Small points and Berkovich metrics
Xinyi Yuan, Shou-Wu Zhang
September 6, 2010
Contents
1 Berkovich space and metrics
2
2 Integrable metrics
6
3 Arithmetic intersections and heights
10
4 Small points and Equidistribution
15
5 Heights for dynamical systems
24
Introduction
The aim of this paper is to develop an intersection theory for metrized line
bundles on varieties over finitely generated fields in terms of Berkovich spaces
and apply it to equidistribution problems of small points.
In §1, we review the basic definition of Berkovich space introduced by
[Be]. We consider more general situation than Berkovich. We also introduce
continuous and norm-equivariant metrics for line bundles, and show that at
so called enclosed points such metric exists uniquely.
In §2, we define our integrable line bundles as limits of line bundles on
integral models over Z with certain topology. We define intersections of these
line bundles with values in §3 as limits of either Gillet–Soulé intersections or
Deligne pairings [GS, GS2, De].
In §4, we study the notion of small points, propose an equidistribution
conjecture, and prove an equidistribution result (Theorem 4.7) about the
1
specialization of small points. Finally, in §5 we apply our theory to polarized
dynamical systems over finitely generated fields, and show that the equidistribution conjecture implies the equidistribution of Galois orbits of preperiodic
points.
The equidistrbution conjecture is known to be true either in number field
case by Yuan [Yu] or the function field case by Faber [Fab] and Gubler [Gu2].
The equidistribution result in Theorem 4.7 is crucial in the consideration of
preperiodic points in another recent paper [YZ] of the authors.
While our work is a continuation of previous works [Zh2, SUZ, Yu] on
varieties over number field but has been inspired by Gubler [Gu1], Moriwaki
[Mo2], and Chambert-Loir [Ch]. The work of Gubler and Chambert-Loir
convince us to work on Berkovich spaces, while Moriwaki’s work suggests a
way to take limits and a Northcott property. The difference between our
treatment and Moriwaki’s is that he uses polarization on the base while we
don’t. Our theory is compatible with base changes, but his theory is not.
1
Berkovich space and metrics
Berkovich spaces
Let X be a scheme and denote by X an the associated Berkovich analytic
space. The definition in [Be] works in this general setting. If X is covered by
affine schemes SpecA then X an as a set is covered by the affinoid (SpecA)an of
multiplicative seminorms on A. For each x ∈ (SpecA)an , the corresponding
norm is a composition
|·|
A −→ κx −→ R
where κx is a residue field and | · | is a valuation (multiplicative norm) on
κx → R. We write the first map as f 7→ f (x). Then the image of f under the
composition is written as |f (x)|, and we also write it as |f |x for convenience.
Then the multiplicative seminorm (on an affinoid) corresponding to x ∈ X an
is just | · |x . The topology on X an is the weakest one such that all (SpecA)an
are open, and such that the function |f (x)| is continuous on X an for all f . If
X is separated then X an is Hausdorff.
We have a continuous map X an → X by sending x to the kernel of the
norm in SpecA. The fiber is simply the the set of norms on the residue fields.
We also have a natural section X ,→ X an by sending a point x to the trivial
norm the residue field κx .
2
Two elements x and y of X an are called norm-equivalent if they have the
same image in X and are induced by equivalent norms on the residue field.
Thus there are numbers a, b > 0 such that | · |ax = | · |by locally. The set
of norm-equivalence classes of in X an is called the set of places of X and is
denoted as M(X). Most of our theory depends only on M(X) despite it
does not have a good topology as X an does.
One important example is (SpecZ)an . It is a tree with vertices V :=
SpecZ ∪ {∞} and the edges Ep for each p ∈ V \ {0} corresponding to norms
| · |tp with t ≥ 0 if p 6= ∞ and 0 ≤ t ≤ 1 if p = ∞. The space (Spec Q)an is
identified with the complement of V \ {0}. The set of places of SpecQ will
be the union of {0} and open parts of Ep . We write (SpecZ)an
p = Ep .
If f : X → Y is a morphism of schemes, then we get a morphism
f an : X an → Y an . If f is proper, so is f an ; if f has connected fibers, then
f an has arcwise connected fibers. If T is a subset of Y an then we can define
T -adic analytic space by base change
fTan :
XTan −→ T.
When Y = Speck and T = {v} corresponds to a valuation of k, we write
XTan as Xvan . If v is the trivial norm, then we simply write it as Xkan .
If X is integral with function field K, then (SpecK)an is identified with
the set of generic points on X an , i.e., the preimage of the generic point in X.
If X is a projective variety over a field k, then for any valuation v of k, Xvan
gives an algebraic compactifiction for (SpecK)an
v .
an
an
Denote X∞
:= X(SpecZ)
,
which
is
the
archimedean
part of X an . Since
an
∞
every archimedean norm of a field k is given by a complex conjugacy class of
embeddings k → C and a norm | · |t (t ∈ (0, 1]) on C, we see that
an
X∞
= (0, 1] × X(C)/complex conjugation.
Here the second factor can be identified with the set of closed points in the
scheme XR .
Enclosed points
For any non-archimedean point x ∈ X an , let C(x) denote the set of the points
norm-equivalent to x. If x ∈
/ X, then we have a natural parameterization
(0, ∞) −→ C(x),
3
t 7→ xt
given by | · |xt = | · |tx . It is clear that C(x) has a limit point x0 ∈ X given by
the trivial norm on κ(x), and thus it is just the image of x in X.
We say that x is enclosed if the limit x∞ ∈ X exists as t → ∞, i.e., the
morphism x
e : Specκ → X can be extended into a morphism SpecRx → X
where Rx is the subring of elements in κx with norm ≤ 1. It is clear that
either x∞ = x if x ∈ X, or x∞ is given a divisor in the Zariski closure x0
of x0 . In this way, x∞ represents the trivial norm on Rx /mx . Conversely,
for any point x0 ∈ X, any divisor x∞ in the Zariski closure x0 defines an
equivalent class of enclosed points.
Let Ẍ an denote the subset of X an consisting of enclosed points. Then
Ẍ an contains X and is called the enclosed locus of X an . For an affine space,
an
¨
(SpecA)
is the set of bounded seminorms on A. Here are some examples
of enclosed locus:
an
¨
• It is easy to have (SpecZ)
= (SpecZ)an .
¨
• If K is a field, then SpecK
is just the point given by the trivial norm.
an
¨
• For a discrete valuation ring R, the enclosed locus (SpecR)
consists
of the line segment described as above.
Proposition 1.1. Let X → Y be a morphism of schemes. Then the induced
map X an → Y an sends enclosed elements to enclosed elements:
Ẍ an −→ Ÿ an .
Moreover, if this morphism is proper, then a point in X an is enclosed if and
only if it has a enclosed image, i.e.,
Ẍ an = XŸanan .
Proof. The first assertion is clear. For the second assertion, use the properness criterion.
If X is an integral scheme with function field K, then Spec(K)an
is the
Ẍ
set of norms induced from divisors of X.
4
Berkovich metrics
Let X be a scheme and L be a line bundle on X. It induces a line bundle Lan
on X an . At each point x ∈ X an , the residue Lan (x) is the same as the residue
of L on the image of x in X. By a Berkovich metric k · k on L we mean a
metric on Lan compatible with norms on OX . More precisely, to each point
x in X an with residue field κ we assign a norm k · kx on the κ-line Lan (x)
which is compatible with the norm of κ in the sense that
kf `kx = |f | · k`kx ,
f ∈ κ,
` ∈ Lan (x).
We say that the metric k · k on L is continuous if for any section ` of Lan
on an open subset U of X an , the function k`(x)k = k`(x)kx is continuous
on x ∈ U . We say that the metric is norm-equivariant if for any two normequivalent points x and y with | · |ax = | · |by , one has k · kax = k · kby .
We denote by Pic(X) (resp. Pic(X)) the Picard group of isomorphism
c an ) (resp.
classes (resp. the category ) of line bundles on X. Let Pic(X
an
Pic(X )) denote the group of isometry classes (resp. the category) of line
bundles on X endowed with continuous and norm-equivariant metrics. Thus
we have a morphism
c an ) −→ Pic(X),
Pic(X
d an ) −→ Pic(X).
Pic(X
The fibers are homogeneous spaces of the group of metrics on OX .
If s is a rational section of a metrized line bundle with divisor D, then
− log ksk define a Green’s function for the divisor Dan in the sense that this
function is continuous and norm-equivariant and has logarithmic singularity
near D. Conversely, given any divisor D and a Green’s function g on X an
with logarithmic singularity along Dan in X an , then e−g defines a metric
d an )
on O(Dan ). The pair (D, g) is called an arithmetic divisor. Let Div(X
denote the group of arithmetic divisors. Then we have just described a
homomorphism
d an ) −→ Pic(X
c an ).
Div(X
Let φ : X → Y be a morphism of schemes and M be a line bundle on
Y with a metric on Y an . Then one has a metric on φ∗ M defined by the
pull-back map.
Analogously, for a morphism X → Y and a subset T of Y an , we can define
T -adic metrics on L by only assigning the metrics on the fibers of Lan above
XTan . In the case T = Ẍ an is the enclosed locus of X an , we have the following
result.
5
Proposition 1.2. Let L be a line bundle on X. Then L has a unique continuous and norm-equivariant Berkovich metric on Ẍ an .
Proof. First let us prove the existence. Let x ∈ Ẍ an be a enclosed point
with ring of integers Rx and maximal ideal mx . Then we have a morphism
SpecRx → X extending the image of x in X. The pull-back of L on SpecRx is
a locally free module M of rank one on Rx , so it is free. We define the metric
on M so that the generator gets value 1. This is well-defined as different
generator differs by an element in Rx× which has norm 1. We left the reader
to check the continuity and the norm-equivariance.
For the uniqueness, assume that we have two continuous and normequivariant metrics k · k and k · k0 of L on Ẍ an . The we have a continuous
function
k · kx
.
φ : Ẍ an −→ R, x 7−→ log
k · k0x
It is norm-equivariant in the sense that for any two norm-equivalent points
x and y with | · |ax = | · |by , one has
aφ(x) = bφ(y).
We need to prove φ = 0 identically.
For any x ∈ Ẍ an and t ∈ (0, ∞), the point xt ∈ Ẍ an given by | · |xt = | · |tx
is norm-equivalent to x by definition. Thus we have
φ(xt ) = tφ(x).
The continuity of φ at x∞ as t → ∞ implies that φ(x) = 0.
Since
an
X∞
= (0, 1] × X(C)/complex conjugation,
an
an archimedean metric of a line bundle L on X∞
is exactly given by a continuous hermitian metric on X(C) invariant under the complex conjugation.
2
Integrable metrics
Conventions on arithmetic models
By a projective arithmetic variety (resp. open arithmetic variety) we mean
an integral scheme, projective (resp. quasi-projective) and flat over Z.
6
By a projective model of an open arithmetic variety U we mean an open
embedding U ,→ X into a projective arithmetic variety X such that the
complement X \ U is the support of an effective Cartier divisor.
Let K be a finitely generated field over Q. By a projective arithmetic
model (resp. open arithmetic model) of K we mean a projective arithmetic
variety (resp. open arithmetic variety) with function field K.
Let K be a finitely generated field over Q and X be a projective variety
over K. By a projective arithmetic model (resp. open arithmetic model) of
X/K we mean a projective and flat morphism X → B where:
• B is a projective arithmetic model (resp. open arithmetic model) of K;
• The generic fiber of X → B is X → SpecK.
Model metrics
Let K be a finitely generated field over Q, X a projective variety over K,
and L a line bundle on X.
Let X → B be a projective model of X/K, and L be a Hermitian line
bundle on X with generic fiber LK isomorphic to L. This model induces a
metric Lan on X an , and thus a metric of Lan on X an . We call the metric of
Lan the model metric induced by L and denote it by k · kL . Thus we have
defined morphisms:
d ) −→ Div(X
d an ),
Div(X
c ) −→ Pic(X
c an ),
Pic(X
d ) −→ Pic(X
d an ).
Pic(X
It is clear that these are injective since they are injective if we replace X an
by X an and since X an is dense in X an .
The unions of the images of these objects over all models (X /B, L) are
denoted by
d an )mod ,
Div(X
c an )mod ,
Pic(X
d an )mod .
Pic(X
D-topology
Let U be an open arithmetic variety. Projective models X of U form a
projective system. Using pull-back morphisms, we define
d
d ),
Div(U)
:= lim
Div(X
−→
X
d
d ),
Pic(U)
:= lim
Pic(X
−→
X
7
c
c ).
Pic(U)
:= lim
Pic(X
−→
X
d
As usual, for each divisor E ∈ Div(U)
we can construct an arithmetic line
d
bundle O(E) in Pic(U).
d ) is
On a projective model X , an arithmetic divisor D = (D, g) ∈ Div(X
effective (resp. strictly effective) if D is an effective (resp. strictly effective)
divisor on X and the Green’s function g ≥ 0 (resp. g > 0) on X (C) − |D(C)|.
d )Q is called effective (resp. strictly
Consequently, a Q-divisor D ∈ Div(X
effective) if for some positive integer n, the multiple nD is an effective (resp.
d )Q , we write
d ). For any D, E ∈ Div(X
strictly effective) divisor in Div(X
d )Q , and
D > E or E < D if D − E is effective. It is a partial order on Div(X
compatible with pull-back morphisms. Thus we can talk about effectivity
d
and the induced partial ordering on Div(U).
Fix a projective model X and an effective Cartier divisor D with support
d
X \ U. It induces a topology on Div(U)
Q as follows. Let g be any Green’s
d ) is strictly
function of D such that the arithmetic divisor D = (D, g) ∈ Div(X
effective. Then a neighborhood basis at 0 is given by
d
d
B(, Div(U)
Q ) := {E ∈ Div(U)Q : −D < E < D},
∈ Q>0 .
By translation, it gives a neighborhood basis at any point. The topology
0
d 0 ) is
does not depend on the choice of X and D. In fact, if D ∈ Div(X
0
another effective divisor with support X \ U, then we can find a third model
0
dominating both X and X 0 . Then we can find r > 1 such that r−1 D < D <
rD.
d
d
c
Let Div(U)
adlc be the completion of Div(U)Q and Pr(U)adlc be the completion of the principal divisors. The group of adelic line bundles on U is
defined to be
c
d
c
Pic(U)
adlc = Div(U)adlc /Pr(U)adlc .
c
Alternatively, we can define Pic(U)
adlc to be the group of isomorphism classes
d
d
of objects in the completion Pic(U)adlc of of Pic(U)
Q with respect to the Dtopology defined by the following neighborhood of the trivial line bundle:
n
o
d
d
d
B(, Pic(U)
)
:=
L
∈
Pic(U)
:
L
'
O(Z)
for
some
Z
∈
B(,
Div(U)
)
.
Q
Q
Q
c adlc is contravariant for projective morphisms.
The functor Pic(·)
Proposition 2.1. The natural morphisms
d
d an ),
Div(U)
−→ Div(U
c
c an ),
Pic(U)
−→ Pic(U
8
d an )
Pic(U) −→ Pic(U
can be extended continuously into morphisms
d
d an
Div(U)
adlc −→ Div(U )Q ,
c
c an
Pic(U)
adlc −→ Pic(U )Q ,
d an )Q .
Pic(U)adlc −→ Pic(U
d
d an
Proof. The image of B(, Div(U)
Q ) in Div(U ) is the space of continuous
d an ) is certainly complete with refunctions bounded by gDan . Thus Div(U
spect to this topology. This gives the extension of the first map. The other
two follows from this one.
c
c
Remark 2.2. The map Pic(U)
Q → Pic(U)ms is injective. Equivalently, the
c
quotient D-topology on Pic(U)
Q is separable. Namely, there is no non-zero
c
element E ∈ Pic(U) such that −D < E < D for any > 0. In fact,
assume that it is true. We can assume that D and E are realized on the same
projective model X . Denote n = dim U − 1. Then for any ample line bundles
c ), we have
L1 , · · · , Ln in Pic(X
(D ± E) · L1 · · · Ln > 0,
∀ > 0.
It follows that
E · L1 · · · Ln = 0.
Then E = 0 by the Hodge index theorem for arithmetical divisors in [Mo1].
Remark 2.3. In [Fal], the definition of the Faltings height of an abelian variety
uses the Petersson metric on the Hodge bundle of the (open) Siegel modular
variety Ag over SpecZ. The metric has logarithmic singularity along the
boundary D. One can check that the corresponding metrized line bundle
c g )ms . Since Ag is only a stack, to make the statement
actually lies in Pic(A
rigorous one needs to put a level structure on it.
Relative case
Let K be a finitely generated field over Q, and X be a projective variety
over K. The set of open arithmetic models U → V of X/K form a projective
system. Define
d
Div(X)
adlc : =
c
Pic(X)
adlc : =
d
lim Div(U)
adlc ,
−→
U →V
c
lim
Pic(U)
adlc ,
−→
U →V
d
Pic(X)
adlc : =
d
lim
Pic(U)
adlc .
−→
U →V
9
Then all these can be embedded to the corresponding objects on X an . Notice
d an ), there are two models (X1 /B1 , L1 ) and
that for any element L in Pic(X
(X2 /B2 , L2 ) with the same underlying bundle L such that
k · kL1 ≤ k · kL ≤ k · kL2 .
If X = Spec(K), we also denote them by
d
Div(K)
adlc ,
c
Pic(K)
adlc ,
d
Pic(K)
adlc .
Integrable metrics
c
We say that an adelic line bundle L ∈ Pic(U)
adlc is semipositive if it can be
given by a Cauchy sequence {(Xm , Lm )}m where each Lm is ample on Xm in
c
the sense of [Zh1]. We say that a line bundle in Pic(U)
adlc is integrable if it
is equal to the difference of two semipositive ones. Denote the subgroup of
c
d
integrable elements by Pic(U)
int . Analogously, we introduce Pic(U)int as a
d
c
d
subcategory of Pic(U)
adlc , and also Pic(X)int and Pic(X)int for any projective
variety X over a finitely generated field.
Remark 2.4. We may extend all the above discussions to the function field
situation. Fix a field k and a finitely generated extension K over k. For a
quasi-projective variety U over k, and a projective variety X over K, we can
define notion
d
Div(U/k)
int ,
d
Pic(U/k)
int ,
c
Pic(U/k)
int ,
d
Div(X/k)
int ,
d
Pic(X/k)
int ,
c
Pic(X/k)
int .
If k itself is finitely generated over Q, then the above objects are base changes
of the “arithmetic objects” defined above. More precisely, if the k-variety U
is the base change of an open arithmetic variety UZ over Z, the objects in the
first row are the base changes of corresponding objects for UZ . For the second
row, view K as a finitely generated field over Q without the constant field.
Then we also have base change from objects for X to X/k in the second row.
3
Arithmetic intersections and heights
Let K be a finitely generated field over Q of transcendental degree d, and
X be a projective variety over K of dimension n. In this section, we will
10
introduce two intersection maps:
n+1
d
d
Pic(X)
int −→ Pic(K)int ,
d+1
d
Pic(K)
int −→ R.
They give definition of heights of subvarieties of X.
Deligne Pairing
Let π : U → V be a flat and projective morphism of open arithmetic varieties
over SpecZ of relative dimension n whose generic fiber is X → SpecK. Then
we have an inductive family given by the Deligne pairing
Pic(U)n+1 −→ Pic(V),
(L1 , · · · , Ln+1 ) 7−→ hL1 , L2 , · · · Ln+1 i.
d
Our goal is to extend this intersection to line bundles in Pic(X)
int . As
the case of [Zh2] and [Mo3], we can not expect the intersection for all adelic
line bundles, but only for the integrable ones.
Proposition 3.1. The above multilinear map extends to a multilinear maps
n+1
d
d
Pic(U)
int −→ Pic(V)int ,
n+1
d
d
Pic(X)
int −→ Pic(K)int .
Proof. By linearity, we only need to extend the image for semipositive line
d
bundles L1 , · · · , Ln+1 in Pic(U)
int for any open model π : U → V of X →
SpecK.
For each i = 1, · · · , n+1, suppose that Li is given by the Cauchy sequence
{(Xm , Li,m )}m with each Li,m ample on models Xm over a projective model
B of V. Here we assume that the integral model Xm is independent of i. It
is always possible. Apply Raynaud’s flattening theorem ([Ra], Theorem 1,
Chapter 4). After blowing up B and replacing Xm by its pure-transform, we
get a flat family πm : Xm → Bm . The goal is to show the convergence of
hL1 , L2 , · · · , Ln+1 i := lim hL1,m , L2,m , · · · , Ln,m i.
m→∞
We can further assume that for each pair m < m0 , the map πm0 dominates
πm and
Li,m0 − Li,m ' O(Zi,m,m0 ),
d
Z i,m,m0 ∈ B(m , Div(U)
Q ).
11
Here {m }m≥1 is a sequence decreasing to zero.
We claim that for any m < m0 ,
d
hL1,m , L2,m , · · · , Ln+1,m i−hL1,m0 , L2,m0 , · · · , Ln+1,m0 i ∈ B(m deg X, Pic(U)
Q ).
where
deg(X) =
n
X
deg(L1,K · L2,K · · · Li−1,K · Li+1,K · · · Ln+1,K ).
i=1
Then the sequence hL1,m , L2,m , · · · , Ln+1,m i is Cauchy in Pic(U)Q under the
D-topology, and thus the convergence follows.
Note that
hL1,m , · · · , Ln+1,m i − hL1,m0 , · · · , Ln+1,m0 i
n
X
=
hL1,m , · · · , Li−1,m , (Li,m − Li,m0 ), Li+1,m0 , · · · , Ln+1,m0 i.
i=1
Fix an i and let `i be a section of Li,m − Li,m0 with divisor Z i,m,m0 . Since all
Lj,m (j < i) and Lj,m0 (j > i) are ample, they have sections `j (j 6= i) so that
div(`j ) for 1 ≤ j ≤ n + 1 intersects properly on Xm0 . In this way, the bundle
hL1,m , · · · , Li−1,m , (Li,m − Li,m0 ), Li+1,m0 , · · · , Ln+1,m0 i
c 1 · div`
c 2 · · · div`
c n+1 ). Since
has a section h`1 , · · · , `n+1 i with divisor π∗ (div`
c i ) = Z i,m,m0 is bounded by m π ∗ D, where D is a fixed boundary divisor
div(`
c j ) (j 6= i) are ample, we see that π∗ (div`
c 1·
of V, and since all other div(`
c 2 · · · div`
c n+1 ) is bounded by
div`
m π∗ L1,m · · · Li−1,m · π ∗ D · Li+1,m0 · · · Ln+1,m0
=m deg(L1,K · · · Li−1,K · Li+1,K · · · Ln+1,K ) D.
It finishes the proof.
Intersection numbers
Let K be a field finitely generated over Q with transcendental degree d. For
c )d+1 →
each projective model X of K, we have an intersection pairing Pic(X
d+1
c
R. It easily extends to a pairing Pic(U)
→ R for any open arithmetic
model U of K.
12
Proposition 3.2. The intersection pairing above extends uniquely to a multilinear and continuous homomorphism
d+1
c
Pic(K)
int −→ R.
c
Proof. It suffices to prove the similar result for Pic(U)
int for any open arithmetic variety U with function field K. We need to define hL1 , · · · , Ld+1 i for
c
any L1 , · · · , Ld+1 ∈ Pic(U)
int .
Let X be a projective model of U. Replacing U by an open subset if
necessary, we may assume that U is the complement of an ample divisor D
in X . Complete D to an ample divisor D, and use it to define the D-topology.
As in the proof in the previous proposition, we may assume that Li is
given by a Cauchy sequence {(Xm , Li,m )}m with each Li,m ample on a projective model Xm dominating X . Assume for any m0 > m,
d
Li,m0 − Li,m ∈ B(m , Pic(U))
with m → 0. For any subset I ⊂ {1, · · · , d + 1}, consider the sequence
αI,m := D
d+1−|I|
Y
Li,m .
i∈I
We want to prove by induction that {αI,m }m is a Cauchy sequence. When I
is the full set, we have the proposition.
There is nothing to prove if I is an empty set. Assume the claim is true
for any |I| < k for some k > 0. Then for any I with |I| = k, we have
d+1−k Y
d+1−k Y
D
Li,m − D
Li,m0
i∈I
=D
d+1−k
i∈I
X Y
Lj,m (Li,m − Li,m0 )
i∈I j∈I,j<i
Y
Lj,m0 .
j∈I,j>i
Its absolutely value is bounded by
Y
d+1−k+1 X Y
m D
Lj,m
Lj,m0
i∈I j∈I,j<i
≤ m D
d+1−k+1
j∈I,j>i
X Y
Lj,m
i∈I j∈I,j<i
13
Y
(Lj,m + m D).
j∈I,j>i
The last term is a linear combination of αI 0 ,m with |I 0 | < k. The coefficients
of the linear combination grow as o(m ). It follows that αI,m is a Cauchy
sequence.
Remark 3.3. The intersection pairing defined by the above two propositions
can be extended to the function field situation described in Remark 2.4. If
the constant field k is finitely generated over Q, then the Deligne pairing in
Proposition 3.1 is compatible with the following “base changes”:
Pic(X) −→ Pic(X/k),
Pic(K) −→ Pic(K/k).
Adelic heights
Let K be a field finitely generated over Q with transcendental degree d and
let X be a projective variety over K of dimension n. Let L be a semipositive
d
element in Pic(X)
int with ample generic fiber L = LK .
For any closed K-subvariety Z of X, define the adelic height of Z as
dim Z+1
L|Zgal
d
∈ Pic(K)
hL (Z) :=
int .
(dim Z + 1) degL (Zgal )
Here Zgal denotes the minimal K-subvariety of X containing Z, L|Zgal ded gal )int , and the self-intersections are taken as the
notes the pull-back in Pic(Z
Deligne pairing in the sense of Proposition 3.1. It gives a map
d
hL : |XK | −→ Pic(K)
int .
Here |XK | denotes the set of closed K-subvarieties of X. In particular, we
have a height map of algebraic points:
d
hL : X(K) −→ Pic(K)
int .
The class of hL modulo bounded functions depends only on the class of L in
Pic(X).
d
d
d
If K is a number field, then Pic(K)
int = Pic(K)adlc = Pic(OK ). We have
d
the degree map deg : Pic(K)
int → R. In that case, deg hL is the same as the
usual height.
14
The height function satisfies the Northcott property. For any D ∈ R and
d
α ∈ Pic(K)
int , the set
{x ∈ X(K) : deg(x) < D, hL (x) < α}
is finite. It follows from the Northcott property of Moriwaki’s height described below.
Moriwaki heights
d
Let H be any element in Pic(K).
For any closed K-subvariety Z of X, define
the Moriwaki height of Z with respect to H as
dim Z+1
d
L|
·H
Z
gal
H
∈ R.
hL (Z) := hL (Z) · H =
(dim Z + 1) degL (Zgal )
d
If both L and H are realized on some model X → B, then hH
is just the
L
height function introduced in [Mo2]. If H is nef and big on B, Moriwaki
prove the height satisfies the Northcott property. Namely, for any D ∈ R
and A ∈ R, the set
(x) < A}
{x ∈ X(K) : deg(x) < D, hH
L
is finite.
Remark 3.4. As in Remark 2.4 and Remark 3.3, we may extend the adelic
heights and Moriwaki heights to the function field situation. The Moriwaki
height has been studied by Faber [Fab] in the case of transcendental dimension one and by Gubler [Gu2] in the general case.
4
Small points and Equidistribution
Let X be a projective variety over a finitely generated field K over Q, and L
be an integrable arithmetic line bundle over X. Then we have height function
hL :
X(K) −→ Pic(K)int .
15
Small points
A sequence {xm } of points in X(K) is called a sequence of adelic small points
if hL (xm ) has limit 0 in Pic(K)int . Recall that the topology in Pic(K)int is
taken as the induced topology on Pic(V)int for open arithmetic varieties V
over Z with function field K.
For a polarization H, we can also define Moriwaki small points in terms
of heights hH
. Sometimes we call it hH
-small to emphasis the dependence on
L
L
the polarization. It is clear that “adelic small” implies “Moriwaki small”.
are equal to the usual height
If K is a number field, both deg hL and hH
L
function hL . Then we can check that both smallness are the equivalent to
the usual one given by hL (xm ) → 0.
Remark 4.1. We can also define the smallness of points in function field
situation. If the constant field k is finitely generated over Q, then we have two
c
notions of small points. The one using heights valued in Pic(K)
int is called
c
the arithmetic smallness, while the one using heights valued in Pic(K/k)int
is called the geometric smallness. It is clear that the arithmetic smallness
implies the geometric smallness.
Equidistribution
Let v be a valuation of K, i.e., a multiplicative norm |·|v : K → R≥0 . It could
be either archimedean or non-archimedean. Then we have the Berkovich
analytic space Xvan and the Chamber-Loir measure
dµL,v =
1
X
c1 (L)dim
v
degL (X)
on Xvan attached to (X, L). Here L = LK is the generic fiber of L on X. Also
for each point x ∈ X(K), we have the measure
µx,v =
1
δO(x)
deg(x)
on Xvan . Here δO(x) is the Dirac measure for the Galois orbit O(x) = Gal(K/K)x
in Xvan .
A sequence {xm }m≥1 of points is said to be generic if for any proper
closed subvariety Y of X contains only finitely many terms of the sequence.
16
A sequence {xm }m≥1 of points is said to be Galois equidistributed on Xvan
with respect to dµL,v if the weak convergence
µxm ,v −→ dµL,v
holds on Xvan .
Conjecture 4.2. Assume that L is semipositive with ample generic fiber LK .
Then hL (X) = 0 if and only if there is a generic sequence of adelic small
points on X(K).
Moreover, if xm is a generic sequence of adelic small points in X(K), then
xm is Galois equidistributed on Xvan with respect to dµL,v for any valuation v
of K.
Remark 4.3. If K is a number field, the conjecture is proved by Yuan [Yu] by
extending the previous work of Szpiro–Ullmo–Zhang [SUZ] for archimedean
place v and point-wise positive c1 (L, k · kv ).
Remark 4.4. If the valuation v is trivial, or more generally in the function field
case where K is is a finitely generated field over a base field k, the conjecture
is proved independently by Faber [Fab] and Gubler [Gu3]. In fact, we can
find a subfield L of K containing k so that K/L has transcendental degree
1. Then the adelic smallness implies Faber’s smallness.
Remark 4.5. An analogue of the conjecture for Moriwaki small points holds
for any place v of Q. More precisely, let U → V be an open arithmetic model
for X/K such that L and H can realized on this model. Assume that H is
dim U
= 0. Moriwaki [Mo2] proved
nef with H
1
1
δxm ∧ c1 (H)dv −→
c1 (L)nv ∧ c1 (H)dv .
deg(xm )
degL (X)
Here n = dim X and d is the transcendental degree of K over Q, xm denotes
an
the Zariski closure in UQ , and the weak convergence is on the space UQ,v
.
Moriwaki proves this for arichimedean places under a positivity assumption
of curvatures of L and H. The general case can be done using the method
in [Yu].
Equidistribution of small 0-cycles
Here we extend Conjecture 4.2 to small 0-cycles. Recall that a 0-cycle is a
finite linear combination of closed points with rational coefficients. So it is
17
of the form
z=
X
ak ∈ Q, zk ∈ X0 ,
ak zk ,
k
where X0 denote the set of closed points on X. We need to define some
notions:
• The degree of z is defined to be
deg z :=
X
ak deg zk .
k
The cycle z is called unitary if ak ≥ 0 and deg z = 1.
• If v is a valuation of K, then the Dirac measure of z on Xvan is defined
to be
X
ak δzk ,v
δz,v :=
k
where δzk ,v is the Dirac measure of zk on Xvan . Notice that the total
mass of this measure is 1.
• For a subvariety Y of X, the Y -degree of z is defined to be
X
degY (z) :=
ak deg(zk ).
zk ∈Y
• The height of z is defined to be
hL (z) :=
X
ak hL|zk i.
k
Let {zm }m be a sequence of unitary 0-cycles on X. We further define the
following notions:
• The sequence {zm } is called a sequence of small 0-cycles with respect
to L if hL (zm ) → 0.
• The sequence {zm } is called a generic sequence if for any subvariety Y
of X, the Y -degree degY (zm ) → 0.
18
• The sequence {zm } is said to be Galois equidistributed on Xvan with
respect to dµL,v if the weak convergence
δzm ,v −→ dµL,v
holds on Xvan .
Conjecture 4.6. Assume that L is semipositive with ample generic fiber LK .
Then hL (X) = 0 if and only if there is a generic sequence of small unitary
0-cycles on X.
Moreover, if zm is a generic sequence of small 0-cycles on X, then zm is
Galois equidistributed on Xvan with respect to dµL,v for any valuation v of K.
It is easy to see that Conjecture 4.6 implies Conjecture 4.2. In fact, for
any small and generic sequence xm ∈ X(K), take zm = xm / deg xm where
xm is the image of xm considered as a morphism xm : SpecK → X.
Next, we show that Conjecture 4.2 implies Conjecture 4.6 when K is a
number field. Let zm is a generic sequence of small 0-cycles on X. A basic
result asserts that the space of semipositive measures of total mass 1 on a
compact Hausdorff space is sequentially compact under the weak topology. In
our case, any subsequence of {δzm ,v }m has a convergent subsequence. Therefore, it suffices to prove that the equidistribution is true for some subsequence
of {zm }.
Order the set of irreducible proper subvarieties of X as
Y1 , Y2 , · · · , Y` , · · · .
It is possible because the set is countable. Write Y`0 = ∪i≤` Yi . Since
degY`0 (zm ) → 0 as m → ∞, inductively replacing zm by a subsequence, we
may assume that
1
∀m.
degYm0 (zm ) ≤ ,
m
After replacing zm by a subsequence, we may assume that
h(zm ) ≤
1
.
m2
Here h = hL = deg hL is the usual height function on X since K is a number
field.
19
In terms of closed points, write
zm =
X
am,i zm,i .
i
For each m, consider the following two partial sums of zm :
X
X
αm =
am,i zm,i ,
βm =
am,i zm,i .
0
zm,i ∈Ym
h(zm,i )≥1/m
It follows that deg(αm ) ≤ 1/m. And
X
h(βm ) ≥
ai deg zm,i ·
h(zm,i )≥1/m
deg βm
1
=
.
m
m
Since h(zm ) ≤ 1/m2 , it follows that deg(βm ) ≤ 1/m.
Now we have decomposition
0
zm = zm
+ γm ,
where
X
0
zm
=
am,i zm,i ,
0 , h(z
zm,i ∈Y
/ m
m,i )<1/m
X
γm =
0
zm,i ∈Ym
am,i zm,i .
or h(zm,i )≥1/m
Then γm ≤ αm + βm and thus deg(γm ) ≤ 2/m. The Dirac measure of
00
=
γm converges to 0 on Xvan . It suffices to prove the Dirac measure of zm
0
0
zm / deg(zm ) converges to dµv .
00
00
d
We still have h(zm
) → 0. It is equivalent to hL (zm
) → 0 in Pic(K)
int
00
since K is assumed to be a number field. So the union ∪m |zm | of supports of
00
00
zm
is generic and small. Apply Conjecture 4.2, any subsequence of ∪m |zm
|
an
00
is equidistributed on Xv . Then it is easy to see that zm is equidistributed.
Reduction of small cycles
Let U → V be an open arithmetic model of X/K over Q. Let s be a point of
V := VQ . Then for any closed cycle Z on X/K, we can define the reduction
Zs := Z|s of Z as the restriction to Xs := Us of the Zariski closure Z of Z in
V.
20
Theorem 4.7. Let v be a place of K and let zm be a generic sequence of
small unitary 0-cycle on X/K. Then for any point s as above, the sequence
of reduction zm,s is also unitary, small, and generic. If either v is trivial on
an
.
Q, or s is a close point, then zm,s is equidistributed on Xs,v
Proof. The properties of being unitary and small are clearly stable under
reduction. The genericity follows from Theorem 4.8 below on the equidistribution of Moriwaki small points on the fibers. The equidistribution follows
from the work of Yuan [Yu] when s is a closed point, and Faber [Fab] and
Gubler [Gu3] when v is trivial on Q.
Equidistribution on fibers of Moriwaki small points
Let (X, K) be as above. Namely, K is a field finitely generated over Q of
transcendental degree d, and X is a projective variety over K of dimension
n.
Let L ∈ Pic(X)int be an arithmetic line bundle. Let U → V be an open
c
arithmetic model of X/K such that L is in Pic(U)
adlc . Then for any closed
point s ∈ VQ , the base change Ls gives an adelic line bundle on Xs := Us in
the sense of [Zh2].
Let B be a projective model of V, and H be a nef hermitian line bundle
d+1
d
= 0. Then B = (B, H) gives a polarization of
on B with HQ
> 0 and H
K in the sense of Moriwaki [Mo2].
Theorem 4.8. Assume that L is semipositive and that the generic fiber LK
is ample on X. For any closed point s ∈ VQ , there is a finite set Ωs of
finite places of Q satisfying the following property. For any generic sequence
{xm } of hH
-small unitary 0-cycles of X, the specialization sequence {xm,s }
L
an
, for all finite
on the fiber Xs is equidistributed on the analytic space Xs,v
places v of Q outside Ωs , with respect to the equilibrium measure of (Xs , Ls ).
In particular, the sequence {xm,s } is generic on Xs .
Proof. As the above argument, it suffices to assume that every xm comes
from a single closed point of X. Note that xm,s may still come from more
than one closed point on Xs . The genericity still means that as a sequence
of unitary 0-cycles.
To obtain the equidistribution, we will apply the variational principle of
Szpiro-Ullmo-Zhang [SUZ]. For more details of the variational principle, we
refer to [Yu, §3]. Let (S, gS ) be an arithmetic 1-cycle on B linearly equivalent
21
d
to H . Assume that the generic fiber s := SQ is contained in VQ . Denote by
ΩS the set of finite places of Q over which S has vertical components. Then
ΩS is the image of the support of the vertical part S − s in Spec(Z).
Let φ be a continuous function on XQanv , it corresponds to an adelic line
bundle O(φ) on XQ over Q in the sense of Zhang [Zh2]. It has trivial generic
fiber, and trivial metric outside v. If φ is a model function, O(φ) can be
obtained by a single integral model of (XQ , OQ ). In general, it is a limit since
any continuous function is a uniform limit of model functions. Conventionally, we write M(φ) = M + O(φ) for any arithmetic line bundle M. For
more details, we refer to [Yu, §3.3].
Lemma 4.9. Let (S, s, ΩS ) be as above, and let φ be a continuous function
on XQanv for a finite place v ∈
/ ΩS . Then for any closed subvariety Z of XK ,
hBL(φ) (Z) − hBL (Z) = hLs (φ) (Z|s ) − hLs (Z|s )
an
Here φ = φ|Xs,Q
is the restriction, and the specialization Z|s is viewed as a
v
closed subvariety of the Q-variety Xs .
Note that Ls and thus Ls (φ) are usual adelic line bundles over the Qvariety Xs inP
the sense of [Zh2]. We remark that s is in general a linear
combination
k ak sk of closed points si in VQ . Then hLs is understood to
P
be i ai hLsi , and so is hLs (φ) . Many intersections and integrations below are
also understood in this sense.
Proof. We can assume that Zgal = Z since both sides depends only on Zgal .
Denote r = dim Z. By definition,
r+1 X
1
r+1
d
−
=
O(φ)|iZ · L|r+1−i
·H ,
Z
(r + 1) degL (Z) i=1
i
r+1 X
1
r+1
i
hLs (φ) (Z|s ) − hLs (Z|s ) =
O(φ)|Z|
· Ls |r+1−i
.
Z|s
s
(r + 1) degL (Z) i=1
i
hBL(φ) (Z)
hBL (Z)
It suffices to show that for each M ∈ Pic(Z)int , the difference
δ(M) := O(φ)i · M
r+1−i
vanishes for all i = 1, · · · , r + 1.
22
d
r+1−i
· H − O(φ)i · Ms
By approximation, we can assume that M is an arithmetic line bundle
on an integral model Z → B of Z. The intersections in
δ(M) = O(φ)i · M
r+1−i
r+1−i
· (S, gS ) − O(φ)i · Ms
are taken over Z. It is easy to obtain
δ(M) = O(φ)i · M
r+1−i
|S−s .
It vanishes since φ is supported over v ∈
/ ΩS and the vertical cycle S − s has
no component lying over v.
For simplicity, we first consider the case that s = SQ is a single point.
Let > 0 be a rational number. The lemma gives
Z
1
B
B
Z
φ c1 (Ls )dim
+ OZ (2 ).
hL(φ) (Z) − hL (Z) = v
degL (Zgal ) Z|an
s,Q
v
Here c1 (Ls )nv denotes the canonical measure of Ls on Z|an
s,Qv introduced by
Chambert-Loir [Ch]. The error term OZ (2 ) depends on Z, but vanishes if
Z is a closed point.
Set Z to be the point xm or the whole variety X. We obtain
Z
B
B
φ µxm |s ,v ;
hL(φ) (xm ) − hL (xm ) = an
Xs,Q
v
hBL(φ) (Xs ) − hBL (Xs ) = Z
φ dµs,v + O(2 ).
an
Xs,Q
v
Here µxm |s ,v denotes the probability Dirac measure associated to the Ga1
X
an
lois orbit of xm |s on Xs,Q
, and dµs,v =
denotes the
c1 (Ls )dim
v
v
degL (Xs )
Chambert-Loir measure of with respect to Ls .
We have the fundamental inequality
sup
inf
Z
x∈X(K)−Z(K)
hBL(φ) (x) ≥ hBL(φ) (X) + O(2 ).
It is a generalized version of [Mo2, Corollary 5.2]. We omit the proof but
leave a few remarks. Moriwaki’s result is based on the arithmetic HilbertSamuel formula due to Gillet-Soule, Bismut-Vasserot and Zhang. The key
23
for our case is the arithmetic bigness result in [Yu, Theorem 2.2, Lemma 3.3],
where a Hilberts-Samuel type estimation is obtained based on the arithmetic
ampleness of Zhang [Zh1].
Now the standard argument of Szpiro-Ullmo-Zhang gives
Z
Z
lim
φ µxm |s ,v =
φ dµs,v .
m→∞
an
Xs,Q
v
an
Xs,Q
v
an
as long as φ is continuous on XQanv . This actually
It is true for φ = φ|Xs,Q
v
an
an
covers all continuous functions on Xs,Q
since Xs,Q
is closed in XQanv . Therev
v
fore, we have proved the weak convergence dµxm |s ,v → dµs,v on the analytic
an
.
space Xs,v
The weak convergence is true under the assumption that (S, gS ) repred
sents H and the generic fiber
P s = SQ is a single point contained in VQ . In
the general case that s = i ai si is a linear combination of closed points si
in VQ , everything above still makes sense by linearity. Then the weak convergence µxm |s ,v → dµs,v implies the weak convergence µxm |si ,v → dµsi ,v for
all si . For any closed point u of VQ , we can find a representative (S, gS ) of
d
H , such that the supported of the 0-cycle SQ contains u. Then we have the
equidistribution µxm |u ,v → dµu,v on the fiber of u for all finite places v ∈
/ ΩS .
It proves the result.
5
Heights for dynamical systems
Invariant metrics for dynamical systems
Let K be a finitely generated field over Q of transcendental degree d. Let
(X, f, L) be a polarized algebraic dynamical system of dimension n over K.
Fix an isomorphism f ∗ L = qL where q > 1 by assumption.
Fix an arithmetic variety B over Z with function field K. Choose any
arithmetic model π : X → B over B and any hermitian line bundle L =
(L, k · k) over X such that (XK , LK ) = (X, L).
fm
For each positive integer m, consider the composition X → X ,→ X .
Denote its normalization by fm : Xm → X , and the induced map to B by
∗
c m )Q . The sequence
πm : Xm → B. Denote Lm = q −1 fm
L, which lies in Pic(X
{(Xm , Lm )}m≥1 is an adelic structure in the sense of [Mo3].
24
In the following, we will show that the sequence {(Xm , Lm )}m≥1 converges
c
to a line bundle Lf in Pic(X)
int . There is an open subscheme V of B such
that U := XV is flat over V and that f : X → X extends to a morphism
fV : U → U with fV∗ LV = qLV . By the construction, we have Xm,V = XV and
Lm,V = LV . We can assume that D := B − V is an effective Cartier divisor
of B by enlarging V if necessary.
d
Theorem 5.1. The sequence Lf = {(Xm , Lm )}m≥1 is convergent in Pic(X)
int .
Furthermore, the limit is semipositive and depends only on the generic fiber
(X, f, L, K).
Proof. We only prove the existence of the limit, since the independence of the
integral models can be proved similarly. Recall that Xm,V = XV and Lm,V =
LV . But these isomorphisms are not given by the morphism fm : Xm → X .
Let π
em : Xem → B be an arithmetic model of X which dominates both
Xm and Xm+1 . More precisely, π
em factors through a birational morphism
φm : Xem → Xm (resp. φ0m : Xem → Xm+1 ) such that φm,V (resp. φ0m,V ) is an
isomorphism to Xm,V (resp. Xm+1,V ). The construction works for m = 0 by
the convention X0 = X .
We first consider the relation between (X , L) and (L1 , X1 ). The difference
∗
φ0 L − φ00 ∗ L1 is an arithmetic Q-line bundle on Xe0 . It is trivial on Xe0,V , and
thus represented by an arithmetic divisor supported on π
e0∗ |D|. Then there
exists r > 0 such that
∗
d
φ∗0 L − φ00 L1 ∈ B(r, Pic(U))
Now we consider Lm − Lm+1 for general m. Without loss of generality, we
can assume that Xem dominates Xe0 via a birational morphism τm : Xem → Xe0 .
Then it is easy to see that
∗
φ∗m Lm − φ0m Lm+1 =
It follows that
1 ∗ ∗
∗
τm (φ0 L − φ00 L1 ).
m
q
∗
φ∗m Lm − φ0m Lm+1 ∈ B(
r d
, Pic(U))
qm
c
In terms of the partial order in Pic(X)
Q , it is just
r d
Lm − Lm+1 ∈ B( m , Pic(U)).
q
It follows that {Lm }m is a Cauchy sequence.
25
Remark 5.2. The arithmetic line bundle Lf is invariant under the pull-back
∗
d
d
f ∗ : Pic(X)
int → Pic(X)int in the sense that f Lf = qLf as in the number
field case.
Canonical height
Let K and (X, f, L) be as above. It gives an f -invariant line bundle Lf in
d
Pic(X)
int . For any closed K-subvariety Z of X, define the canonical height
function of Z as
d
hf (Z) := hLf (Z) ∈ Pic(K)
int .
We can also define the canonical height by Tate’s limiting argument:
hf (Z) = lim
m→∞
1
h(X ,L) (f m (Z)).
m
q
Here (X , L) is any initial model of (X, L) as in the construction of Lf above.
d
Then one can check that it is convergent in Pic(K)
int and compatible with
the previous definition.
Proposition 5.3. Let Z be a closed subvariety of X over K. Then:
d
(a) The height hf (Z) ≥ 0 under the partial order in Pic(K)
int .
(b) The height is f -invariant in the sense that hf (f (Z)) = q hf (Z).
d
(c) The height hf (Z) = 0 in Pic(K)
int if Z is preperiodic under f . The
inverse is also true if Z is a point.
Proof. Since Lf is semipositive, the height hf (Z) ≥ 0 under the partial order.
The formula hf (f (Z)) = qhf (Z) follows from the projection formula and the
invariance of Lf . Thus hf (Z) = 0 if Z is preperiodic under f . The second
statement of (c) follows from the Northcott property.
Since all preperiodic points have height 0, our equidistribution conjecture
for small points gives the following:
Conjecture 5.4. Let (X, f, L) be a dynamical triple defined over a finitely
generated field K over Q. Let v be a place of K. Let xm be a generic
sequence of preperiodic points. Then the Galois orbit of xm is equidistributed
with respect to the equilibrium measure dµLf ,v on Xvan .
26
References
[Be]
V. Berkovich, Spectral theory and analytic geometry over nonArchimedean fields. Mathematical Surveys and Monographs, 33.
American Mathematical Society, Providence, RI, 1990. x+169 pp.
[Ch]
A. Chambert-Loir, Mesures et équidistribution sur les espaces de
Berkovich, J. Reine Angew. Math. 595 (2006), 215–235.
[De]
P. Deligne, Le déterminant de la cohomologie. Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), 93C177, Contemp.
Math., 67, Amer. Math. Soc., Providence, RI, 1987.
[Fal]
G. Faltings, Endlichkeitssäze für abelsche Varietäten
Zahlkörpern. Invent. Math. 73 (1983), no. 3, 349–366.
über
[Fab] X. W. C. Faber, Equidistribution of dynamically small subvarieties
over the function field of a curve. Acta Arith. 137 (2009), no. 4, 345–
389.
[GS]
H. Gillet, C. Soulé, Arithmetic intersection theory, Publ. Math. IHES,
72 (1990), 93-174.
[GS2] H. Gillet, C. Soulé, Characteristic classes for algebraic vector bundles
with hermitian metric I, II, Annals of Math. 131 (1990), 163-203.
[Gu1] W. Gubler, Local heights of subvarieties over non-Archimedean fields.
J. Reine Angew. Math. 498 (1998), 61–113.
[Gu2] W. Gubler, The Bogomolov conjecture for totally degenerate abelian
varieties. Invent. Math. 169 (2007), no. 2, 377–400.
[Gu3] W. Gubler, Equidistribution over function fields. Manuscripta Math.
127 (2008), no. 4, 485–510.
[Mo1] A. Moriwaki, Hodge index theorem for arithmetic cycles of codimension one. Math. Res. Lett. 3 (1996), no. 2, 173–183.
[Mo2] A. Moriwaki, Arithmetic height functions over finitely generated
fields, Invent. Math. 140 (2000), no. 1, 101–142.
27
[Mo3] A. Moriwaki, The canonical arithmetic height of subvarieties of an
abelian variety over a finitely generated field. J. Reine Angew. Math.
530 (2001), 33–54.
[Ra]
M. Raynaud, Flat modules in algebraic geometry. Compos. Math. 24
(1972), 11–31.
[SUZ] L. Szpiro, E. Ullmo, S. Zhang, Équidistribution des petits points,
Invent. Math. 127 (1997) 337–348.
[Yu]
X. Yuan, Big line bundle over arithmetic varieties. Invent. Math. 173
(2008), no. 3, 603–649.
[YZ]
X. Yuan, S. Zhang, Calabi–Yau theorem and algebraic dynamics,
preprint.
[Zh1] S. Zhang, Positive line bundles on arithmetic varieties, Journal of the
AMS, 8 (1995), 187–221.
[Zh2] S. Zhang, Small points and adelic metrics, J. Alg. Geometry 4 (1995),
281–300.
28