . .∪Zr. Following
the convention above, define
J (i)p =
f c1(L)
p c1(L0)
so that
(5.37) Jp =
J (i)p .
By Corollary 5.6, there exist constants α, β > 0 and an integer R ≥ 0, depending
only on A, {AQ}Q, B, C, M and N such that, for all p ∈ {0, . . . , d},
(5.38) |J (i)p | ≤ α degL Zi + β · (degL Zi) · log
log ‖s‖−20
c1(L0)
degL Zi
If ‖ · ‖ is good along D, then we can take R = 1. Now recall that the function
logR is increasing and concave on ]eR−1,+∞[, so that
degL Zi
degL Z
log ‖s‖−20
c1(L0)
degL Zi
≤ logR
degL Zi
degL Z
log ‖s‖−20
c1(L0)
degL Zi
= logR
log ‖s‖−20
c1(L0)
degL Z
≤(2R+ 2)R logR
log ‖s‖−20
c1(L0)
degL Z
(5.39)
For the last inequality we used that x + y ≤ 2xy for real x, y ≥ 1. The lemma
follows combining (5.37)–(5.39).
6 Arakelovian heights
In this section we turn to an arithmetic situation and deal with arakelovian
heights on arithmetic varieties. We prove Theorem 1.3, which can be seen as an
arithmetic counterpart of the bounds in §5. A remarkable and straightforward
outcome is the finiteness property of arakelovian heights with respect to pre-log-
log hermitian ample line bundles, as well as the existence of a universal lower
bound (Corollary 1.4).
6.1 Heights attached to pre-log-log hermitian line bundles
Let K be a number field and OK its ring of integers. Write S = SpecOK .
Throughout this section we work with a fixed arithmetic variety π : X → S
of relative dimension n. We recall this means that X is a flat and projective
scheme over S , with regular generic fiber XK = X ×S SpecK of pure di-
mension n. The set of complex points X (C) of X has a natural structure of
complex analytic manifold, and it can be decomposed as
X (C) =
σ:K →֒C
Xσ(C).
Complex conjugation induces an antiholomorphic involution
F∞ : X (C) −→ X (C).
We fixD ⊂ XK a divisor, such that D(C) ⊂ X (C) has simple normal crossings.
Write U = X (C) \D(C).
Notation 6.1 ([2]). We define Z
U (X ) to be the free abelian group generated
by the irreducible reduced subschemes Z ⊆ X of codimension p, such that Z(C)
intersects D(C) properly. We call Z
U (X ) the group of cycles of codimension
p, intersecting D(C) properly. A cycle z is said to be vertical if its components
are supported on closed fibers X℘, ℘ ∈ S \ {(0)}. A cycle z is said to be
horizontal if its irreducible components are flat over S . We denote by Z
U (XK)
the subgroup of Z
U (X ) of horizontal cycles.
Definition 6.2. A pre-log-log hermitian line bundle on X , with singularities
along D, is a couple L = (L , ‖ · ‖) formed by
i. a line bundle (invertible sheaf) L on X ;
ii. a pre-log-log hermitian metric ‖·‖ on the line LC on X (C), with singular-
ities along D(C), and invariant under the action of complex conjugation
F∞: F
∞‖ · ‖ = ‖ · ‖.
In [2], [3], Burgos, Kramer and Kühn attach a height morphism to a pre-
log-log hermitian line bundle L ,
U (X ) −→ R,
generalizing the height morphism for smooth hermitian line bundles introduced
by Bost, Gillet and Soulé in [1]. We refer the reader to the cited bibliography
for the precise definition and basic properties of h
, both in the smooth and
pre-log-log case. For our purposes, it will be enough to state the following
propositions summarizing the main features of h
Proposition 6.3. Let L be a pre-log-log hermitian line bundle on X , with
singularities along D. The height h
satisfies the following three properties:
H1. if PK : SpecK → X is a K-valued point whose image does not belong to
D, and P : S → X denotes its extension to S , then
(P∗S ) = d̂eg(P
L )(4)
(4)The arithmetic degree ddeg of a hermitian line bundle M = (M, ‖ · ‖) over SpecOK is
defined as follows: if s is a non zero global section of M, then ddeg(M) = log ♯
σ:K →֒C log ‖s‖σ .
H2. if z is a vertical cycle supported on a closed fiber X℘, then
(z) = log(N℘) deg
where N℘ denote the norm of the ideal ℘;
H3. let z ∈ Z
n+1−p
U (XK) be irreducible and reduced. Let s be a rational section
of L ⊗N which does not identically vanish on z. If (div(s).z)(C) intersects
D(C) properly, then
(z) = h
(div(s).z)−
log(‖s‖L⊗N ) c1(L )
Proof. This follows from the definition of h
and the extended arithmetic in-
tersection theory in [2].
Remark 6.4. i. The convergence of the integral in H3 is implicit in the state-
ment.
ii. If LK is an ample line bundle, then an easy inductive argument shows that
the properties H1, H2 and H3 actually characterize h
Proposition 6.5. Let L be a line bundle on X , ‖ · ‖ a pre-log-log hermitian
metric on L , with singularities along D, and ‖ · ‖0 a smooth hermitian metric
on L . Write L = (L , ‖ · ‖), L 0 = (L , ‖ · ‖0) and ‖ · ‖ = e
−f/2‖ · ‖0, where
f : X (C) \D(C) → R is a pre-log-log function, with singularities along D(C).
For any cycle z ∈ Z
n+1−p
U (X ) we have
(z) = h
(z) +
f c1(L )
k c1(L 0)
p−1−k.
Proof. This is contained in [3], Theorem 4.1.
Remark 6.6. Proposition 6.5 allows to recover Proposition 6.3 once it is known
for smooth hermitian line bundles. In this case the properties H1, H2 and H3
are already established in [1].
Let F |K be a finite extension of fields and write T = SpecOF . Base
changing by T → S , we get an arithmetic variety XT → T , together with a
finite flat morphism g : XT → X of degree [F : K]. If D ⊆ XK is an effective
divisor such that D(C) ⊆ X (C) has simple normal crossings, then DF ⊆ XF is
an effective divisor and DF (C) ⊆ XT (C) has simple normal crossings as well.
Let L be a pre-log-log hermitian line bundle on X , with singularities along
D. The pull-back g∗L of L to XT is a pre-log-log hermitian line bundle,
with singularities along DF . Since g is flat, for every cycle z on X there
is a well defined pull-back cycle g∗(z). Observe that if z(C) intersects D(C)
properly, then g∗(z)(C) intersects DF (C) properly. Namely, the correspondence
z 7→ g∗(z) induces a morphism
U (X ) −→ Z
V (XT )
z 7−→ g∗(z),
(6.1)
where V = XT (C) \DF (C). This morphism maps Z
U (XK) into Z
V (XF ).
Lemma 6.7. Let L be a pre-log-log hermitian line bundle on X , with singu-
larities along D. Let F |K be a finite extension of fields and T = SpecOF .
Write g : XT → X for the finite flat projection induced by T → S . Let
w ∈ Z
V (XF ) be an irreducible and reduced cycle and set z = g(w)red. Let δ be
the degree of g |w. Then we have the equality
(w) = δh
Consequently, for the morphism g∗ of (6.1), we have
hg∗L (g
∗(z)) = [F : K]h
for every z ∈ Z
U (X ) and every p.
Proof. This follows for instance from the case of smooth metrics (see [1], §3.1.4
and Proposition 3.2.1) and Proposition 6.5, since g|w(C) : w(C) → z(C) is gener-
ically smooth and finite of degree δ, so that
f c1(L )
k c1(L 0)
p−1−k
f c1(L )
k c1(L 0)
p−1−k.
Notation 6.8. Let L be a pre-log-log hermitian line bundle, with singularities
along D. Let z ∈ Z
U (XK) with degLK z 6= 0. We define its normalized height
to be
(z) =
[K : Q] deg
Lemma 6.9. Let g : XT → X be as before. Let w ∈ Z
V (XF ) be an irreducible
and reduced cycle and set z = g(w)red. For the normalized height we have
(w) = h̃
(z) and h̃
(g∗z) = h̃
Proof. For every w ∈ Z
n+1−p
V (XF ) irreducible and reduced and z = g(w)red, we
have the equalities
deg(g∗L )F w =
[F : Q]
L )p−1
[F : Q]
c1(L )
[F : K]
where δ is the degree of g|w. It follows that deg(g∗L )F (g
∗z) = deg
z. Com-
bined with Lemma 6.7, we get h̃g∗L (w) = h̃L (z) and h̃g∗L (g
∗z) = h̃
Remark 6.10. The normalization h̃
just introduced is not the standard one.
However it appears naturally in the statement and proof of Theorem 1.3.
6.2 Proof of the main theorem
We now proceed to prove Theorem 1.3. The argument mainly relies on the
bounds established in §5 , and more concretely the situation studied in §5.3.
However, in the reduction steps we will need the following Bertini’s type theo-
rem. The proof is essentially well known, but we include it in the Appendix for
lack of reference.
Proposition 6.11. Let X be a nonsingular projective scheme over an alge-
braically closed field k. Let D ⊆ X be a divisor with simple normal crossings.
Let L be an ample line bundle on X. Then there exist an integer N > 0 and
global sections s1, . . . , sr ∈ H
0(X,L⊗N ) such that supp(div s1),. . . , supp(div sr)
are divisors with simple normal crossings and the following equality of schemes
holds:
D = (supp(div s1) ∩ . . . ∩ supp(div sr))red.
Under the hypothesis of Theorem 1.3, since D(C) has simple normal cross-
ings, DK has also simple normal crossings in XK . We will apply Proposition
6.11 through the following straightforward corollary.
Corollary 6.12. There exist a finite extension K ′|K, a positive integer N and
global sections s1, . . . , sr ∈ H
0(XK′ ,L
K′ ) such that
B1. supp(div sj)(C) is a divisor with simple normal crossings in X (C), for
every j = 1, . . . , r;
B2. DK′ = (supp(div s1) ∩ . . . ∩ supp(div sr))red.
The next two lemmas provide the final reductions before the proof of Theo-
rem 1.3.
Lemma 6.13. It is enough to proof Theorem 1.3 in the following situations:
i. after some finite extension K ′ | K;
ii. L is very ample and z ∈ Z
U (XK) is irreducible and reduced.
Proof. The first claim i is clear. For the proof of ii, we first note that the
statement of Theorem 1.3 for L ⊗N already implies the statement of the theorem
for L , since h̃
⊗N (z) = Nh̃
(z) and h̃
(z) = Nh̃
(z). Hence we assume
LK is very ample. We then proceed in two steps.
Step 1. We can suppose that L is very ample. Indeed, there exists some model
(Y ,A ) of (XK ,LK) with A very ample. The metrics ‖ · ‖, ‖ · ‖0 on L induce
metrics on A , and we write A and A 0 for the corresponding hermitian line
bundles. For every effective cycle z ∈ Z
U (XK), we write z̃ for the corresponding
effective and horizontal cycle on Y . By Proposition 3.2.2 in [1], there exists a
positive constant C, independent of z, such that
(6.2)
(z̃)− h
∣∣ ≤ C deg
Since A C and L C are isometric, Proposition 6.5 and (6.2) together give
(z̃)− h
(z̃)− h
∣∣ ≤ C deg
Step 2. We can suppose that z is irreducible and reduced. Indeed, suppose that
we have shown the existence of constants α, β, γ > 0 and R ∈ Z≥0 such that,
for every w ∈ Z
U (XK) irreducible and reduced, we have h̃L 0(w) + γ ≥ 1 and
(6.3)
∣∣∣h̃
(w)− h̃
∣∣∣ ≤ α+ β logR
(w) + γ
After possibly increasing γ we can suppose that h̃
(w) + γ > eR−1, for every
irreducible and reduced w ∈ Z
U (XK). Let us now consider w =
i∈I wi ∈
U (XK) \ {0} where the wi ∈ Z
U (XK) are irreducible and reduced. Then (6.3)
yields
∣∣∣h̃
(w) − h̃
∣∣∣ ≤
∣∣∣h̃
(wi)− h̃L 0(wi)
≤ α+ β
(wi) + γ
Since h̃
(wi) + γ > e
R−1 for all i and logR is concave on ]eR−1,+∞[, we
conclude
(wi) + γ
≤ logR
(wi) + γ)
= logR
(w) + γ
This completes the proof.
By Corollary 6.12 and Lemma 6.13, after possibly extending K and choosing
a suitable model of (XK ,LK), we can suppose that L is very ample and there
exist sections s1, . . . , sr ∈ H
0(XK ,LK) with the properties B1, B2 above (with
K ′ = K). After possibly multiplying the sections sj by a sufficiently divisible
integer, we can even suppose that s1, . . . , sr ∈ H
0(X ,L ). We denote Ej =
supp(div sj)K . We fix these data until the end of the proof.
Lemma 6.14. Let z ∈ Z
U (XK) be irreducible and reduced. Let F be a finite
extension of K over which all the irreducible components of zK are defined.
Let T = SpecOF and g : XT → X be the finite flat projection induced by
T → S . Write g∗(z) =
i∈I zi, zi irreducible, reduced and flat over T . Then
z(C) intersects D(C) properly if, and only if, every zi(C) intersects one of the
Ej,F (C) properly.
Proof. Straightforward.
Now we can complete the proof of Theorem 1.3.
Proof of Theorem 1.3. First of all, for every integer M ≥ 0 we construct a
function ΘM , as in §4, for the complex analytic variety X (C) and the divisor
with simple normal crossings D(C).
Observe that we can suppose that the metric ‖ · ‖0 satisfies c1(L 0) > 0 and
0 ≤ e
−e for every j = 1, . . . , r. Indeed, by [1], Proposition 3.2.2 (or also
Proposition 6.5 for smooth metrics), a change of smooth metric causes only
bounded variations of the normalized height.
We introduce the pre-log-log functions
Q = (log log ‖sj‖
Q : X (C) \ Ej(C) −→ R.
For everyQ we fix a positive constantAQ > 0 such that dd
Q )+AQ c1(L 0) ≥
0, A0 = 1, for all j = 1, . . . , r (see Proposition 4.9).
Write ‖ · ‖ = e−f/2‖ · ‖0, where f : X (C) \ D(C) → R is a pre-log-log
function (resp. P-singular if ‖ · ‖ is good). Attached to c1(L 0) we perform
a decomposition f = ϕ − ψ as in Theorem 4.3. Recall that ϕ, ψ are positive
pre-log-log (resp. P-singular) functions along D(C), with
ωϕ :=dd
c(−ϕ) +BΘN(dd
c(−Θ1) +A c1(L 0)) ≥ 0,
ωψ :=dd
c(−ψ) + BΘN(dd
c(−Θ1) +A c1(L 0)) ≥ 0,
for some A,B > 0 and N ∈ Z≥0. If ‖ · ‖ is good, then we can take N = 1.
For every j = 1, . . . , r, D ⊆ Ej . By compactness of X (C) there exist constants
C > 0, M ∈ Z≥0 such that
ϕ ≤ Cℓ
M , ψ ≤ Cℓ
M ,Θ1 ≤ Cℓ
1 ,ΘN ≤ Cℓ
for all j ∈ {1, . . . , r}.
Let z ∈ Z
n+1−p
U (XK) be irreducible and reduced. Denote by F an extension
of K such that all the irreducible components of zK are defined over F . Let
T = SpecOF , g : XT → X be as before. Decompose g
∗(z) =
i∈I zi, with
the zi irreducible and flat over T . By Lemma 6.14, for every zi there exists
j = j(i) ∈ {0, . . . , r} such that zi(C) intersects Ej,F (C) properly. By Corollary
5.7 and Proposition 6.5 we have
(6.4)
∣∣∣h̃g∗L (zi)− h̃g∗L 0(zi)
∣∣∣ ≤ α+β logR
zi(C)
log g∗‖sj‖
[K : Q] deg(g∗L )F zi
for α, β > 0, R ∈ Z≥0 depending only on A, {AQ}Q, B, C,M and N . Moreover,
if ‖ · ‖ is good, then we can take R = 1. Applying the property H3 of heights
(see Proposition 6.3), we rewrite (6.4) as
(6.5)∣∣∣h̃g∗L (zi)− h̃g∗L 0(zi)
∣∣∣ ≤ α+ β logR
g∗L 0
(zi)− 2h̃g∗L 0(div(g
∗sj).zi)
To derive this inequality we point out that
deg(g∗L )F (div(g
∗sj).zi) = deg(g∗L )F zi,
so that
hg∗L 0(div(g
∗sj).zi)
[F : Q] deg(g∗L )F zi
= h̃g∗L 0(div(g
∗sj).zi).
By Theorem 1.2 there exists a positive constant κ > 0 such that, for every effec-
tive cycle w 6= 0 on X , we have h̃
(w) > −κ. Combined with Lemma 6.9 this
yields h̃g∗L 0(div(g
∗sj).zi) > −κ, because div(g
∗sj).zi is effective. Therefore,
from (6.5) we deduce
∣∣∣h̃g∗L (zi)− h̃g∗L 0(zi)
∣∣∣ ≤α+ β logR
2h̃g∗L 0(zi) + 2κ+ 2e
≤α+ 2Rβ logR
h̃g∗L 0(zi) + κ+ e
(6.6)
where we applied the trivial inequalities log 2 ≤ 1 and x + y ≤ 2xy for real
x, y ≥ 1. Now h̃g∗L 0(zi)+κ+ e
R+1 > eR−1 and logR is concave on ]eR−1,+∞[.
From (6.6) we derive
∣∣∣h̃g∗L (g
∗z)−h̃g∗L 0(g
deg(g∗L )F (zi)
deg(g∗L )F (g
∣∣∣h̃g∗L (zi)− h̃g∗L 0(zi)
≤α+ 2Rβ logR
deg(g∗L )F (zi)
deg(g∗L )F (g
h̃g∗L 0(zi) + κ+ e
=α+ 2Rβ logR
h̃g∗L 0(g
∗z) + κ+ eR+1
(6.7)
By Lemma 6.9, h̃
(g∗z) = h̃
(z) and h̃
g∗L 0
(g∗z) = h̃
(z). Hence (6.7) is
equivalent to
(6.8)
∣∣∣h̃
(z)− h̃
∣∣∣ ≤ α+ 2Rβ logR
(z) + κ+ eR+1
The constants α, 2Rβ, γ := κ + eR+1 > 0, R ∈ Z≥0 (R = 1 if ‖ · ‖ is good) in
(6.8) depend only on L and L 0, and not on z. This concludes the proof of the
theorem.
7 Examples
7.1 Automorphic vector bundles on toroidal compactifi-
cations
The first natural examples of good hermitian vector bundles are provided by the
theory of (fully decomposed) automorphic vector bundles on locally symmetric
varieties, and their extensions to smooth toroidal compactifications. These have
been firstly worked by Mumford in his proof of Hirzebruch’s proportionality
principle in the non-compact case [14]. In this section we quote from loc. cit.
the main construction and Mumford’s theorem. As an application, we briefly
discuss the example of modular forms.
Let B be a bounded symmetric domain. We can write B = G/K, where
G is a semi-simple adjoint group and K is a maximal compact subgroup. De-
note KC, GC the complexifications of K and G. Inside GC there is a parabolic
subgroup of the form P+ · KC (P+ being its unipotent radical), such that
K = G∩ (P+ ·KC) and G · (P+ ·KC) is open in GC. Then B̌ := GC/G · (P+ ·KC)
is a rational projective variety and there is a G-equivariant immersion B →֒ B̌
compatible with the complex structure of B. Let E0 be a G-equivariant vector
bundle on B attached to a representation σ : K → GLr(C). We complexify
σ and extend it to P+ · KC, by letting it act trivially on P+. This extension
induces a GC equivariant analytic vector bundle Ě0 on B̌, with ι
∗(Ě0) = E0.
This way we get a holomorphic structure on E0 (which depends on the chosen
extension of σ to P+ ·KC).
Let Γ be a neat arithmetic subgroup of G acting on B. Then X = Γ\B is a
smooth quasi-projective complex variety. The vector bundle E0 descends to a
holomorphic vector bundle E on X . Such a vector bundle is called fully decom-
posed automorphic vector bundle. Since K is compact, there is a G-invariant
hermitian metric h0 on E0, thus inducing a hermitian metric h on E.
Theorem 7.1. Let X be a smooth toroidal compactification of X with D =
X \X a divisor with normal crossings. Then the automorphic vector bundle E
extends to a vector bundle E1 over X, such that h is good along D.
The following proposition may be interesting for some arithmetic purposes.
Proposition 7.2. Suppose that E0 = ωB is the canonical bundle of B. Then
E1 = ωX(D) and coincides with the pull-back of an ample line bundle O(1) on
the Baily-Borel compactification X∗ of X. The global sections of O(n) naturally
correspond to the modular forms whose automorphy factor is the nth power of
the jacobian.
Remark 7.3. Under the hypothesis of the proposition, the line bundle ωX(D)
is not ample in general.
Equip the line bundle E0 = ωB with the hermitian metric h0 induced by the
Kähler-Einstein metric on B, say with Einstein constant −1. The existence and
uniqueness is guaranteed by a result of Mok and Yau [13]. Since the Kähler-
Einstein metric is invariant under automorphisms, h0 is G-equivariant. By The-
orem 7.1 h0 induces a good hermitian metric h on E1 = ωX(D), with singulari-
ties along D. Observe that this metric is induced by the Kähler-Einstein metric
on X with Einstein constant −1, by uniqueness. Since the Kähler-Einstein met-
ric has negative Ricci curvature, c1(ωX(D), h) ≥ 0 on X . Together with the
fact that ωX(D) is the pull-back of an ample line bundle O(1) on X
∗, this can
be shown to be enough for the main theorem to hold, as soon as X and X
are defined over a number field K and we have chosen suitable models X of
X, X ∗ of X∗, etc. over SpecOK . Suppose that O(1) extends to an ample
line bundle A on X ∗, that there is a morphism π : X → X ∗ extending the
natural projection X → X∗ and put L = π∗(A). The line bundle L is a model
of ωX(D) and it can be endowed with the good hermitian metric induced by
h. Then Corollary 1.4 hold for L , provided we restrict to effective horizontal
cycles. The proof follows the same lines as for pre-log-log hermitian ample line
bundles, and it will be detailed elsewhere.
7.2 Some natural hermitian vector bundles on the moduli
space of curves
Let g ≥ 2 be an integer and Mg the moduli space of complex stable curves of
genus g. We denote by π : Cg → Mg the universal curve. By Mg we mean the
open subset of Mg parametrizing smooth curves, and we write Cg = π
−1(Mg).
The boundary ∂Mg = Mg \ Mg, which classifies singular stable curves, is a
divisor with normal crossings. We write ∂ Cg = π
−1(∂Mg), which is a divisor
with normal crossings, too. For the sake of simplicity we neglect that Mg and
Cg are actually V -manifolds, and we work as if they were complex analytic
manifolds (see [17] for the definition of V -manifold and the description of the
moduli space of curves as a V -manifold).
The first example concerns ωCg/Mg , the relative dualizing sheaf of π. Every
fiber of π|Cg admits a unique complete hyperbolic metric of constant negative
curvature −1. By Teichmüller’s theory these metrics glue together and define
a smooth hermitian metric on ω∨
Cg/Mg
. We get a smooth hermitian metric on
ωCg/Mg . By a theorem of Wolpert [18] this metric extends to a good hermitian
metric on ωCg/Mg , with singularities along ∂ Cg. It is well known that ωCg/Mg
is relatively ample ([5], Corollary to Theorem 1.2).
Let us now consider (ΩMg , hWP ) the cotangent bundle of Mg with the Weil-
Petersson metric. Recall that if p is a point of Mg representing a Riemann
surface R, then ΩMg ,p is isomorphic to the space of holomorphic quadratic
differentials on R. If R is written as H/Γ (H Poincaré’s upper half plane and
Γ ⊆ PSL2(R) a discrete subgroup), then the metric hWP on the stalk ΩMg ,p is
the usual scalar product
〈ϕ, ψ〉 =
ϕ(z)ψ(z)y2dxdy
on automorphic forms of weight (2,0) for the group Γ. By a result of Trapani [16],
(detΩMg , dethWP ) extends to a good hermitian line bundle ωMg (log ∂Mg),
with singularities along ∂Mg. Moreover Trapani shows that ωMg (log ∂Mg)
admits a smooth hermitian metric with positive curvature form. Therefore, its
pull-back to the moduli space of curves of genus g with level n structure (n ≥ 3)
is ample.
In an ambitious program pioneered by [11], [12], Liu, Sun and Yau study
the goodness and bounded geometry of several natural Kähler metrics on the
moduli space of curves. The interested reader is referred to loc. cit. for precise
statements.
7.3 Kähler-Einstein metrics on quasi-projective varieties
7.3.1 Complex theory
The main references we follow are [10], [15], and [19].
Let M be a complex analytic manifold of dimension n and Ω a smooth
positive (n, n)-form on M , namely a volume form. For every analytic chart
(V ; z1, . . . , zn) of M write
Ω|V = ξV
dzk ∧ dz̄k
The functions {ξV }V define a smooth hermitian metric ‖ · ‖Ω on the canonical
line bundle ωM . By definition, the Ricci form of Ω is the real (1,1)-form
RicΩ = c1(ωM , ‖ · ‖Ω)
which is locally given by
RicΩ |V= dd
c log ξV .
The generalized Fefferman operator J acting on volume forms is defined as
J : Ω 7−→
(RicΩ)n
Theorem 7.4 (Kobayashi [10]). Let X be a compact complex analytic manifold
and D ⊆ X a divisor with simple normal crossings. Suppose that the line bundle
ωX(D) is ample on X. Then there exists a unique complete Kähler-Einstein
metric gKE on X \D with constant negative Ricci curvature -1. If ΩKE is the
volume form of gKE, then gKE is characterized by being complete on X \D and
by the equation
J(ΩKE) = 1.
Proposition 7.5. Let X be a compact complex analytic manifold and D ⊆
X a divisor with simple normal crossings. Suppose that ωX(D) is ample on
X. Let U = X \ D and endow ωU with the smooth hermitian metric ‖ · ‖KE
induced by ΩKE. Then (ωU , ‖ · ‖KE) extends to a good hermitian line bundle
(ωX(D), ‖ · ‖KE), with singularities along D.
The proof of Proposition 7.5 follows easily from the more precise growth
properties established in the proof of Theorem 7.4. For the sake of completeness
we now deepen some of the details involved. In the sequel we fix X and D as
in the proposition.
Definition 7.6. Let M be a complex analytic manifold of dimension n. Let
V ⊆ Cn be an open subset. A holomorphic map φ : V → M is called a
quasicoordinate if rank(dvφ) = n for every v ∈ V . Then (V, φ) is called a local
quasicoordinate of M .
Let x ∈ D and (V = V (x); z1, . . . , zn) be an analytic chart of X centered at
x, by means of which V gets identified with ∆r1 ×∆
1 and V \D with ∆
(r = r(x)). For every η ∈ (0, 1)r, define the quasicoordinate
φη : Vη = (∆3/4)
r ×∆s1 −→ V \D
v = (v1, . . . , vn) 7−→ (φη,1(v), . . . , φη,n(v))
where
φη,k(v) =
if 1 ≤ k ≤ r,
vk if k > r.
Observe that
η∈(0,1)r
φη(Vη).
We now construct a quasicoordinate covering V = {(Vβ , φβ)}β , containing ex-
actly:
• all the quasicoordinates {(Vη, φη)}η∈(0,1)r , for (V = V (x); z1, . . . , zn),
x ∈ D, as above. Denote by W the union of the images of all these
quasicoordinates. This is an open neighborhood of D;
• a finite coordinate covering of the compact subset X \W .
We introduce the Hölder space of Ck,α-functions on U = X \D, with respect to
the quasicoordinate covering V .
Definition 7.7 (Hölder spaces). Let k ≥ 0 be an integer and α ∈ (0, 1). The
Ck,α-norm with respect to V of a function u ∈ Ck(U) is
|u‖k,α,V = sup
(V,φ)∈V
‖φ∗(u)‖k,α
= sup
(V,φ)∈V
|p|+|q|≤k
∣∣∣∣∣
∂|p|+|q|
∂ vp ∂ vq
φ∗(u)(v)
∣∣∣∣∣
+ sup
v,v′∈V
|p|+|q|=k
|v − v′|−α
∣∣∣∣∣
∂|p|+|q|
∂ vp ∂ vq
φ∗(u)(v)
|p|+|q|
∂ vp ∂ vq
φ∗(u)(v′)
∣∣∣∣∣
We define the space of Ck,α-functions on U as
Ck,α(U) = {u ∈ Ck(U) | ‖u‖k,α,V < +∞},
which is seen to be a Banach space with respect to the norm ‖ · ‖k,α,V .
Definition 7.8. We define Rr,s(U) as the space of (r, s)-differential forms ω on
U such that, for all quasicoordinate (V, φ) ∈ V ,
φ∗(ω) =
|p|=r
|q|=s
(apdv
p + bqdv
‖ap‖k,α, ‖bq‖k,α < +∞
for all multi-indices p, q with |p| = r, |q| = s and all k ≥ 0, α ∈ (0, 1).
If (v1, . . . , vn) are the standard coordinates on V ⊂ C
n and p = (i1, . . . , ir),
q = (j1, . . . , js) are ordered multi-indices, we wrote
dvp = dvi1 ∧ . . . ∧ dvir , dv
q = dvj1 ∧ . . . ∧ dvjs .
Lemma 7.9. i. Rr,s(U) is a C-vector space;
ii. Rr,s(U) ∧Rr
′,s′(U) ⊆ Rr+r
′,s+s′(U);
iii. ∂ Rr,s(U) ⊆ Rr+1,s(U) and ∂̄ Rr,s(U) ⊆ Rr,s+1(U).
Proof. Immediate to check from the definition of Rr,s(U).
Recall that PD denotes the sheaf of Poincaré forms on X with singularities
along D (Definition 2.8).
Lemma 7.10. We have an inclusion Rr,s(U) ⊆ Γ(X,PD)
(r,s), where the su-
perscript stands for the (r, s) part with respect to the usual bigrading of complex
differential forms.
Proof. We localize near the divisor D and consider a quasicoordinate φη : Vη →
V \ D in V . Hence V has coordinates z1, . . . , zn and V \ D is the divisor
z1 . . . zr = 0. For simplicity we consider the differential form
ω = h
z1 log(|z1|−1)
∧ . . . ∧
zr log(|zr|−1)
∧ dzr+1 ∧ . . . dzn
and suppose that φ∗η(ω) has finite C
k,α-norm for all k ≥ 0 and α ∈ (0, 1). We
have to prove that h is bounded on the image of φη. From the definition of φη,
a straightforward computation shows that
φ∗η(ω) = φ
1− |vi|2
|vi − 1|
(vi − 1)2
dv1 ∧ . . . ∧ dvn.
The hypothesis implies the coefficient of dv1 ∧ . . .∧ dvn has bounded sup-norm.
Since |vi| ≤ 3/4 for i = 1, . . . , r, this immediately yields the boundedness of
φ∗(h).
Corollary 7.11. Let u : U = X \D → C be a smooth function. If u ∈ R0,0(U),
then u is a P-singular function with singularities along D.
Proof. By Lemma 7.9, du ∈ R1,0(U)⊕R0,1(U) and ddcu ∈ R1,1(U). By Lemma
7.10, du and ddcu have Poincaré growth with singularities along D. Hence u is
a P-singular function with singularities along D.
Theorem 7.12 (Kobayashi [10]). Let X be a compact complex analytic manifold
and D ⊆ X a divisor with simple normal crossings. Suppose that ωX(D) is
ample on X. Let D1, . . . , Dm be the irreducible components of D and si ∈
Γ(X,O(Di)) sections with div si = Di, for all i = 1, . . . ,m. Let Ω be a volume
form on X. There exist suitable smooth hermitian metrics on the line bundles
O(Di), that we write ‖ · ‖ for simplicity, with ‖si‖ small enough, and a function
u ∈ R0,0(U) such that the volume form ΩKE of the Kähler-Einstein metric on
ΩKE = e
u Ω∏m
i=1 ‖si‖
2 log(‖si‖2)2
With Theorem 7.12 at hand, we can prove Proposition 7.5.
Proof of Proposition 7.5. We may localize at an analytic chart (V ; z1, . . . , zn)
of X by means of which V gets identified with ∆r1 ×∆
1 and V \D = ∆
For simplicity suppose that Di ∩ V gets identified with zi = 0, for i = 1, . . . , r
and Di ∩ V = ∅ for i > r. A local analytic frame of ωX(D)|V is
, . . . ,
, dzr+1, . . . , dzn.
Write ‖si‖
2 = |zi|
2hi for i = 1, . . . , r. Then by Theorem 7.12 we can write
ΩKE|V = e
hk log(‖sk‖2)2
‖sk‖2 log(‖sk‖2)2
dzk ∧ dz̄k
where γ is a smooth positive function. Observe also that the functions hk as
well as the second product are smooth positive functions. We are thus reduced
to prove that u and log(log(‖si‖
2)2) are P-singular functions with singularities
along D. On one hand, Lemma 7.11 proves that u is P-singular. On the other
hand, log(log(‖si‖
2)2) is P-singular by Lemma 4.6.
7.3.2 Arithmetic theory
Let K be a number field and X a nonsingular projective variety over K. Let
D ⊆ X be a reduced effective divisor such that D(C) ⊆ X(C) has simple
normal crossings. Suppose that ωX(D) is an ample divisor on X . Then, for
every σ : K →֒ C, ωXσ,C(Dσ,C) is ample and there exists a unique Kähler-
Einstein metric on Xσ(C) \ Dσ(C) with constant negative Ricci curvature -1
(see Theorem 7.4). By Proposition 7.5, these metrics induce good hermitian
metrics on the lines ωXσ,C(Dσ,C), with singularities along Dσ(C), respectively.
The collection of these metrics, for varying σ : K →֒ C, is invariant under the
action of complex conjugation F∞. Indeed, let gKE,σ be the Kähler-Einstein
metric on Xσ(C) \Dσ(C). Then F
∞(gKE,σ) defines a complete Kähler metric
on Xσ(C) \Dσ(C). Let ΩKE,σ be the volume form of gKE,σ, so that F
∞ΩKE,σ
is the volume form of F ∗∞(gKE,σ). Since RicF
∞ΩKE,σ = F
∞ RicΩKE,σ, we find
J(F ∗∞ΩKEσ) = F
∞J(ΩKE,σ) = 1.
By uniqueness we derive F ∗∞(gKE,σ) = gKE,σ. We write ωX(D)KE for the
resulting good hermitian line bundle, with singularities along D.
Let now (X ,L ) be any model of (X,ωX(D)KE) over S = SpecOK . Then
Theorem 1.3 can be applied for any choice of smooth metric ‖ · ‖0 on L . If L
is ample over X , then Corollary 1.4 applies to (X ,L ). In this situation L
verifies the finiteness and the universal lower bound properties.
8 Appendix
The appendix is aimed to prove Proposition 6.11.
Proof of Proposition 6.11. Decompose D into smooth irreducible components,
D = D1 ∪ . . . ∪ Dm. Let D
∗ be the Weil divisor D1 + . . . + Dm. Denote by
F the family of nonsingular subschemes of X consisting of X itself and all the
irreducible components of the intersections
where I runs over all the subsets of {1, . . . ,m}. Since L is ample, there ex-
ists some positive integer N such that L⊗N and L⊗N(−D∗) are very ample.
Consider the exact sequence
0 → L⊗N (−D∗) → L⊗N → L⊗ND → 0.
Taking global sections, we find the exact sequence
(8.1) 0 → Γ(X,L⊗N(−D∗))
→ Γ(X,L⊗N )
→ Γ(X,L⊗ND ).
For every Y ∈ F , Bertini’s theorem ([9], Chapter II, Theorem 8.18) provides us
with a dense open subset UY of the projective space P = P(Γ(X,L
⊗N(−D∗)))
such that, for any t ∈ UY , supp(div t) intersects Y transversally. Since F is
finite and the open subsets UY are dense, the intersection
is a non-empty open subset of P. Therefore we can take t1, . . . , tr ∈ U such
that supp(div t1) ∩ . . . ∩ supp(div tr)=∅. Let t1, . . . , tr ∈ Γ(X,L
⊗N(−D∗)) be
representatives of t1, . . . , tr, respectively. Let s1 = j(t1), . . . , sr = j(tr) be their
images by the morphism j of (8.1). Since p(si) = 0, D ⊆ supp(div si) for all
i = 1, . . . , r. Actually, for every i = 1, . . . , r, we have
div si = div ti +D
Therefore, for the support of div si we find
supp(div si) = supp(div ti) ∪D.
By the choice of the sections ti (ti ∈ U), supp(div si) is a divisor with simple
normal crossings. Finally, since supp(div t1) ∩ . . . ∩ supp(div tr) = ∅, we have
an equality of reduced closed subschemes of X
D = (supp(div s1) ∩ . . . ∩ supp(div sr))red.
The proof of the proposition is complete.
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G. Freixas i Montplet, Département de Mathématiques, Univer-
sité Paris-Sud, Bâtiment 425, 91405 Orsay cedex, France
E-mail address : gerard.freixas@math.u-psud.fr
	Introduction
	Differential forms with logarithmic singularities
	Logarithmically singular hermitian vector bundles
	Global bounds for real log-log growth (1,1)-forms
	Bounding height integrals
	Arakelovian heights
	Examples
	Appendix
