Vojta 1996

Invent. math.
126, 133–181 (1996)

c Springer-Verlag 1996
Integral points on subvarieties of semiabelian

varieties, I ?
Paul Vojta
Department of Mathematics, University of California, Berkeley, CA 94270, USA
e-mail: vojta@math.berkeley.edu
Oblatum 7-VII-1993 & 18-XII-1995
Dedicated to Professor Wolfgang M. Schmidt on the occasion of his sixtieth birthday
Let X be a closed subvariety of an abelian variety A, and assume that both are
defined over some number field k . Then a conjecture of Lang [L 1] states that
the set of rational points is as small as one might reasonably expect:
S
Theorem 0.1. The set X (k ) is contained in a finite union Bi (k ), where each Bi
is a translated abelian subvariety of A contained in X .
In [F 1], Faltings proved this in the special case where X ×k k̄ contains no
nontrivial translated abelian subvarieties of A ×k k̄ ; the conclusion in that case
simplifies to the assertion that X (k ) is finite. He proved this in general in [F 2].
This proof is also described in detail in [V 4]; we will follow the latter exposition
closely here.
In this paper we generalize Theorem 0.1 to cover the corresponding statement
for integral points on closed subvarieties of semiabelian varieties:
Theorem 0.2. Let k be a number field, with ring of integers R. Let S be a finite
set of places of k , containing the set of archimedean places, and let RS be the
localization of R away from (non-archimedean) places in S . Let X be a closed
subvariety of a semiabelian variety A; assume both are defined over k . Let X
be a model for X overSSpec RS . Then the set X (RS ) of RS -valued points in X
equals a finite union Bi (RS ), where each Bi is a subscheme of X whose
generic fiber Bi is a translated semiabelian subvariety of A.
A future paper will address similar questions for certain open subvarieties of
A.
Theorem 0.2 partially proves a conjecture of Lang ([L 2], p. 221): Let A
be a semiabelian variety, and let Γ be a finitely generated subgroup of A. Let
? Partially supported by NSF grants DMS-9001372 and DMS-9304899.

134 P. Vojta
Γ be the division group of Γ ; i.e., the group of all x ∈ A such that nx ∈ Γ

for some positive integer n. Then Lang conjectures that the intersection of Γ
with any closed subvariety X of A is contained in the union of finitely many
translated semiabelian subvarieties of A contained in X . Theorem 0.2 does not
apply to this more general conjecture, but it is equivalent to a similar statement
where one does not take the division group. Indeed, the set of integral points on
A is a finitely generated group. More recently, M. McQuillan [McQ] has proved
Lang’s conjecture in full generality, by using methods of M. Hindry to reduce
the general statement to the special case proved here.
I doubt that this result can be generalized to a larger class of group varieties:
consider, for example, Pell’s equation on A2 ∼ = G2a .
By a standard result on subvarieties of abelian varieties (Theorem 4.2), The-
orem 0.1 gives an affirmative answer, in the case of subvarieties of abelian
varieties, to a question posed by Bombieri [N 2]: if the variety X is of general
type, then is the set X (k ) contained in a proper Zariski-closed subset? Similarly,
in the semiabelian case, Theorem 0.2 provides a partial answer to ([V 1], 4.1.2).
Moreover, by the Kawamata structure theorem (Theorem 4.3), the nontrivial
Bi occurring in the conclusion of Theorem 0.2 must lie in a proper subvari-
ety which is geometrical in nature. This supports a conjecture of Lang which
strengthens the question posed by Bombieri: Lang conjectures in [L 3] that if X
is of general type then the higher dimensional components of X (k ) must lie in a
subvariety which is independent of k .
Section 14 proves a corollary of Theorem 0.2 which generalizes ([V 1],
Theorem 2.4.1). The proof essentially reduces to showing that the given variety
embeds into a semiabelian variety.
Corollary 0.3. Let X be a projective variety defined over a number field k , and
let ρ denote its Picard number. Let D be an effective divisor on X , also defined
over k , which has at least dim X − h 1 (X , OX ) + ρ + 1 geometrically irreducible
components. Then any set of D-integral points on X is not dense in the Zariski
topology.
Acknowledgements. I thank S. Lang, M. Nakamaye, G. Faltings, and D. Bertrand for fruitful dis-
cussions concerning this paper. I especially thank the referee for many helpful comments on the
manuscript.
1. Notation
We use the notational conventions of ([V 4], Sect. 5), which for convenience are
summarized here.
For places v and absolute values k · k on a number field k , we use the
conventions of ([V 1], 1.1); in particular, for a place v corresponding to a real or
complex embedding σ : k ,→ C, the absolute value kx kv equals |σ(x )| or |σ(x )|2 ,
respectively, and the product formula reads
Integral points on subvarieties of semiabelian varieties, I 135
Y
kx kv = 1
v
for all nonzero x ∈ k . Let Cv denote the completion of the algebraic closure of
the completion kv of k at v; this field is algebraically closed. If v is archimedean
then it is isomorphic to C. The absolute values k · k extend from kv to Cv .
For ρ ≥ 0 and for a given v let
Dρ = {z ∈ Cv | kz k < ρ} and ∂ Dρ = {z ∈ Cv | kz k = ρ} .
Note that these differ from the usual notation if v is a complex place, and that
∂ Dρ is not the topological boundary if v is non-archimedean. Let D = D1 .
We will also use the notations and conventions of arithmetic schemes, as in
for example [V 2] or [V 3], except that complex conjugate pairs of fibers at an
archimedean place will be identified, to conform with the above convention on
absolute values. This is possible because in the Gillet-Soulé theory, all objects at
complex conjugate places are assumed to be taken into each other by conjugating.
In particular, we assume X is an integral arithmetic scheme which is quasi-
projective and flat over Spec R with generic fiber X . This assumption on X
differs from that used in the statement of Theorem 0.2, but the change does not
affect the set X (RS ). The exact choice of X is made in the beginning of Sect.
10.
Throughout this paper we will refer to Q-divisors (divisors with rational coef-
ficients) and Q-divisor classes. The latter are taken to be elements of Div(X ) ⊗ Q
modulo principal (Z-)divisors, as opposed to Pic(X ) ⊗ Q. If D is a Q-divisor,
then writing O (dD) shall implicitly assume that d is sufficiently divisible to
cancel all of the denominators in D.
If g is a Green function or Weil function with respect to a divisor D, then
we say D = div(g) and Supp g = Supp div(g).
In this paper, a variety is an integral scheme of finite type over a field. All
schemes in this paper are assumed to be separated. As in [V 4], we use the
notation line sheaf and vector sheaf to mean invertible sheaf and locally free
sheaf, respectively.
We use N = {0, 1, . . .}.
And finally, on any product (such as An or X n ), let pri denote the projection
onto the i th factor.
2. Structure of semiabelian varieties
A semiabelian variety is a group variety A for which there exists an exact se-
quence
0 → Gµm → A−→A0 → 0 ,
ρ
(2.1)
where A0 is an abelian variety. By ([I 2], Lemma 4), A is commutative. In general
the kernel of ρ need not be a split torus, but we may assume this to be the case
by enlarging the number field k ; this will not weaken Theorem 0.2.
136 P. Vojta
Lemma 2.2. For fixed µ and A0 , the set of semiabelian varieties (2.1), modulo
isomorphisms fixing the factors Gµm and A0 , is in 1–1 correspondence with the set
Pic0 (A0 )µ , via the function taking a tuple (M1 , . . . , Mµ ) to the product
(2.2.1) P0 (OA0
⊕ M1 ) ×A0 . . . ×A0 P0 (OA
0
⊕ Mµ ) ,
where P0 denotes the open subset of P(OA0 ⊕ Mm ) obtained by removing the sec-
tions corresponding to the projections onto each factor of OA0 ⊕ Mm . Moreover,
Pic(A) ∼= Pic(A0 ).
Proof . When µ = 1 the first assertion follows by ([L 2], Ch. 11, Sect. 6); the
general case then follows by ([S 1], Ch. VII, Sect. 1, (10)). The second assertion
is then a consequence of ([H 2], II, Ex. 7.9a). See also ([S 1], Ch. VII) for a
treatment of this topic in full generality.
The group law on A can be described in terms of this construction; see the
proof of Proposition 2.6.
The fact that the Mm lie in Pic0 is vital here: it implies that, although A
may not equal a product of A0 and Gµm , it is close enough to a product that some
of the properties of the product still apply.
We will also frequently use a completion of A to a proper variety Ā, which
will be chosen as the completion
(2.3) Ā := P(OA0 ⊕ M1 ) ×A0 . . . ×A0 P(OA0 ⊕ Mµ ) .
The morphism ρ : Ā → A0 extends in the obvious manner. This completion was

originally defined by Serre, ([S 2], 1.3). It has a canonical exact sequence
(2.4) 0 → Pic(A0 ) → Pic(Ā) → Zµ → 0 ,
where Zµ = Pic((P1 )µ ). Also let [∞]m and [0]m denote the divisors corresponding
(respectively) to the projections
(2.5) OA0 ⊕ Mm → OA0 and OA0 ⊕ Mm → Mm .

Pµ
By (2.2.1) the divisor Ā \ A is the sum m=1 ([0]m + [∞]m ). Also, note that

O [∞]m − [0]m ∼ = ρ∗ Mm ,
so that in particular we have the numerical equivalence
[0]m ≡ [∞]m , m = 1, . . . , µ .
Note also that by ([I 1], Theorem 2), any morphism of semiabelian varieties
is the composition of a group homomorphism and a translation. Thus, in the
wording “translated semiabelian subvariety,” it is not necessary to specifically
state that the group law on the subvariety is obtained from the group law on A.
The above completed semiabelian varieties have a natural choice of Green
function for the divisors [0]m and [∞]m .
Proposition 2.6. Let Ā be the completion of a semiabelian variety A defined over

a local or global field k . Then for m = 1, . . . , µ there is a unique Weil function
λm for [0]m − [∞]m satisfying
(2.6.1) λm (P + Q) = λm (P ) + λm (Q)
for all P , Q ∈ A(Cv ) and all places v of k . Moreover, if v is archimedean, then

λm is C ∞ .
Proof . We may assume µ = m = 1; the general case follows by pulling back to

A. Thus we may assume that A = P0 (OA0 ⊕ M), and omit m from the notation.
It is well known that points on P(OA0 ⊕ M) correspond bijectively
to pairs
consisting of a point P 0 ∈ A0 and a surjection (OA0 P 0 ) ⊕ (M P 0 ) OP 0 , up
to multiplication by a nonzero constant. Let s be a local generator for M in
a neighborhood of P 0 ; then points in Ā(Cv ) lying outside the support of [∞]
correspond to pairs (P 0 , z ), where P 0 ∈ A0 (Cv ) and z ∈ Cv , corresponding to a
surjection (f1 , f2 s) 7→ zf1 − f2 . Regarding z as a rational function on Ā, we then
have the equality of divisors
(2.6.2) (z ) = [0] − [∞] − ρ∗ (s) .
For divisors D Lon A0 , let λD denote a Néron function as in ([L 2], Ch. 11, 1.1
and 1.5). Let Γ = v R denote the group of Mk -constants; then Néron functions
have the properties:
1. λD+D 0 = λD + λ0D mod Γ ;
2. λ(f ),v = − log kf kv mod Γ ; and
3. λφ∗ D = λD ◦ φ mod Γ for all morphisms φ between abelian varieties for
which φ∗ D is defined.
Moreover, these functions are unique modulo Γ . Then we may define
(2.6.3) λ(P ) = (− log kz kv + λ(s) ◦ ρ) − (− log kz (0)kv + λ(s) (0)) ,
provided s is chosen so as to generate M at 0 ∈ A0 . By (1) and (2) and (2.6.2),

this definition does not depend on the choice of s.
Before showing (2.6.1), we first describe the group law on A(Cv ) explicitly.
Let P1 and P2 be points in A(Cv ), and let s be a rational section of M which
generates M in neighborhoods of 0, ρ(P1 ), ρ(P2 ), and ρ(P1 ) + ρ(P2 ). Then for
i = 1, 2 there exist zi ∈ Cv such that Pi corresponds to ρ(Pi ) ∈ A0 (Cv ) and the
surjection (f , gs) 7→ zi f − g. Let z0 be the element of Cv for which 0 ∈ A(Cv )
corresponds to 0 ∈ A0 (Cv ) and the surjection (f , gs) 7→ z0 f − g. Finally, for
i = 1, 2 let τi : Q 7→ Q + Pi denote translation by Pi . The theorem of the square
then gives an isomorphism
τ1∗ τ2∗ M ∼
= τ1∗ M ⊗ τ2∗ M ⊗ M −1
which varies algebraically in P1 and P2 and which is the obvious isomorphism
if P1 = 0 or P2 = 0. Therefore there is a rational function u defined by
138 P. Vojta
s ⊗ τ1∗ τ2∗ s
u= .
τ1∗ s ⊗ τ2∗ s
We claim that P1 + P2 corresponds to ρ(P1 ) + ρ(P2 ) and the surjection
(f , gs) 7→ (z1 z2 /z0 u(0))f − g .
Indeed, this defines a rational map A × A 99K A. Replacing s by s 0 = hs changes

z to z 0 = z /h and u to u 0 = u · (hτ1∗ τ2∗ h)/(τ1∗ h · τ2∗ h). Therefore this map is
independent of the choice of s and hence it extends to a morphism on all of
A × A. Checking its value on 0 × A and A × 0 then shows that it must be the
group law.
Now by (2.6.3), the identity (2.6.1) is equivalent to
λ(s) (ρ(P1 ) + ρ(P2 )) = λ(s) (ρ(P1 )) + λ(s) (ρ(P2 )) − λ(s) (0) − log ku(0)kv .
But the proof of the theorem of the square, viewed in the context of Néron
functions, gives exactly this identity.
At times it will be convenient to use a multiplicative version of λm : the
function αm := e −λm satisfies
(2.7) αm (P + Q) = αm (P )αm (Q)
for all P , Q ∈ A(Cv ) and all places v of k .

Therefore, for archimedean v, the functions
αm 1
(2.8) − log and − log
1 + αm 1 + αm
can be taken as Green functions for [0]m and [∞]m , respectively. Since the
metrics on Mm defined by Néron functions are flat,
(2.9) αm dd c αm = d αm ∧ d c αm ,
and the curvatures of the above Green functions are both equal to
(1 + αm )dd c αm − d αm ∧ d c αm
dd c log(1 + αm ) =
(1 + αm )2
(2.10)
dd αm
c
= .
(1 + αm )2
Also let αm,i = pr∗i αm on Ān .
3. The divisor
Once and for all, fix an ample symmetric divisor class L0 on A0 , let
µ
X
L1 = [0]m + [∞]m ,
m=1
and let
L = ρ∗ L0 + L1 .
The following lemma implies that L is ample on Ā.
Lemma 3.1. Let ρ: Ā → A0 be a morphism of complete schemes, all of whose
closed fibers are isomorphic. Let L1 be a nef divisor class on Ā whose restrictions
to closed fibers of ρ are the same under the above isomorphisms, and which is
ample on those fibers. Let L0 be an ample divisor class on A0 . Then L := ρ∗ L0 + L1
is ample.
Proof . This is a straightforward application of Seshadri’s criterion for ampleness

([H 1], Ch. I, Sect. 7). For curves C on Ā and points P ∈ C let mP (C ) denote
the multiplicity of P on C , and let m(C ) = supP ∈C mP (C ). Seshadri’s criterion
implies that there exists δ > 0 depending only on Ā, L0 , and L1 , such that (a) if
C lies in a fiber of ρ, then (L1 . C ) ≥ δm(C ), and (b) otherwise, (L0 . ρ(C )) ≥
m(ρ(C )). Now for any curve C on Ā we have (L . C ) ≥ δm(C ): if C lies in a
fiber of ρ then this follows from (a); otherwise
(L . C ) ≥ (ρ∗ L0 . C ) = (L0 . ρ(C )) ≥ δm(ρ(C )) ≥ δm(C )
since L1 is nef.
For this section, X1 , . . . , Xn will be closed subvarieties of A and X 1 , . . . , X n
will denote their closures in Ā.
Let s = (s1 , . . . , sn ) be a tuple of positive integers, and let i and j be integers
with 1 ≤ i < j ≤ n. We let si · pri −sj · prj denote the morphism from An to A
(or from An0 to A0 if it is clear from the context) defined using the group law.
In the semiabelian case this leads to a problem, because the group law does not
extend to a morphism Ā × Ā → Ā unless µ Q= 0. Q
Therefore we will need a blow-up of X i . Let ψs : X i 99K Ān(n−1)/2 be
the rational map whose components are the restrictions of si · pri −sj · prj as i
and j vary over integers with 1 ≤ i < j ≤ n. Let Ws be the closure of the Q graph
of this rational map. We have a proper birational morphism πs : Ws → X i .
For n-tuples s of positive integers and for rational δ we define
X X X
n
(3.2) Lδ,s = (ρn )∗ (si ·pri −sj ·prj )∗ L0 + (si2 ·pri −sj2 ·prj )∗ L1 +δ si2 pr∗i L
i <j i <j i =1
as a Q-divisor class on Ws . Also let

X X
n X
n
(3.3) Mδ,s = (ρn )∗ (si ·pri −sj ·prj )∗ L0 +(n −1) si2 ·pr∗i L1 +δ si2 ·pr∗i L .
i <j i =1 i =1
Q
It is a Q-divisor class on X i .
The major part of the proof of Theorem 0.2 will consist of showing that
there exists some > 0 and d ∈ N such that O (dL−,s ) has certain properties,
uniformly in s. Here d depends on s, but may not.
140 P. Vojta
Let ` be a positive integer. Multiplication by ` extends to a morphism

µ` : Ā → Ā; moreover, µ∗` [0]m = `[0]m and µ∗` [∞]m = `[∞]m for all m. This,
together with the theorem of the cube, implies that Lδ,s and Mδ,s are homoge-
neous of degree 2 in s1 , . . . , sn (up to pulling back by the obvious morphism
Ws → W`s ). Thus it is natural to extend these definitions to tuples s of positive
rational numbers: for such s, let ` be the lowest common denominator. Then
define Ws = W`s , Lδ,s = `−2 Lδ,`s , and Mδ,s = `−2 Mδ,`s .
Ideally, one would prefer to work entirely with Lδ,s , but for technical reasons
it is easier to introduce Mδ,s (see (10.1)). We now describe the divisor class
πs∗ Mδ,s − Lδ,s .
Definition 3.4. Let s be a tuple of positive integers and let 1 ≤ m ≤ µ and
1 ≤ i , j ≤ n be integers. We define
(3.4.1)
ij
Qs,m = si2 ·pr∗i ([0]m +[∞]m )+sj2 ·pr∗j ([0]m +[∞]m )−(si2 ·pri −sj2 ·prj )∗ ([0]m +[∞]m )
as a divisor on Ws . This is homogeneous of degree 1 in s (up to pulling back, as

before). Therefore if s is a tuple of positive rational numbers with lowest common
denominator `, we define the Q-divisor Qs,m ij
= `−1 Q`s,m
ij
.
Proposition 3.5. The Q-divisor Qs,m

ij
is effective. Its support is contained in the
exceptional set of πs .
Proof . We may assume that s is a tuple of integers. To shorten notation, let

a = si2 and b = sj2 . Using the expression (3.4.1), the divisor can be given the
Green function
αm,i
a
αm,j
b
αm,i
a
/αm,j
b
− log 2 − log 2 + log 2
1 + αm,i
a
1 + αm,j
b
1 + αm,i
a
/αm,j
b
2
αm,i
a
+ αm,j
b
= − log 2 2 .
1 + αm,i
a
1 + αm,j
b
The result then follows immediately from the fact that this function is bounded
from below and is smooth except near the sets
pr∗i [0]m ∩ pr∗j [0]m and pr∗i [∞]m ∩ pr∗j [∞]m .
These sets have codimension two, so they must come from the exceptional set
of πs .
Thus we have
µ X
X
(3.6) Lδ,s = πs∗ Mδ,s − ij
Qs,m ,
m=1 i <j
where the last term is effective. For the bulk of the proof it will be convenient
to regard Lδ,s as a subsheaf of πs∗ Mδ,s and replace the notion of section of Lδ,s
with the notion of section of Mδ,s satisfying certain vanishing conditions.
4. Reductions
First of all, we may assume that X is relatively closed in A and geometrically

irreducible. In the latter case this may involve extending the ground field k , but
such a change will not weaken the theorem.
The next set of reductions follows from some standard results on subvarieties
of abelian varieties defined over C, which carry over directly to the semiabelian
case.
Definition 4.1. Let B (X ) be the identity component of the subgroup
{a ∈ A | X + a = X }
in A. Then the restriction of the quotient map A → A/B (X ) to X exhibits X as a

fibering with fiber B (X ). This map is called the Ueno fibration associated to X .
It is trivial when B (X ) is.
Theorem 4.2 ([N 1], Sect. 4). If X has trivial Ueno fibration then it is of loga-
rithmic general type.
Theorem 4.3 ([N 1], Lemma 4.1). The union Z (X ) of all nontrivial translated
semiabelian varieties of A contained in X is a finite union of irreducible subvari-
eties of X , each of which has nontrivial Ueno fibration.
By a simple Galois theoretic argument, if X and A are defined over k , then
so are B (X ) and Z (X ).
The general plan, then, is the same as in ([V 4], Sect. 10): we may assume
/ X . It then suffices to show that
that B (X ) is trivial; this implies that Z (X ) =
X (RS ) \ Z (X ) is finite. To do so, let
n = dim X + 1
and choose points P1 , . . . , Pn in X (RS )\Z (X ) satisfying conditions CP (c1 , c2 , 1 ):

4.4.1. hL (P1 ) ≥ c1 .
4.4.2. hL (Pi +1 )/hL (Pi ) ≥ c2 ≥ 1 for all i = 1, . . . , n − 1.
4.4.3. P1 , . . . , Pn all point in roughly the same direction in A(RS ) ⊗Z R, up to a
factor 1 − 1 (see (13.2) and (13.3)).
The main part of the proof involves closed subvarieties X1 , . . . , Xn P
of X . We
start with X1 = . . . = Xn = X and successively find collections with dim Xi
strictly smaller. At each stage, X1 , . . . , Xn are assumed to satisfy conditions
CX (c3 , c4 , P1 , . . . , Pn ):
142 P. Vojta
4.5.1. Each Xi contains Pi (and hence has trivial Ueno fibration since Xi *
Z (X )).
4.5.2. Each Xi is geometrically irreducible and defined over k .
4.5.3. The degrees deg Xi satisfy deg Xi ≤ c3 .
4.5.4. The heights h(Xi ) will be bounded by the formula
X
n
h(Xi ) X
n
1
≤ c4 .
hL (Pi ) hL (Pi )
i =1 i =1
Here and throughout the proof, constants c and ci depend only on X , A, n, k ,

S , L, and sometimes the tuple (dim X1 , . . . , dim Xn ), but not on Pi , Xi , or si .
Eventually, this inductive process reaches the point where some Xi is zero
dimensional; i.e., Xi = Pi . As in ([V 4], 10.6), this leads to an upper bound on
hL (P1 ), contradicting (4.4.1).
The following gives the rigorous description of the main step of the proof.
∀ c3 , c4 and ∀ δ1 , . . . , δn ∈ N
∃ c1 , c2 , 1 , c30 , c40 such that

∀ P1 , . . . , Pn ∈ X \ Z (X ) (k ) satisfying CP (c1 , c2 , 1 ) and
∀ X1 , . . . , Xn ⊆ X satisfying CX (c3 , c4 , P1 , . . . , Pn ) and dim Xi = δi ∀ i
∃ X10 , . . . , Xn0 with Xi0 ⊆ Xi ∀ i and Xi0 =
/ Xi for some i ,
and satisfying CX (c30 , c40 , P1 , . . . , Pn ) .

√ In Sects. 12 and 13, si will be taken to be rational numbers close to 1
hL (Pi ). The main step of the proof starts by constructing a small section of
O (dL−,s ) for all large and sufficiently divisible integers d .
5. Self-intersections of Lδ,s
Q
Lemma 5.1. If n ≥ dim X + 1, then the rational map f : Xi 99K Ān(n−1)/2
given by (x1 , . . . , xn ) 7→ (xi − xj )i <j is generically finite.
Proof . If X is a closed subvariety of A and P ∈ Xreg , then the tangent space TX ,P

may be identified with a linear subspace of the tangent space TA,0 at the origin of
A via translation. Via this identification, the intersection of all such TX ,P equals
TB (X ),0 . (This fact is proved by passing to the analytic category and using the
universal covering space; details are left to the reader.)
QSince all Xi have trivial Ueno fibration, there exists a point Q = (Q1 , . . . , Qn )
∈ Xi such that f is smooth at Q, such that Qi lies in (Xi )reg for all i , and such
that
\n
TXi ,Qi = (0) .
i =1
Then any tangent to the fiber of f at Q must be zero, so f is a finite map there.

P
dim Xi
Proposition 5.2. If n ≥ dim X + 1, then (L0,1 ) > 0.
P ∗
Proof . Note that L0,1 = i <j (pri − prj ) L. Thus it is the pull-back, to some
Q
blowing-up of X i , of an ample divisor class on Ān(n−1)/2 via a generically
finite morphism (see ([Kl], Ch. 1, Sect. 2, Proposition 6)).
The remainder of this section is devoted to proving a homogeneity result in
s.
Lemma 5.3. Fix an embedding of k into C. Then the cohomology class in
H∂1,1 n
¯ (A0 ) corresponding to the divisor class
Pij := (pri + prj )∗ L0 − pr∗i L0 − pr∗j L0
is represented over A0 (C)n by a form in
pr∗i E 1,0 (A0 ) ⊗ pr∗j E 0,1 (A0 ) + pr∗i E 0,1 (A0 ) ⊗ pr∗j E 1,0 (A0 ) ⊆ E 1,1 (An0 ) .
Proof . See ([V 4], Lemma 11.3).

By counting degrees we immediately obtain:
Proposition 5.4. Let X1 , . . . , Xn be closed subvarieties of A0 . Then any intersec-
tion product
Y eij Y n
pr∗i L0i
e
Pij .
i <j i =1
Q
of maximal codimension on Xi vanishes unless
X X
2ei + eji + eij = 2 dim Xi , i = 1, . . . , n .
j <i j >i
Consequently, since
(5.4.1) (si · pri −sj · prj )∗ L0 = si2 pr∗i L0 + sj2 pr∗j L0 − si sj Pij ,
it follows that if µ = 0 then the highest self-intersection number of Lδ,s is homo-

geneous of degree 2 dim Xi in each si .
The argument in the semiabelian case is not as straightforward.
Theorem 5.5. The highest self-intersection number of Lδ,s is homogeneous of
degree 2 dim Xi in each si .
Proof . First note that this self-intersection number is independent of the scheme
on which Lδ,s is taken to be defined. Indeed, given any birational morphism,
Chow’s moving lemma allows us to move the corresponding 0-cycle away from
the exceptional set.
144 P. Vojta
P
dim Xi
As before, we begin by writing the self-intersection number (Lδ,s ) as a
sum of terms, each of which is a product of either Pij , pr∗i L0 , pr∗i ([0]m + [∞]m ),
or (si2 ·pri −sj2 ·prj )∗ ([0]m +[∞]m ). SuchQproducts can be evaluated by integrating
suitably chosen Chern-like forms over Xi . For Pij and pr∗i L0 we use the same
forms as in the proof of Lemma 5.3; i.e., obtained from a translation-invariant
metric on O (L0 ). For the terms pr∗i ([0]m + [∞]m ) we use (2.10). Finally, for the
terms (si2 · pri −sj2 · prj )∗ ([0]m + [∞]m ) we use
2 s 2 /e
si /e
dd c αm,i αm,j
j
(5.5.1) 2e · 2
si2 /e sj /e 2
1 + αm,i αm,j
for e ∈ Z, e > 0.
Fixing e for the moment, each of these terms now has a (1, 1)-form attached
to it; this defines a (1, 1)-form Ξ corresponding
Q to Lδ,s . Now Ξ is not necessarily
smooth over any scheme birational to X i , so in general it is not a Chern form
for Lδ,s . However, it is sufficiently close to a Chern form in the sense that the
integral of its top exterior power still equals the highest self-intersection number
Q. This is proved
of Lδ,s as follows. Let a = n(n − 1)/2 and recall the rational map
ψs : X i 99K Āa used in defining Ws . Let f : Āa → Āa be the morphism given
on each factor by multiplication by e. Let V be a desingularization of Ws ×Āa Āa ,
and let g : V → Ws be the projection.
 −→
V Āa
g f
y y
ψs
Ws −→ Āa
Then g is generically finite. Moreover the forms (5.5.1) come from forms on Āa
which pull back via f to smooth forms which are indeed Chern forms associated
to Green functions as in (2.10). Thus g ∗ Ξ is a Chern form representing g ∗ Lδ,s
and therefore the equality between the integral and the intersection number holds
after pulling back to V . By formal properties of intersection theory (see ([Kl], Ch.
1, Sect. 2, Proposition 6)) and integration, the desired property therefore holds
on Ws . Note in particular that the integral in question is independent of e; the
proof proceeds by breaking the integral into parts and for each part taking the
limit as e → ∞.
However, if one breaks this integral further into subterms in the naı̈ve way,
one obtains divergent integrals. Therefore some care is needed.
Replacing each X i with a desingularization such that L1 pulls back to a normal
crossings divisor does not affect the integral. Therefore it suffices to work on a
bounded open set Ω ⊆ CN such that each αm,i is of the form
Y
N
ρm,i |zj |2fmij ,
j =1
where fmij ∈ Z and ρm,i is a nonzero smooth function on the closure of Ω. We

may further assume that for each I ⊆ {1, . . . , N }, the boundary of Ω intersects
the coordinate subspace defined by zi = 0 ∀ i ∈ I in a set of measure zero in
CN −#I . We also assume each zi comes from a function on some X j .
To fix notation, let 0 ≤ p ≤ N and let Ψ be a smooth (N − p, N − p)-form,
which we may assume to be of the form
(smooth function) · dzi1 ∧ . . . ∧ dziN −p ∧ d z̄j1 ∧ . . . ∧ d z̄jN −p .
Let F = (fij ) be a p × N matrix with entries in Z, and for i = 1, . . . , p let

Y
N
γi = |zj |2fij .
j =1
For all such i let ρi be a positive smooth function, bounded away from zero on
the closure of Ω, and let βi = ρi γi . We will consider integrals
Z 1/e 1/e
e∂ ∂β
¯ e∂ ∂β
¯ p
Ψ∧ 1
1/e
∧ ... ∧ 1/e
,
Ω (1 + β1 )2 (1 + βp )2
where βi are of the form αja,m /αkb ,m (cf. (5.5.1)). Thus, as in (2.9),
e∂ ∂β
¯ 1/e 1 β 1/e ∂β ∂β
¯
= · · ∧ ,
(1 + β 1/e )2 e (1 + β 1/e )2 β β
and therefore the integral can be rewritten as
Z 1/e 1/e
β1 βp ∂β1 ∂β¯ 1 ∂βp ∂β
¯ p
(5.5.2) e −p ... Ψ∧ ∧ ∧...∧ ∧ .
Ω (1 +
1/e
β 1 )2 (1 +
1/e
β p )2 β1 β1 βp βp
Lemma 5.5.3. For each positive e ∈ Z, let φe : Ω → C be a function which is
measurable and bounded uniformly in e and z ∈ Ω. Let p, F , γi , ρi , and βi be
as above, and let
Yp
κi = |zj |2fij .
j =1
Then for each e the integral

Z 1/e 1/e
β1 βp
e −p φe · 1/e
... 1/e
(5.5.3.1) Ω (1 + β1 )2 (1 + βp )2
∂γ1 ∂γp d z̄1 d z̄N
· ∧ ... ∧ ∧ dzp+1 ∧ . . . ∧ dzN ∧ ∧ ... ∧
γ1 γp z̄1 z̄N
converges to a value bounded uniformly in e. Moreover, let F 0 be the matrix

consisting of the first p columns of F and assume that φ := φe is independent of
e and is C 1 on the closure of Ω. Then as e → ∞ these integrals approach the
finite limit
146 P. Vojta
(5.5.3.2) Z
(−1)N (N −1)/2 (2π)N 0 dd c |zp+1 |2 dd c |zN |2
√ (det F ) φ ∧ . . . ∧
( −1)N Ω∩{z1 =...=zp =0} |zp+1 | |zN |
Z
κ1 κp dd |z1 |
c 2
dd |zp |
c 2
· ... ∧ ... ∧ .
D p (1 + κ 1 ) 2 (1 + κ p )2 |z |2
1 |zp |2
Proof . Since
∂γi X dzj
N
= fij ,
γi zj
j =1
it follows that (5.5.3.1) equals

Z 1/e 1/e
−p β1 βp
e φe · 1/e
... 1/e
Ω (1 + β1 )2 (1 + βp )2
dz1 dzp d z̄1 d z̄N
· (det F 0 ) ∧ ... ∧ ∧ dzp+1 ∧ . . . ∧ dzN ∧ ∧ ... ∧ .
z1 zp z̄1 z̄N
Hence if F 0 is singular then the integral vanishes.

Otherwise there exist t1 , . . . , tp ∈ [−1, 1] such that, letting
X
p
j = 2 ti fij for j = 1, . . . , N ,
i =1
we have j > 0 for j = 1, . . . , p. But now note that for x > 0 and t ∈ [−1, 1],
x
≤ xt .
(1 + x )2
In particular, we apply the facts that
1/e
βi t /e
1/e 2
≤ βi i , i = 1, . . . , p
(1 + βi )
to bound the absolute value of the integral by

(5.5.3.3) ÃN !
(2π)N | det F 0 | Y
sup ρi (z )ti /e
ep
i =1 ∈Ω
z
Z Yp
Y
N
dd c |z1 |2 dd c |zN |2
· |φe | |zj |j /e · |zj |1+j /e · ∧ ... ∧ .
Ω |z1 | 2 |zN |2
j =1 j =p+1
The first assertion of the lemma is then clear, since

Z
dd c |z |2
|z |/e ≤ O(e) .
D |z 2 |
The second assertion follows from the fact that for a region Ω 0 ⊆ Cp , for
positive smooth ρi : Ω 0 → R, for smooth φ : Ω 0 → C, and for κi as above,
Z
1 (ρ1 κ1 )1/e (ρp κp )1/e dd c |z1 |2 dd c |zp |2
lim p φ(z ) ... ∧ ... ∧
e→∞ e Ω0 (1 + (ρ1 κ1 ) )
1/e 2 (1 + (ρp κp ) ) |z1 |
1/e 2 2 |zp |2
Z
κ1 κp dd c |z1 |2 dd c |zp |2
= φ(0) . . . ∧ . . . ∧ .
Dp (1 + κ1 )2 (1 + κp )2 |z1 |2 |zp |2
This is proved by replacing zi with zie and applying straightforward arguments.
By (5.5.3.3), we may then use Fubini’s theorem to reduce the second assertion
of the lemma to the above limit.
Corollary 5.5.4. The same conclusions hold with (5.5.3.1) replaced by
Z 1/e 1/e
−p β1 βp
e φe · 1/e
. . . 1/e
(5.5.4.1) Ω (1 + β1 )2 (1 + βp )2
∂β1 ∂βp d z̄1 d z̄N
· ∧ ... ∧ ∧ dzp+1 ∧ . . . ∧ dzN ∧ ∧ ... ∧ .
β1 βp z̄1 z̄N
Proof . We use the identity

∂βi ∂ρi ∂γi
= + , i = 1, . . . , p
βi ρi γi
and expand (5.5.4.1) into a sum of 2p integrals. Each such integral can be written
as a sum of integrals of the form (5.5.3.1) by expanding out any smooth forms
∂ log ρi in terms of dz1 , . . . , dzp , incorporating them into φe , and permuting the
indices 1, . . . , p. This gives the convergence. But now each term involving a
d log ρi vanishes as e → ∞, due to the extra factor 1/e which appears when p
decreases. This proves the assertion on taking a limit.
Continuing with the proof of Theorem 5.5, consider again the integral (5.5.2).
Breaking Ψ into its components, we may assume that
Ψ = ψ dzi1 ∧ . . . ∧ dziN −p ∧ d z̄j1 ∧ . . . ∧ d z̄jN −p .
Permuting coordinates, we may assume {i1 , . . . , iN −p } = {p +1, . . . , N }. Then we
may also assume that {j1 , . . . , jN −p } = {p + 1, . . . , N }. Indeed, if, for example,
1 ∈ {j1 , . . . , jN −p }, then φ in (5.5.4.1) vanishes along z1 = 0 since Ψ has a
term d z̄1 , while (5.5.4.1) only requires d z̄1 /z̄1 . This causes the limit (5.5.3.2) to
vanish. Thus we are reduced to considering
(5.5.5)
Z Z
κ1 κp ∂κ1 ∂κp ∂κ ¯ 1
Ψ· ... ∧ ... ∧ ∧ ∧ ...
Dp (1 + κ1 ) (1 + κp ) κ1 κp κ1
2 2
Ω∩{z1 =...=zp =0}
Z
∂κ
¯ p
0 2
... ∧ = (det F ) Ψ
κp Ω∩{z1 =...=zp =0}
Z
κ1 κp dz1 dzp d z̄1 d z̄p
· ... ∧ ... ∧ ∧ ∧ ... ∧ .
Dp (1 + κ1 ) (1 + κp ) z1
2 2 zp z̄1 z̄p
148 P. Vojta
Now consider how this expression changes as the matrix F varies. Suppose
fij = sj2 gij . Then det F 0 is quadratic in each of s1 , . . . , sp . Also, letting
Y
N
ωi = |zj |2gij ,
j =1
we find that the factor

Z
κ1 κp dz1 dzp d z̄1 d z̄p
... ∧ ... ∧ ∧ ∧ ... ∧
D p (1 + κ 1 )2 (1 + κ p )2 z
1 zp z̄1 z̄p
Z
1 ω1 ωp dz1 dzp d z̄1 d z̄p
= 2 ... ∧ ... ∧ ∧ ∧ ... ∧
s1 . . . sp2 Dp (1 + ω1 )2 (1 + ωp )2 z1 zp z̄1 z̄p
1/s 2
by replacing each zj with zj j . Thus the expression (5.5.5) is quadratic in each
of s1 , . . . , sp . As in (5.4.1), however, the form Ψ is also quadratic in each of
sp+1 , . . . , sN . Therefore, keeping track of which Xi each zj comes from gives the
theorem.
6. A lower bound on h 0
The goal of this section is to prove a lower bound on h 0 (Ws , O (dL−,s )) for some
fixed > 0 and sufficiently large (and divisible) d > 0.
Lemma 6.1. For all (rational) δ > 0, the Q-divisor class Lδ,s is ample.
Proof . We may regard Ws as a closed subscheme of Ān+n(n−1)/2 in an obvious

way, and Lδ,s extends in an obvious manner. The lemma then follows by applying
Lemma 3.1 to the morphism Ān+n(n−1)/2 → A0
n+n(n−1)/2
.
Proposition 6.2. There exist constants c > 0 and > 0, depending only on
X , Ā, L, dim X1 , . . . , dim Xn , and the bounds on deg Xi , such that for all tuples
s = (s1 , . . . , sn ) of positive rational numbers,
P Y
n
2 dim Xi
h (Ws , O (dL−,s )) ≥ cd
0 dim Xi
si
i =1
for all sufficiently large d (depending on s).
Proof . By Lemma 6.1, Lδ,s is ample. Riemann-Roch therefore implies that, as

d → ∞,
(Ldim
δ,s
Ws
)
h 0 (Ws , O (dLδ,s )) = d dim Ws (1 + o(1)) .
(dim Ws )!
Choose ` such that `L is very ample; then for each index i let Hi be the subscheme
of Ws cut out by some section of Γ (X i , O (`L)). As before,
dim Ws −1
(Hi . Lδ,s )
h 0 (Hi , O (dLδ,s )H ) = d dim Ws −1 (1 + o(1))
i (dim Ws − 1)!
dim Ws −1
(pr∗i L . Lδ,s )
= `d dim Ws −1 (1 + o(1)) .
(dim Ws − 1)!
These two estimates replace the first two estimates in the proof of ([V 4], Propo-
sition 11.5); the proof then continues as in that case, with a little extra care
because of the variable `.
7. Generalized Weil functions
This section gives some preliminary results on Weil functions in preparation for
Sect. 9.
For a general reference on Weil functions, see [L 2] or [L 4]. Instead of
` X × Mk (for a scheme X of finite type over k ), however, we will
working over
work over v X (Cv ). This will be denoted by X (Mk ). Also, Weil functions will
be normalized so that − log kf k is a Weil function for the principal divisor (f ).
The results of [L 2] carry over into this situation.
Definition 7.1. A generalized Weil function on a scheme X of finite type over
k is an equivalence class of pairs (U , g). Here U is a dense Zariski-open subset
of X and g : U (Mk ) → R is a function such that there exists a scheme X e and a
e
proper birational morphism Φ: X → X such that g ◦ Φ extends to a Weil function
for some divisor De on X e . Pairs (U , g) and (U 0 , g 0 ) are equivalent if g = g 0 on
0 e is an effective divisor. The support
(U ∩ U )(Mk ). We say that g is effective if D
of g, written Supp g, is defined as the set Φ(Supp D).e
Proposition 7.2. Generalized Weil functions on a scheme X form an abelian

group under addition. If φ: X 99K Y is a dominant rational map and g is a gen-
eralized Weil function on Y , then φ∗ g (defined in the obvious way) is a generalized
Weil function on X .
Proof . Obvious.
Proposition 7.3. Let g1 and g2 be generalized Weil functions on a proper scheme
X /k . Then g3 := min(g1 , g2 ) is also a generalized Weil function on X . If g1 and
g2 are effective, then so is g3 , and Supp g3 ⊆ Supp g1 ∩ Supp g2 .
Proof . Since min(g1 , g2 ) = g2 + min(g1 − g2 , 0), we may assume that g2 = 0. By

blowing up X , we may assume that g1 is a Weil function. Moreover, we may
further blow up X to the point where components occurring with positive mul-
tiplicities in div(g1 ) do not intersect those occurring with negative multiplicities.
This is accomplished as follows. Let D = div(g1 ), and let U1 , . . . , Un be a finite
cover of X by open affines such that D = (fi ) on Ui for each i . Then we replace
150 P. Vojta
X with the graph of the rational map X 99K (P1 )n given by (f1 , . . . , fn ). The first
assertion then follows by standard properties of Weil functions.
To prove the other assertion, assume g1 and g2 are effective. Let Φ : X e →X
∗ ∗
be a proper birational morphism P such that Φ g1 and Φ g2 are Weil functions.
Moreover, writing div(Φ∗ gi ) = niD · D for i = 1, 2 we may assume that prime
divisors D for which n1D > n2D do not meet prime divisors P with n1D < n2D .
Then Φ∗ g3 is also a Weil function, associated to the divisor min(n1D , n2D ) · D.
This easily gives Supp Φ∗ g3 = Supp Φ∗ g1 ∩ Supp Φ∗ g2 . Pushing it down to X
gives the desired inclusion (which may become strict).
Proposition 7.4. Let f : X → Y be a morphism of proper schemes over k and let
g be a generalized Weil function on X whose restriction to a generic closed fiber
of f is effective. Then there exists a generalized Weil function g 0 on Y such that
Supp g 0 does not contain f (X ) and such that f ∗ g 0 ≤ g. If f is surjective then we
may choose g 0 such that Supp g 0 = f (Supp g).
Proof . Replacing X with a suitable nonsingular blowing-up, we may assume that

g is a Weil function, relative to a divisor D. The condition then implies that the
restriction of D to the generic fiber of f is effective. Then there exists a divisor
D 0 on Y such that D − f ∗ D 0 is effective on X . Let g 0 be a Weil function for D 0
on Y ; then the L
desired inequality holds up to the addition of an Mk -constant (i.e.,
an element of v R). We may then adjust g 0 by such a constant to obtain the
inequality without constants. To prove the last assertion, we may assume after
blowing up X and Y that X and Y are nonsingular, g is a Weil function (as
above), and that f (Supp g) is a divisor. Then the above choice of D 0 may be
made such that Supp D 0 = f (Supp D).
Definition 7.5. Let X be a variety. A min-min generalized Weil function on X
is a generalized Weil function on X which can be written in the form
(7.5.1)
min(− log kφ1 k, . . . , − log kφn k) − min(− log kψ1 k, . . . , − log kψm k) + cv
for some Mk -constant (cv ), where φ1 , . . . , φn , ψ1 , . . . , ψm are nonzero rational

functions on X . If g is a min-min generalized Weil function, then we also say that
g is of min-min type. A min-min Weil function is a Weil function of min-min
type.
Definition 7.6. Let g be a generalized Weil function on a variety X and let F ⊆ X

be a finite set. We say that g is nicely defined at F if Supp g is disjoint from F ,
if g is of min-min type, and if g can be written as an expression (7.5.1) in which
φ1 , . . . , φn , ψ1 , . . . , ψm are all regular at all P ∈ F , and in which some fixed φi
and some fixed ψj are nonzero at all P ∈ F .
Proposition 7.7. Min-min generalized Weil functions on a variety X form a

subgroup of the group of generalized Weil functions on X . The same assertion
holds for generalized Weil functions nicely defined at some fixed finite subset
F ⊆ X.
Proof . Additivity of min-min generalized Weil functions follows from the identity
min ai + min bj = min(ai + bj ) .

i j i ,j
The other assertions are trivial.

Proposition 7.8. If φ : X 99K Y is a dominant rational map of varieties and g is
a min-min generalized Weil function on Y , then φ∗ g is also of min-min type. If φ
is regular at all P ∈ F for some finite subset F ⊆ X and g is nicely defined at
φ(F ), then φ∗ g is nicely defined at F .
Proof . Obvious.
Proposition 7.9. If g1 and g2 are min-min generalized Weil functions on a given
variety, then so are min(g1 , g2 ) and max(g1 , g2 ). If g1 and g2 are nicely defined at
some finite subset F ⊆ X , then so are min(g1 , g2 ) and max(g1 , g2 ).
Proof . The assertions regarding min(g1 , g2 ) follow from the identity

min min ai − min bj , min ck − min d`
i j k `

= min min(ai + d` ), min(bj + ck ) − min(bj + d` ) .
i ,` j ,k j ,`
The result for max(g1 , g2 ) is similar.

Proposition 7.10. Let X be a projective variety, D a Cartier divisor on X , and
F ⊆ X a finite set disjoint from Supp D. Then there is a Weil function g with
divisor D which is nicely defined at F .
Proof . First assume X = Pn and D is the hyperplane at infinity. Let x1 , . . . , xn

be the standard coordinate functions on X \ D. By applying a suitably chosen
automorphism of Pn fixing D, we may assume that x1 is nonzero at all P ∈ F .
Then g := max1≤i ≤n (log kxi k) has the required properties. By Proposition 7.8
this extends to the case where X is arbitrary and D is very ample. The general
case then follows by writing D as a difference of two very ample divisors not
passing through F .
8. Metrics at non-archimedean places
This section introduces metrics on line sheaves at non-archimedean places, to

parallel the theory at infinite places.
152 P. Vojta
Definition 8.1. Let K be a local field with valuation ring R, let X be a proper
scheme over Spec R, let L be a line sheaf on X , let U be a Zariski-open subset
of X ×R K , and let γ ∈ Γ (U , L ). For closed points P ∈ U we define kγ(P )k as
follows. Let K1 = K (P ) and let R1 be its valuation ring. The valuative criterion
of properness implies that P extends to a section σ : Spec R1 → X over Spec R.
Then σ ∗ γ is a rational section of σ ∗ L ; letting g be a generator of σ ∗ L we have
σ ∗ γ = ag for some a ∈ K1 . We then define kγ(P )k = kak; this is independent of
the choice of g.
This defines a metric on L , in the sense that if f is a function that is regular
at P then k(f γ)(P )k = kf (P )k · kγ(P )k.
Proposition 8.2. With notation as above, the function P 7→ kγ(P )k is continuous
on U (K ) (in the topology induced by the valuation).
Proof . Fix a point P0 ∈ U (K ). Let V be an open neighborhood of the point

where P0 passes through the special fiber of X ; we may assume that L is trivial
on V . Then there exists a rational function f on V , regular on U ∩ V , such that
kγ(P )k = kf (P )k for all P ∈ (U ∩ V )(K ) entirely contained in V . Since all P
in a sufficiently small neighborhood of P0 satisfy this condition, it follows that
P 7→ kγ(P )k is continuous in this neighborhood. This implies continuity.
b by
Definition 8.3. If U and γ are as above, then we define kγ(P )k on U K
continuity.
Lemma 8.4.
(a) The above definition is functorial: if, in addition to the above notation,
f : X2 → X is a morphism of proper schemes over Spec R and P2 ∈
b , then kf ∗ γ(P )k = kγ(f (P ))k.
f −1 (U ) K 2 2
b .
(b) If γ ∈ Γ (X , L ) then kγ(P )k ≤ 1 for all P ∈ X K
(c) If a ∈ K then kaγ(P )k = kak · kγ(P )k.
(d) If L2 is another line sheaf on X and γ2 ∈ Γ (U , L2 ) then
k(γ ⊗ γ2 )(P )k = kγ(P )k · kγ2 (P )k .
Proof . Obvious.
We also note that the converse of (b) holds if X is normal.
Definition 8.5. If X is a proper scheme over a localization of the ring of integers
of a number field, then we define kγ(P )kv for non-archimedean places v by base
change to the completed local ring at v.
Lemma 8.4 holds also in the context of number fields.
Proposition 8.6. Let R be as in Definition 8.1 (resp. 8.5), let X be a proper

scheme over Spec R, let L be a line sheaf over X , and let γ ∈ Γ (U , L ) be a
nonzero global section over an open subset U . Then − log kγk (resp. − log kγkv )
defines a Weil function on X relative to the divisor associated to γ.
Proof . If L = OX then this is trivial. If L and γ are trivial on the generic fiber
then this follows from parts (b) and (d) of Lemma 8.4. By combining these two
facts with Chow’s lemma and functoriality, we may reduce the problem to the
case where X = PnR , L = O (1), and γ = x0 . In that case it can be checked by
direct computation.
9. An analytic result
This section proves an analytic result which will be needed in Sects. 10 and 11.
For the latter section, it will be important to establish bounds having a uni-
formity as v varies over all places of a number field. This uniformity is provided
by the formalism of Weil functions.
We start with some lemmas.
Lemma 9.1. Let X ⊆ PN be a smooth projective variety of dimension n and let

P0 ∈ X . Then for generic linear subspaces L of codimension n − s + 1, the linear
projection from L induces a smooth morphism p : U → Pn−s for a Zariski-open
U ⊆ X such that X \ U meets the fiber p −1 (p(P0 )) at only finitely many points.
Moreover, L meets X transversally. Finally, if s = 0 then U contains the entire
fiber over p(P0 ).
Proof . By a minor adaption of the proof of Bertini’s theorem, one can show
that the generic hyperplane passing through P0 crosses X transversally except at
finitely many points. This holds even if X has finitely many singular points. Thus,
by induction, the generic linear subspace L0 of codimension n − s containing
P0 meets X transversally except at finitely many points. If L ⊆ L0 is any linear
subspace with P0 ∈ / L and dim L = dim L0 − 1, then the corresponding projection
satisfies the first assertion of the lemma, by ([H 2], III 10.4(iii)). The second
assertion is satisfied for a generic choice of L within L0 , by Bertini’s theorem.
To prove the last assertion, we first assume that X is a curve (possibly re-
ducible) and P0 ∈ X , and show that the generic hyperplane H through P0 crosses
X transversally. Indeed, it is sufficient that H is not tangent to X at P0 and that
it avoid the (finitely many) singular points and the points Q ∈ X \ {P0 } such that
the line P0 Q is tangent to X at Q. Any irreducible component of X containing
infinitely many such Q must be a line through P0 , which the generic hyperplane
avoids. For such generic projections, p is étale at all of p −1 (p(P0 )). This proves
the last assertion.
154 P. Vojta
Lemma 9.2. Let Y ⊆ X be affine schemes of finite type over k and let F be
a finite subset of Y such that X and Y are regular at all P ∈ F and such that
dim OP ,X − dim OP ,Y is independent of P for P ∈ F . Let r equal this constant.
Then there exist f1 , . . . , fr ∈ O (X ) which generate the sheaf of ideals I of Y in
X in a neighborhood of F .
Proof . If r = 0 then this is immediate. If r > 0 then for each P ∈ F there exists
gP ∈ I which lies in the maximal ideal P mP ,X ⊆ OP ,X , but not in m2P ,X . There
exists a suitable linear combination f1 := φP gP for φP ∈ O (X ) which lies in
mP ,X \ m2P ,X for all P . Let X 0 = Spec O (X )/(f1 ); by induction on r there exist
f¯2 , . . . , f¯r ∈ O (X 0 ) generating the sheaf of ideals of Y in X 0 . Lifting the f¯i to
fi ∈ OX for i = 2, . . . , r then gives the required factors.
For vectors z = (z1 , . . . , zr ) ∈ Crv , we define kzkv = max kzi kv if v is
non-archimedean; otherwise we use the standard definition kzk = (|z1 |2 + . . . +
|zr |2 )[kv :R]/2 .
Definition 9.3. Let Y be a projective scheme over k and let g be an effective
generalized Weil function on Y . For each place v let
Λv (g) = {(P , z) ∈ Y (Cv ) × Crv | kzk < e −g(P ) }
and
Υv (g) = {(P , z) ∈ Y (Cv ) × Crv | kzi k < e −g(P ) , i = 1, . . . , r} .
Here, if P ∈ Supp g then we take g(P ) = ∞ so that e −g(P ) = 0.
Note that these two definitions coincide if v is non-archimedean. Strictly
speaking, the value of r should be specified in the notation, but its value will
always be clear from the context. Often these sets will be identified with subsets
of PrY (Cv ).
The goal of the rest of this section is to construct certain rigid analytic maps
with domain Λv (g) or Υv (g). For our purposes, though, it suffices to regard them
as maps such that, for all P ∈ Y (Cv ) with P ∈ / Supp g, the restriction to the disc
or polydisc Λv (g) ∩ {P } × Crv or Υv (g) ∩ {P } × Crv is given by a power series.
The following lemma does most of the work that will be needed.
Lemma 9.4. Let p : Γ → Y be a morphism of equidimensional projective k -
schemes with a regular section σ : Y → Γ , let q : Γ → PrY be a generically
finite morphism such that q ◦ σ equals the canonical section of the natural map
π : PrY → Y with image Y × {[1 : 0 : . . . : 0]}, and let z = (z1 , . . . , zr ) denote
the coordinate functions on Ar = Pr \ {x0 = 0}. Let F be a finite subset of the
image of σ such that Γ is regular at all P ∈ F and such that q is étale in a
neighborhood of F . Then:
(a) There is an effective generalized Weil function g1 on Y with support disjoint
from p(F ) such that, for each v, the map Γ (Cv ) → PrY (Cv ) has a rigid analytic
partial section θv : Λv (g1 ) → Γ whose image contains σ(Y \ Supp g1 ).
(b) Let φ1 , . . . , φs be rational functions on Γ which are regular on F . Then g1 can

be chosen such that there exist effective generalized Weil functions g1∗ , . . . , gs∗
on Y such that for all v, all (P , z) ∈ Λv (g1 ), and all i ,
∗
(9.4.1) kφi (θv (P , z)) − φi (θv (P , 0))k ≤ e gi (P ) · kzk .
Moreover, Supp gi∗ is disjoint from p(F ) for all i .
Proof . First consider non-archimedean places v. Fix a finite cover of Y by open

affines such that p(F ) lies in each open set. Let V be any element of this cover,
and let y1 , . . . , y` be a generating set for O (V ) over k . After adjusting yi by
constant factors, we may assume that for all non-archimedean v the sets
{Q ∈ V (Cv ) | kyi (Q)k ≤ 1 for all i }
cover Y (Cv ) as V varies over the chosen cover. For each such V fix an open
affine U ⊆ q −1 (V ) containing F and let x1 , . . . , xM be a generating set for O (U )
over k . We may assume that xi = yi ◦ p for i = 1, . . . , `. By Lemma 9.2, there
exist polynomials f`+1 (X ) = f`+1 (X1 , . . . , Xn ), . . . , fM −r (X ) which generate the
sheaf of ideals of Γ in AM near all P ∈ F . We may assume that the coefficients
of f`+1 , . . . , fM −r all lie in R (the ring of integers of k ). Let fM −r+1 , . . . , fM be
polynomials in R[X1 , . . . , XM ] which equal ai · zi ◦ q on U for some ai ∈ k ∗ and
all i = 1, . . . , r.
Fix a non-archimedean place v and Q ∈ V (Cv ) with kxi (Q)k ≤ 1 for all i . For
i = 1, . . . , ` let fi (X ) = Xi − xi (Q); then all of f1 , . . . , fM lie in Rv [X1 , . . . , XM ],
where Rv is the valuation ring of Cv . Let J denote the matrix (∂fi /∂Xj )1≤i ,j ≤M .
All entries in this matrix lie in R[X1 , . . . , XM ] and are independent of v and Q.
The assumption that q is étale near F implies that det J = / 0 at all P ∈ F .
For an M × M matrix A with entries in Cv we let kAk = infkbk=1 kAbk. It
follows that if A is nonsingular then kAk−1 equals the largest absolute value of
an entry of A−1 . If all entries of A lie in Rv , then kAk ≥ k det Ak.
By ([V 4], Corollary 15.13a), there is a rigid analytic lifting of q over the
subset kzk < kJ (σ(Q))k2 / max kai k of π −1 (Q) which maps 0 to σ(Q). (More-
over, as Q varies, the lifting varies rigid analytically since the convergents vary
algebraically in Q.)
Let gV = max 0, − log k det J ◦ σk2 / max kai k . This is an effective gen-
eralized Weil function on Y . Its support is disjoint from F since det J (P ) = / 0
for all P ∈ F . By construction, for all non-archimedean v there exists a unique
lifting of q over
Λv (gV ) ∩ π −1 ({Q ∈ V (Cv ) | kyi (Q)k ≤ 1 for all i }) .
Let g1 be the maximum of all gV . Then g1 satisfies part (a) for all non-
archimedean v.
The proof for archimedean places v is essentially the same, except that it
requires a little more care due to the archimedean property of v. We will work
156 P. Vojta
with | · | instead of k · k for consistency with [V 4]. We define |A| for an M × M

matrix A by the same formula as before. We then have
| det A|
|A| ≥ .
M ! · max |aij |M −1
We will use ([V 4], Lemma 15.8 and Corollary 15.13b), with c = 1/3. We
will use the same open subsets U and V as before; for each such V let C be a
compact subset of V (Cv ) such that the union of all these C cover Y (Cv ). Let
Ã M M !
X X ∂ 2 f(α) c0

B = max sup vi wj , sup J (α0 ) ;
∂Xi ∂Xj 3 α0 ∈σ(C )
i =1 j =1
here the first supremum is taken over the set of all points α ∈ U (Cv ) of distance
≤ 1 from σ(C ) and the set of all unit vectors u and v in CM . In the second term
in the above maximum,
∞ i 2 −1
X
i
0 4 3
(9.4.2) c = .
3 8
i =0
(This second term ensures that ρ < 1 in the statement of ([V 4], Lemma 15.8).)
This variant of Hensel’s lemma then gives a unique complex analytic map θv
over the set ( )
|J (σ(Q))|2
(Q, z) ∈ C × C |z| <
r
.
3B
with the desired properties. Letting

∂f
i
D = sup max (α0 ) ,
α0 ∈σ(C ) 1≤i ,j ≤M ∂X j
we have
| det σ(Q)|
|J (σ(Q))| ≥
M ! · D M −1
and therefore g1 will also satisfy part (a) for the archimedean places after adding
the constants [kv : R] log 3B (M ! · D M −1 )2 to g1,v for v | ∞.
To prove (b), we may assume that φ1 , . . . , φs were included among the gen-
erators x1 , . . . , xM of O (U ) in the proof of part (a). Then the application of
Hensel’s lemma bounds the variation in φi . Indeed, if v is non-archimedean,
then ([V 4], (15.5)) implies that
kzk
kφi (θv (P , z)) − φi (θv (P , 0))k ≤ ,
kJ (P )k
and in the archimedean case the same inequality holds up to multiplication by
c 0 or c 02 by ([V 4], (15.11)) and (9.4.2). So we may take gi∗ = − log k det J (P )k,
with [kv : R] log c 0 added at archimedean places.
Definition 9.5. Let A be a nonsingular projective variety over k . An extended

model of A is a model A of A which is proper and flat over Spec R plus, for each
v | ∞, a hermitian metric on the tangent bundle of A(Cv ). For a given extended
model we define distance functions dv (·, ·) on A(Cv ) for each v as follows. If
v | ∞ then dv is the distance relative to the chosen metric on the tangent space of
A(Cv ). For non-archimedean v, suppose P and Q lie in A(Cv ). Let Rv denote the
valuation ring of Cv ; then P and Q define sections of A ×R Rv . If these sections
do not meet on the closed fiber, then let dv (P , Q) = 1. Otherwise let B be the
local ring of A ×R Rv at the point where these sections meet the closed fiber,
and let dv (P , Q) = supφ∈B kφ(P ) − φ(Q)k.
Remark 9.6. If v is non-archimedean and U is an open affine in A×R Rv which

contains the images of the sections corresponding to P and Q, then the above
local ring B may be replaced with O (U ). This holds even if the corresponding
sections do not meet on the closed fiber.
Lemma 9.7. Let f : Z → X be a birational projective morphism of integral

schemes, quasi-projective and of finite type over k . Then there exists a coherent
sheaf of ideals I on X such that f : Z → X is isomorphic to the blowing-up of
X with respect to I . Moreover, there exists a finite collection I1 , . . . , Is of such
sheaves of ideals such that the intersection of the corresponding subschemes of
X equals the set over which f fails to be an isomorphism.
Proof . The first assertion follows from the proof of ([H 2], II 7.17). It is stated
there only for quasi-projective varieties over k , but the proof can be adapted
to the present situation as follows. First replace ([H 2], II 5.20) with ([H 2],
III Remark 8.8.1). Next, when proving that S and T agree in all large enough
degrees in Step 2, replace the use of finiteness of the integral closure with an
adaptation of the proof of ([H 2], III 5.2).
To prove the second assertion, it suffices to show that given any point P ∈ X
such that f is an isomorphism over a neighborhood of P , one can choose I such
that its corresponding subscheme of X does not contain P . This can be done by
choosing the map OX → J ⊗ M n to be an isomorphism at P .
Proposition 9.8. Let A be a nonsingular projective variety over k , let A and A0
be two projective models for A over R, and let dv and dv0 be the corresponding
distance functions on A(Cv ) for non-archimedean v. Then there exists an Mk -
constant (cv ) such that
dv (P , Q) ≤ e cv dv0 (P , Q)
for all non-archimedean v and all P , Q ∈ A(Cv ).
Proof . It will suffice to show that if there exists a morphism ψ : A → A0

which restricts to the identity on A, then there exists (cv ) such that
dv0 (P , Q) ≤ dv (P , Q) ≤ e cv dv0 (P , Q)
158 P. Vojta
for all v, P , and Q as above.

The first inequality holds because ψ induces a homomorphism of local rings
on A0 into their corresponding local rings on A.
To prove the second inequality, Lemma 9.7 implies that there exist sheaves of
ideals I1 , . . . , Ir on A0 suchTthat for all i , A is isomorphic to the blowing-up
of A0 at Ii , and such that i Z (Ii ) is contained in the special fiber of A0 .
This latter condition implies that the sheaf of ideals (I1 , . . . , Ir ) contains some
nonzero a ∈ R.
Let P , Q ∈ A(Cv ) for some non-archimedean v. We will show that
dv0 (P , Q)
(9.8.1) dv (P , Q) ≤ .
kakv
If P and Q correspond to different points on the special fiber of A0 ×R Rv then

this is obvious since dv (P , Q) = dv0 (P , Q) = 1. Otherwise let Spec B be an open
affine such that Spec B ⊗R Rv contains this point on the special fiber. Write B 0 =
B ⊗R Rv . For all i let Ii be the ideal in B 0 corresponding to Ii and let Ii0 = Ii ⊗R Rv .
Since Rv is flat over R (it is a torsion free module over the local ring, which is
principal), Ii0 ⊆ B 0 for all i , and
Lthe restriction of ψ 0 : A ×R Rv → A0 ×R Rv
0 −1 0 0 n
to (ψ ) (Spec B ) equals Proj n≥0 (Ii ) for all i . Since a ∈ (I1 , . . . , Ir ), there
exists some i and some b ∈ Ii such that kb(P )k ≥ kakv . Let b1 , . . . , bs be
generators for Ii ; we may assume that kb1 (P )k ≥ kbj (P )k for j = 1, . . . , s;
therefore kb1 (P )k ≥ kakv . Now if kb1 (Q)k = / kb1 (P )k then kb1 (P ) − b1 (Q)k ≥
kakv , and if kbj (Q)k > kb1 (P )k for some j then kbj (Q) − bj (P )k ≥ kakv ; in
either case we have dv0 (P , Q) ≥ kakv which implies (9.8.1) since dv ≤ 1 always.
So we may assume that kb1 (Q)k ≥ kbj (Q)k for all j and that kb1 (P )k = kb1 (Q)k.
In that case the open affine Spec B 0 [b2 /b1 , . . . , bs /b1 ] ⊆ A ×R Rv contains the
liftings of the sections determined by P and Q. Thus
dv (P , Q) = sup kφ(P ) − φ(Q)k

φ∈B 0 [b2 /b1 ,...,bs /b1 ]
Ã !
bj (P ) b (Q)
= max sup kφ(P ) − φ(Q)k, max
j
0 j b1 (P ) − b1 (Q)
φ∈B
Ã
1
= max dv0 (P , Q), max kbj (P ) − bj (Q)k ,
kb1 (P )k j
!
1
max kbj (Q)kkb1 (P ) − b1 (Q)k
kb1 (P )k2 j
dv0 (P , Q)
≤
kb1 (P )k
d 0 (P , Q)
≤ v .
kakv
Thus (9.8.1) holds.

Lemma 9.9. Given a nonsingular projective variety A with an extended model,

a morphism ψ : Γ → A, and an Mk -constant (cv ), one may choose g1 in Lemma
9.4 such that
(9.9.1) dv (ψ(θv (P , z)), ψ(θv (P , 0))) < e cv .
Proof . We may assume that A is embedded into PN in such a way that ψ(F )
is disjoint from the hyperplane at infinity. Let x1 , . . . , xN denote the coordinates
on AN ⊆ PN . We may assume that x1 , . . . , xN were included among the φi of
part (b) of Lemma 9.4; let g1∗ , . . . , gN∗ be the corresponding generalized Weil
functions.
First consider archimedean v; fix such a v. There exists a constant cv0 such
that
0
dv (P , Q) ≤ e cv max kxi (P ) − xi (Q)k
i
for all P , Q ∈ (A ∩ AN )(Cv ). By (9.4.1), inequality (9.9.1) will hold for v if
g1 ≥ max gi∗ + cv0 − cv .

i
Now let v be non-archimedean. Taking the closure of A in PNR defines a

different model for A; relative to this model we have
(9.9.2) dv (P , Q) ≤ max kxi (P ) − xi (Q)k

i
for all P , Q ∈ (A ∩ AN )(Cv ) such that the right-hand side is strictly less than
one. If kxi (P )k ≤ 1 for all i then this holds because the sections on PNRv
corresponding to P and Q do not meet the hyperplane at infinity, so the in-
equality follows by Remark 9.6. Otherwise we may assume kx1 (P )k ≥ kxi (P )k
for all i ; preceding assumptions then imply that kx1 (P )k = kx1 (Q)k > 1 and
kx1 (Q)k ≥ kxi (Q)k for all i . Then the sections are contained in the open affine
Spec Rv [1/x1 , x2 /x1 , . . . , xN /x1 ], and (9.9.2) follows from the inequality

xi (P ) xi (Q)
max(kxi (P )kkx1 (P ) − x1 (Q)k, kx1 (Q)kkxi (P ) − xi (Q)k)
x1 (P ) − x1 (Q) ≤ kx1 (P )k2
and from a similar inequality for 1/x1 . Then (9.9.1) follows from (9.9.2) if
g1 ≥ max gi∗ + cv0 − cv ;

i
in this case cv0 comes from Proposition 9.8.

Lemma 9.10. Let π : Γ → C be a projective morphism whose generic fiber is
smooth, and let D1 , . . . , Dr be divisors on Γ whose restrictions to the generic
fiber of π are prime, are smooth, and meet transversally. Let gD,1 , . . . , gD,r be
Weil functions for D1 , . . . , Dr , respectively. Then there exists effective generalized
Weil functions g1 and g2,1 , . . . , g2,r on D1 ∩. . .∩Dr and generalized Weil functions
g3,1 , . . . , g3,r on Γ whose support does not contain D1 ∩ . . . ∩ Dr . Letting
160 P. Vojta
Σv = {(P , z) ∈ (D1 ∩ . . . ∩ Dr )(Cv ) × Crv | kzi k < e −g1 (P ) , i = 1, . . . , r} ,
there is also an injection θv : Σv ,→ Γ (Cv ). These objects have the following

properties for all places v of k . From now on we omit v from the notation.
i. There exists an Mk -constant (cv ) such that for all i and all (P , z) ∈ Σ with
/ 0,
zi =

(9.10.1) gD,i (θ(P , z)) + log kzi k − g2,i (P ) ≤ cv .
ii. if Q ∈ Γ satisfies Q ∈ / Supp g3,i and gD,i (Q) > g3,i (Q) for all i then Q =
θ(P , z) for some (P , z) ∈ Σ and P ∈ / Supp g2,i for any i .
iii. Supp g3,i ⊇ Supp g1 ∪ Supp g2,i .
Moreover:
(a) For any prescribed P0 ∈ D1 ∩ . . . ∩ Dr with π(P0 ) suitably generic, the above
choices can be made such that P0 is not in the support of any of the above
generalized Weil functions.
(b) For any prescribed generalized Weil functions g10 , . . . , gm0 on Γ with P0 ∈/
Supp gj0 for any j , g1 may be chosen sufficiently large such that there exists
an Mk -constant (cv ) with
0
(9.10.2) gj (θ(P , z)) − gj0 (P ) ≤ cv
for all j and all (P , z) ∈ Σ.
Proof . We assume ξ is sufficiently generic that C is regular at ξ; π −1 (ξ) is

regular; and D1 , . . . , Dr meet π −1 (ξ) transversally, and remain prime, remain
smooth, and still meet transversally there.
If v is archimedean and real let Nv = 1; if archimedean and complex, let
Nv = 2; otherwise let Nv = 0.
Let Γ ⊆ PNC for some N . Let d be the relative dimension of π. Let L be a
linear subspace of codimension d − r + 1 satisfying the conclusions of Lemma
9.1 for X = Γ and s = r and also for X = D1 ∩ . . . ∩ Dr and s = 0. After blowing
up L in PNC (and replacing Γ with its strict transform in the blow-up), the linear
projection from L extends to a morphism p : Γ → Y := PdC−r whose restriction
to D1 ∩ . . . ∩ Dr is étale at F := p −1 (p(P0 )) ∩ D1 ∩ . . . ∩ Dr and which is smooth
on Γ except at a set which meets the fiber containing P0 at only finitely many
points. We also assume that L is chosen such that Supp gi0 is disjoint from F for
all i .
Let z1 , . . . , zr be rational functions on Γ whose principal divisors equal
D1 , . . . , Dr , respectively, near F . By Bertini’s theorem (writing Di as a difference
of two very ample divisors) we may assume that all components of all polar and
zero divisors of the zi are distinct and have multiplicity 1 on the fiber of p con-
taining P0 , and that their union is a normal crossings divisor on that fiber. After
further blowing up Γ (but leaving a neighborhood of P0 unchanged) these func-
tions define a morphism q : Γ → PrY , which is étale at q −1 (q(P0 )). Then Lemma
` a partial section of the canonical map Γ ×Y (D1 ∩. . .∩Dr ) → Y defined

9.4 gives
over v Λv (g4 ); this can then be composed with the projection to Γ . Moreover
g4 is effective and Supp g4 is disjoint from F .
Next consider (b). We may assume that g10 , . . . , gm0 are nicely defined at F . It
will suffice to consider the case m = 1, and (by Proposition 7.10) to assume that
g10 is of the form
g10 = min(− log kφ1 k, . . . , − log kφs k) ,
with all φi regular at all P ∈ F and φ1 (P ) =

/ 0 for all P ∈ F . We now assume
g4 was constructed so that (9.4.1) holds for φ1 , . . . , φs . Replace g4 with the
generalized Weil function
max(g4 , max gi∗ − log kφ1 k + Nv log 2) .

i
At non-archimedean places, (9.4.1) and the fact that kz k < e −g4 imply that
the right-hand side of (9.4.1) is strictly less than kφ1 (P )k. If kφi (P )k ≥
kφ1 (P )k then this implies that kφi (θv (P , z))k = kφi (P )k; otherwise it implies
that kφi (θv (P , z))k ≤ kφ1 (P )k. Thus g10 (θv (P , z)) = g10 (P ). At archimedean
places, a similar argument shows that the right-hand side of (9.4.1) is less than
kφ1 (P )k/2Nv , which in turn gives
|g10 (θv (P , z)) − g10 (P )| ≤ Nv log 2 .
Thus if g10 is nicely defined at F then (9.10.2) holds with cv = Nv log 2. (For
arbitrary g10 , (cv ) will be different.)
Now let g1 = g4 + (Nv /2) log r; this implies that Υv (g1 ) ⊆ Λv (g4 ) for all v.
Note that Supp g1 is still disjoint from F .
The above also implies (9.10.1), since gD,i + log kzi k is a generalized Weil
function on Γ for each i .
Next consider condition (ii). Let U be an open affine subset of Γ such that
q U is étale, such that U contains q −1 (q(P0 )), such that p(U ) is contained in an
open affine V ⊆ Y , and such that Di = (fi ) on U , for some f1 , . . . , fr ∈ O (U ).
Let Y 0 = q −1 (Y × [1 : 0 : . . . : 0]). Another application of Lemma 9.4 gives
an effective generalized Weil function g5 on Y 0 such that the injection Y 0 ⊆
Γ extends for each v to a rigid analytic partial section θv0 : Λv (g5 ) → Γ (Cv ).
Moreover, there exists a generalized Weil function g6 on Y 0 such that
(9.10.3) kf1 (θv0 (P , z)) − f1 (θv0 (P , 0))k ≤ e g6 (P ) · kzk
holds for all (P , z) ∈ Λv (g5 ). Furthermore, the coordinate functions on U

are bounded as in (9.4.1). And finally, Supp g5 and Supp g6 are disjoint from
q −1 (q(P0 )).
By∗Proposition 7.4 there exists a generalized Weil function g7 on Y such that
q Y 0 g7 ≥ g5 (identifying Y with Y × [1 : 0 : . . . : 0]), but q(P0 ) ∈ / Supp g7 .
∗
We may also assume that q D ∩...∩D g7 ≥ g1 . Likewise there exists g8 on
∗ 1 r
Y such that q 0 g8 ≥ g6 and q(P0 ) ∈
Y
/ Supp g8 . Let Y 00 be the closure of
162 P. Vojta
Y 0 \(D1 ∩. . .∩Dr ). By the choice of L made previously, no irreducible component

of Y 00 is contained
∗in D1 . Therefore there is a generalized Weil function g9 on
Y such that q Y 00 g9 ≥ − log kf1 k, but q(P0 ) ∈/ Supp g9 .
Let g10 = max(g8 + g9 + Nv log 2, g7 ). Then
g3,1 := max(gD,1 +log kf1 k+p ∗ g9 +Nv log 2, gD,1 +log kz1 k+p ∗ g10 +(Nv /2) log r, 0)
and
g3,i := max(gD,i + log kzi k + p ∗ g10 + (Nv /2) log r, 0) , i = 2, . . . , r
satisfy the requirement of part (ii). To see this, first note that neither Di nor
Supp(− log kzi k) contain any fiber of p, so Supp g3,i ⊇ p −1 (Supp g10 ) for all i .
Now suppose Q ∈ Γ (Cv ) for some v, and Q ∈ / Supp g3,i for any i . If q(Q) ∈
/
Λv (g10 ) then q(Q) ∈
/ Υv (g10 + (Nv /2) log r) and therefore, trivially,
− log kzi (Q)kv ≤ g10 (p(Q)) + (Nv /2) log r
for some i ; thus g3,i (Q) ≥ gD,i (Q). Otherwise Q = θv0 (P , z) for some P ∈ Y 0
and, by (9.10.3),
kf1 (θv0 (P , z)) − f1 (θv0 (P , 0))k ≤ e g8 (q(P ))−g10 (q(P )) ≤ e −g9 (q(P ))−Nv log 2
and therefore
kf1 (Q)k < e −g9 (p(P )) 2Nv
if and only if Q lies in the image of the θv constructed earlier. In particular, if
Q is outside the image of θv then − log kf1 (Q)k ≤ g9 (p(Q)) + Nv log 2, which
again implies g3,i (Q) ≥ gD,i (Q). Thus (ii) holds.
Finally, we may increase g3,i such that Supp g3,i ⊇ Supp g1 ∪ Supp g2,i for
all i . This gives (iii).
A divisor with simple normal crossings is a divisor whose components are
smooth, meeting transversally. Here we allow the components to be multiple.
Proposition 9.11. Let π : Γ → C be a projective morphism to a projective variety
over k , let D be a Cartier divisor on Γ which is effective on the generic fiber,
and let gD be a Weil function for D. Let L1 , . . . , LL be line sheaves on Γ and let
0 0
c1,v , . . . , cL,v and c1,v , . . . , cL,v be constants such that for all i : 0 < ci ,v ≤ ci0,v
for all v, ci ,v = ci ,v for almost all v, and ci ,v < ci0,v if v is archimedean. Then
0
there exists a generalized Weil function g on C with the following properties.

For each Q ∈ Γ (Cv ) there exists an integer r ≥ 0 and a power series map
φ : Dr → π −1 (ξ)(Cv ) such that
i. the image of φ contains Q;
ii. there exist positive integers f1 , . . . , fr (depending on φ) such that φ∗ D equals
f
the principal divisor (z11 . . . zrfr );
iii. for all z ∈ D with z1 . . . zr =
r
/ 0,

gD (φ(z)) + f1 log kz1 k + . . . + fr log kzr k ≤ g(ξ) ;
and
iv. for all i and all v there exist sections γi ∈ Γ (Dr , φ∗ Li ) whose norms satisfy
ci ,v ≤ kγi k ≤ ci0,v
on all of Dr .
Moreover, as φ varies, the tuple (f1 , . . . , fr ) takes on only finitely many values.
If in addition the generic fiber of π is smooth, if the restriction of D to that generic
fiber has simple normal crossings, and if ξ is suitably generic, then the image of
φ crosses D transversally.
Proof . First, we immediately reduce to the case where the generic fiber of π
is smooth, and the restriction of D to this fiber is a divisor with simple normal
crossings.
Let D1 , . . . , Dr be the components of D. Applying the lemma to D1 ∩. . .∩Dr
with various P0 gives various g1,j , g2,i ,j , and g3,i ,j such that ∩j ∪i Supp g3,i ,j does
not meet D1 ∩. . .∩Dr on the generic fiber of π. Then the generalized Weil function
min max min(g3,i ,j , gD,1 , gD,2 , . . . , gD,r )

j i
has no support along the generic fiber, so by Proposition 7.4 it is bounded

from above by some generalized Weil function g 0 coming from C . Therefore
if gD,1 , . . . , gD,r are all greater than or equal to g 0 at some point Q, then there
exists some j and some P ∈ D1 ∩ . . . ∩ Dr as in condition (ii) of Lemma 9.10.
By condition (iii) of Lemma 9.10, g1 + g2,i is also bounded from above (as well
as from below) at P for each i ; regarding the map φ as being a family of poly-
discs and dilating the polydisc attached to P then gives the map required for the
proposition.
If, however, some of gD,1 , . . . , gD,r is less than g 0 , one can apply Lemma
9.10 with a smaller value of r (taking care by (9.10.2) that the discarded gD,i
remain small in that polydisc).
Part (iv) of the proposition can be guaranteed by fixing a local generator for
each Li at P0 and applying (9.10.2) to the logarithm of its metric.
Proposition 9.12. Given a nonsingular projective variety A with an extended
model, a morphism ψ : Γ → A, and an Mk -constant (cv ), one may choose g, r,
and φ in Proposition 9.11 so that
dv (ψ(φ(z)), ψ(φ(0))) < e cv
for all z ∈ Dr .
Proof . This follows immediately from Lemma 9.9 by choosing g1 appropriately

in Lemma 9.10.
164 P. Vojta
10. Models and complexes
This section defines models over R for A, Ā, X , Xi , and the line sheaves M−,s to
models over Spec R. Extending L−,s would lead to technical difficulties, how-
ever, so instead of doing that we spend the bulk of this section working around
that difficulty.
To begin, we choose a model A¯ for Ā, as follows. Fix a model A0 for A0
such that the line sheaf O (L0 ) and the line sheaves Mm of Sect. 2 extend as
line sheaves to A0 . Then the construction (2.3) gives a model A¯ for Ā. Again
let [∞]m and [0]m correspond to the projections (2.5); they are the closures in
A¯ of the corresponding divisors in Ā. Let A denote the complement of their
union; it is therefore a model for A.
Let X be the closure of X in A; it is a model for X . Of course, this
model is not necessarily the same as the model X chosen in Theorem 0.2. To
fix this discrepancy, the extended line sheaves Mm can be tensored with some
fractional ideal in R; then finitely many modified models obtained from these
modified Mm will have the property that the union of their sets of integral points
will contain the set of integral points X (RS ) from Theorem 0.2. Or, one can
also fix this discrepancy by enlarging the set S of exceptional primes. In either
case, we assume from now on that X is the closure of X in the model A
constructed above, and that X is the closure of X in A. ¯
Now let V0 be the model for A0 constructed as in ([V 4], Sect. 13); we may
n
assume V0 dominates A0n . Let
V1 = (A¯ ×Spec R . . . ×Spec R A)

¯ ×An V0 .
0
Let X1 , . . . , Xn be closed subvarieties of X satisfying (4.5.1–4.5.4). Let X i be the

closure of Xi in X , and let Xi and X i be Q the closures of Xi and X i in A and
A, ¯ respectively. Let V be the closure of X i in V1 .
Since the Mm extend as line sheaves to A0 , the extended divisors [0]m
and [∞]m on A¯ are Cartier. These extensions, together with (5.4.1) and ([V 4],
Lemma 13.2), define an extension of Mδ,s to a line sheaf on V . Let be as in
Proposition 6.2.
A logical next step might then be to define a model Ws for the scheme Ws
defined in Sect. 3. One would then extend L−,s to that model and construct
a suitably small section γ ∈ Γ (Ws , dL−,s ). However, this presents a number
ij
of difficulties since the natural extensions of the divisors Qs,m to Ws may con-
tain fiber components over Spec R. Instead, we will identify Γ (V , dM−,s ) and
Γ (Ws , dL−,s ) with R-submodules of Γ (Ws , d πs∗ M−,s ) via πs and (3.6), and work
with sections
(10.1) γ ∈ Γ (V , dM−,s ) ∩ Γ (Ws , dL−,s ) .
Such a section will be constructed in Sect. 12, for sufficiently large d . This
will be done by applying geometry of numbers arguments to various terms in
a modified Faltings complex. This section and the next develop the machinery
necessary for dealing with the vanishing conditions in that complex.
Let ` be an integer such that O (`L) and O (`L0 ) are generated by their global
sections over Ā and A0 , respectively. Then the sheaves
X X
n ∗ ∗ 2 ∗
(10.2) O ` (ρ ) (si · pri +sj · prj ) L0 + `(n − 1) si pri L
i <j
and
(10.3) X
X
n ∗ ∗ n ∗ ∗
O ` (ρ ) (si · pri −sj · prj ) L0 ⊗O ` (ρ ) (si · pri +sj · prj ) L0
i <j i <j
X
∼
=O 2`(n − 1) si2 pr∗i ρ∗ L0
are generated by their global sections over Ān , and remain so when pulled back
to Ws . Letting
L0 = 2(n − 1)ρ∗ L0 + (n − 1)L1 + (n − 1 − )L
and
L00 = 4(n − 1)ρ∗ L0 + (n − 1)L1 + (n − 1 − )L ,
we can create a Faltings complex as in ([V 4], Sect. 9):
(10.4)
X a β X b
α
0 → Γ (V , dM−,s )−→Γ V , d si2 pr∗i L0 −→Γ V , d si2 pr∗i L00 .
Here the maps α and β are defined by tensoring with sections of the sheaves in
(10.2) and (10.3), respectively. Also a and b are independent of d and s.
To construct the desired section, then, we use the Faltings complex to con-
struct a suitable set of sections of Γ (V , dM−,s ), and then show that one can
obtain a section of O (dL−,s ) by imposing certain vanishing conditions on the
sections of O (dM−,s ). These vanishing conditions will be treated in the next
section; the remainder of this section compares the modules Γ (V , dM−,s ) and
Γ (Ws , dL−,s ). First of all, the difference between the two divisor classes in ques-
tion is an effective divisor, so the module Γ (Ws , dL−,s ) can be regarded as a
submodule of Γ (Ws , dM−,s ).
We now compare their metrics. This will be done in the slightly more general
context of singular metrics on O (dM−,s ). These will be metrics of the form
k·k
(10.5) k · k0 = Q P
exp(−eij pr∗i gij )
J
j =1 i ∈Ij
Here each Ij is a nonempty subset of {1, . . . , n}, each gij is some effective Green
function on Ā, and each eij is a positive integer. Moreover, Ij , gij , and eij /dsi2
166 P. Vojta
are independent of d and s. In the case at hand, we use the Green functions (2.8)
and the inequality
2 2
(α + β)2 1 1 α β
(10.6) + +
(1 + α)2 (1 + β)2 1+α 1+β 1+α 1+β
to find a singular metric of the form (10.5) which is equivalent

P to the metric on
O (dL−,s ). Here all constants are of the form exp cd si2 , with c depending
only on Ā. In particular note that if γ ∈ Γ (V , dM−,s ) lies in Γ (V , dL−,s ) then
its singular metric is bounded. This immediately gives the following corollary of
Proposition 6.2:
Proposition 10.7. Let Γ 0 (V , dM−,s ) be the subset of Γ (V , dM−,s ) consist-
ing of sections γ whose norm k · k0 is bounded. Then there exist c and as in
Proposition 6.2 such that for all tuples s,
P Y
n
dimk Γ 0 (V , dM−,s ) ≥ cd dim Xi 2 dim Xi
si
i =1
for all sufficiently large d (depending on s).

We now compare these singular metrics with the original one.
Lemma 10.8. Let v ∈ S , let I1 , . . . , IJ be as above, let ìjk (j = 1, . . . , J , i ∈ Ij ,
k = 1, . . . , r) be nonnegative integers, let ` = max ìjk , and let
Y
J XY
r
(10.8.1) Ψ= kzk kìjk .
j =1 i ∈Ij k =1

Then every power series f : Dr → Cv such that kf (z1 , . . . , zr )k Ψ is bounded
satisfies
kf k
sup ≤ c ` sup kf k
Dr Ψ Dr
for some constant c depending only on r and I1 , . . . , IJ .
Proof . Let Q ∈ Dr and fix ρ ∈ (1/2, 1) with ρ ≥ max kzk (Q)k. It will suffice to
show that
kf (Q)k kf k
(10.8.2) ≤ c ` sup .
Ψ (Q) (∂ Dρ )r Ψ
After rearranging coordinates, we may assume that kzk (Q)k < ρ for k =
1, . . . , s and kzk (Q)k = ρ for k = s + 1, . . . , r. We prove (10.8.2) by induction on
s. If s = 0 then we are done. Otherwise, find integers mk ≥ 0, numbers ωk ∈ Cv
with ρ ≤ kωk k ≤ 2, and ξ ∈ D such that
zk (Q) = ωk ξ mk
for k = 1, . . . , r. Note that the restrictions on kωk k imply that mk > 0 if and
only if k ≤ s. Letting ξ vary defines a curve in Dρ on which
r
Y
J
Ψ #Ij · kξkm
j =1
for some integer m ≥ 0; moreover the constants are of the form c ` (even if
m m
` = 0). By the maximum principle applied to f (ω1 ξ1 1 , . . . , ωk ξk k )/ξ m , we may
move Q to a point with one more of its coordinates lying on ∂ D, but affecting
kf k Ψ by at most a factor of c ` . This then gives (10.8.2) by induction.
Lemma 10.9. Let I1 , . . . , Ij and gij (j ∈ Ii ) be as above. For i = 1, . . . , n let
πi : Γi → Ci be morphisms of projective varieties, and let ψi : Γi → Ā be mor-
phisms. Then there exist generalized Green functions g1 , . . . , gn on C1 , . . . , Cn ,
respectively, and an Mk -constant (cv ) with the following properties. For each
v, forQeach ξ1 , . . . , ξn in C1 , . . . , Cn with ξi ∈ / Supp gi for all i , and for each
P ∈ X i (Cv ) (where Q X i := π −1
(ξ i )) there exists an integer r ≥ 0 and a power
series map φ: Dr → X i (Cv ) such that:
i
i. the image of φ contains P ; P

ii. all φ∗ pr∗i ψi∗ gij are of the form τ − k fijk log kzk k with fijk ∈ Z and |τ | ≤
gi (ξi ) on Dr ; and
iii. For all rational δ and all tuples s of positive rational numbers, let Mδ,s be the
Q-divisor class on Q Γi defined by ψ1 , . . . , ψn and (3.3). Then for sufficiently
divisible d > 0, φ∗ O (dM−,s ) has a powerP 2 series section on D whose metric
r
is bounded from below by exp(−cv d si ) and from above by 1.

Moreover, as φ varies, the tuple (fijk ) takes on only finitely many values. If
in addition
∗
Pthe generic fibers of π1 , . . . , πn are smooth, if the restrictions of all
ψi Supp j gij to those generic fibers have simple normal crossings, and if
ξ1 , . . . , ξn are suitably generic, then the images of φi cross the above divisors
transversally.
Proof . By Proposition 9.11, for i = 1, . . . , n there exist integers ri ≥ 0, power

series maps φi : Dri → X i (Cv ), and generalized Weil functions gi on Ci such
that:
(a) the image of φi contains Pi for Pall i ;
(b) all φi gij are of the form τ − k fijk log kzk k with fijk ∈ Z and |τ | ≤ gi (ξi )
∗
on Dri ; and
(c) for all i the sheaves φ∗i ρ∗ O (L0 ) and φ∗i O (L1 ) have sections on Dri whose
metric lies in the interval (1/2, 1) if v is archimedean or is identically equal
to 1 if v is non-archimedean.
Indeed, (a) follows from condition (i) of Proposition 9.11, (b) follows from
(iii), and (c) follows from (iv). Moreover, for any prescribed extended model of
A0 and for any prescribed c > 0, ri and φi can be chosen such that
168 P. Vojta
dv (ρi (ψi (φi (z))), ρi (ψi (φi (0)))) < c
for all z ∈ Dri and all i . This holds by Proposition 9.12.

Q
P The above maps φi combine to give a map φ : D → X i (Cv ) (with r =
r
ri ). This map automatically satisfies (i) and (ii) above. To show (iii), it suffices
to show that for all i and j with 1 ≤ i < j ≤ n, the Poincaré sheaf φ∗ Pij has a
section on Dr whose metric lies in the interval (1/2, 1) if v is archimedean or is
identically 1 if v is non-archimedean. To see this, fix any extended model for A0
and let dvn denote the corresponding distance on A0 (Cv )n . If v is archimedean,
then a compactness argument shows that there exists c > 0 such that for all
P ∈ A0 (Cv )n there exists a rational section of Pij whose metric on the set
{Q ∈ A0 (Cv )n | dvn (P , Q) < c}
lies in the range (1/2, 2). If v is non-archimedean, an analogous constant c > 0

exists by Proposition 9.8: if we replace dvn with the distance associated to the
model W0 , then we may take c = 1. Now (iii) is immediate, by (3.3).
Note that the tuple s only occurs in (iii).
Proposition 10.10. Let γ ∈ Γ (V , dM−,s ) and let v ∈ S . If kγk0v is bounded,
then
X X
− log kγk0sup,v ≥ − log kγksup,v + cd si2 h(Xi ) + c 0 d si2 .
Here c and c 0 depend only on A, X , dim Xi (i = 1, . . . , n), and the bounds on

deg Xi (i = 1, . . . , n).
Proof . For this

P proof, an admissible
P multiplicative constant is a constant of the
form exp(cd si2 h(Xi ) + c 0 d si2 ). Here c and c 0 depend only on A, X , dim Xi
for all i , and the bounds on the degrees of Xi , but not on s.
Q
Let P = (P1 , . . . , Pn ) be a point
Q on X i (Cv ) where kγk0 comes close to
its maximum, and let φ : D → r
X i (Cv ) be as in Lemma 10.9. Here Ci is
an appropriate Chow variety, Γi is the corresponding family of varieties, and
ξi is the point on Ci corresponding to X i . If ξi ∈ Supp gi , then we proceed by
Noetherian induction.
Let
YX
Ψ= exp(−eij φ∗ pr∗i gij ) .
j i ∈Ij
By condition (iii) of Lemma 10.9, it suffices to show that if f is a power series

on Dr satisfying
kf k Ψ ,
then
kf k
(10.10.1) sup ≤ sup kf k
Dr Ψ Dr
up to an admissible multiplicative constant. By condition (ii), replacing Ψ with the

expression (10.8.1) changes it by at most an admissible multiplicative constant;
after doing so (10.10.1) follows immediately from Lemma 10.8.
11. Vanishing conditions
This section describes how to use derivative conditions to define the set
Γ 0 (V , dM−,s ) of sections γ ∈ Γ (V , dM−,s ) whose metric (10.5) is bounded.
For this section let L2 be the union of the supports of the Green functions
pr∗i gij in (10.5). By an embedded resolution of singularities we may replace each
Q
X i with a smooth proper X ei such that the support of L2 on X ei is a divisor with
simple normal crossings; i.e., components of the support are smooth and cross
transversally.
Definition 11.1. Let R≥0 denote the set of nonnegative real numbers. If e, f ∈
RN≥0 , then we say e ≤ f if the inequality holds for all components. Then:
(a) A leading set in RN≥0 is a subset σ such that e ∈ σ and f ≤ e implies f ∈ σ.
(b) A leading set is boundedly generated if it can be written as a union of sets
pr∗I σI , as I varies over subsets of {1, . . . , N }, where each σI is a bounded
subset of R#I ≥0 . (Here prI denotes the projection to R≥0 obtained by throwing
#I
out coordinates not in I .)

(c) A leading set σ is boundedly generated of Q multiweight ≤ (d1 , . . . , dN ) if
moreover each set σI as above is a subset of i ∈I [0, di ].
This is a big generalization of the notion of the index. The situation here
is complicated by the fact that the sheaf defined by k · k0 is a tensor product
of sheaves defined by the index, but with varying sets of multiplicities and
involving different subsets of the variables. This happens because the restrictions
ei can involve several components of L2 with different
of pr∗i [0]m and pr∗i [∞]m to X
multiplicities.
In this section and the next, such leading sets will be used to indicate the
required vanishing of the corresponding power series coefficients. The reason
for the σI is that one can ensure the vanishing of T all coefficients in pr∗I σI by
considering finitely many partial derivatives along i ∈I {zi = 0}. The orders of
these derivatives also need to be bounded; hence the notion of multiweight.
Lemma 11.2. Fix an embedding ofQk into C. Then there exists a constant c
with the following property. Let U ⊆ X ei (C) be a coordinated open subset such
that the support of L2 is contained in the union of the coordinate hyperplanes,
and such that all coordinates zk are pull-backs of local coordinates from X ei (k ) .

Let γ0 be a local generator of O (dM−,s ) U . Then there exists a leading subset
σ ⊆ RN≥0 , boundedly generated of multiweight ≤ (cdsi2(1) , . . . , cdsi2(N ) ), such that
a local section γ ∈ Γ (U , dM−,s ) lies in Γ 0 (U , dM−,s ) if and only if
170 P. Vojta
r1 rN
∂ ∂ γ
(11.2.1) ... (0, . . . , 0) = 0
∂z1 ∂zN γ0
for all (r1 , . . . , rN ) ∈ σ ∩ NN .
Proof . Up to a bounded function, each gij in (10.5) satisfies
X
N
gij = −fijk log kzk k
k =1
for some integers fijk . Let σ be the complement of the set of all (r1 , . . . , rN ) ∈ RN≥0
such that
ÃN !
YN
rk
Y Y eij fijk
(11.2.2) tk max tk
i ∈Ij
k =1 j k =1
for all (t1 , . . . , tN ) ∈ [0, 1]N .

With this σ, if γ satisfies (11.2.1) then γ ∈ Γ 0 (U , dM−,s ). Conversely, if
r ∈ σ then the derivative (11.2.1) must vanish, by an appropriate use of Cauchy’s
inequalities.
Next consider the question of the boundedness of σ. We show that σ is
boundedly generated of multiweight ≤ (d1 , . . . , dN ), where
X
dk = max eij fijk .
i ∈Ij
j
Indeed, we show that if r ∈ σ and r1 ≥ d1 , then N × (r2 , . . . , rN ) ⊆ σ (and

likewise for the other coordinates of r). This follows from the fact that if r1 ≥ d1
then (11.2.2) holds for all (t1 , . . . , tN ) if and only if it holds for all (1, t2 , . . . , tN ).
To finish the proof, first note that if fijk = / 0 then i = i (k ). Since eij is a fixed
2
multiple of dsi as d and s vary, it follows that all dk are fixed multiples of dsi (k ) .
Moreover, as U varies, the integers fijk have only finitely many possibilities,
so this multiple may be taken independent of U . Likewise, it can be taken
independent of the X ei if their degrees are bounded.
The conditions (11.2.1) are equivalent to the vanishing of certain partial
derivatives along the components of L2 . The remainder of this section proves
uniform upper bounds on the heights of those vanishing conditions as the Xi
vary.
In order to obtain these uniformities, we assume that the Xei used above have
been defined in the following manner. As in ([V 4], Sect. 16) let Ci be a projective
arithmetic scheme over Spec R, let Ci denote its generic fiber over Spec k , and
let Γi be a family of Pprojective varieties over Ci . We perform an embedded
resolution of L2,i := j Supp gij on the generic fiber of Γi over Ci , and extend
Γi so as to dominate the original Γi . We may also assume that each irreducible
component Dj0 of L2,i extends to a Cartier divisor on Γi . For each Dj0 choose
metrics at all archimedean places (i.e., choose Green forms for the cycles Dj0 ).
For ξi ∈ Ci let X ei be the fiber in Γi and let X fi be the closure of X ei in Γi .

f
(Note that Xi may be singular.) Then for ξ1 , . . . , ξn in nonempty Zariski-open
subsets of Ci , X ei will be irreducible and smooth, and the restriction of L2,i to X ei
0
will still be a divisor with simple normal crossings with the Dj still irreducible.
Outside of these Zariski-open subsets of Ci , the situation can be handled by
Noetherian induction.
Q situation we have partial derivatives, as follows. Let γ be ∗a global
In this sec-
tion on X ei of O (dM−,s ), let D1 , . . . , DJ be some subset of the pr D 0 , and let
i j
` = (`1 , . . . , `J ) be a tuple of nonnegative integers describing a leading term.
Then we have a global section
Y \
D` γ(D1 ∩ . . . ∩ DJ ) ∈ Γ Xei ∩ Dj , O (dM−,s − `1 D1 − . . . − `J DJ )T .
D j
The chosen metrics on Dj determine metrics k · kv on O (Dj ) for archimedean

v; for non-archimedean v assign metrics k · kv by Definition 8.5. Proposition
8.6 then determines Weil functions gD,1 , . . . , gD,J for D1 , . . . , DJ . Similarly, the
extension of M−,s to V defines metrics k · kv on O (dM−,s ) for all v. The
product of these metrics then gives a metric on O (dM−,s − `1 D1 − . . . − `J DJ )
at all v.
We now determine upper bounds for the heights of these partial derivatives.
Q
Lemma 11.3. There exist effective generalized Weil functions g1 , . . . , gJ on Ci
and an Mk -constant (cv ) such that the inequality
− log kD` γ(D1 ∩ . . . ∩ DJ )ksup,v
X
≥ − log kγksup,v − `1 g1,v (ξ) − . . . − `J gJ ,v (ξ) − cv d si2
holds for all tuples s, all `1 , . . . , `J ∈ N, all sufficiently divisible d , and all places
v of k . Moreover, if each Dj is the pull-back of a metrized divisor on Γi (j ) , then
each gj can be taken as the pull-back of a generalized Weil function on Ci (j ) .
Q
Proof . Let v be a place and let P ∈ X ei (Cv ) be a point where the supremum
on the left is attained. Applying Lemma 10.9 with the gij ’s corresponding to
gD,1 , . . . , gD,J (appropriately collated) gives an integer r ≥ 0 and a power series
Q
map φ : Dr → Xei (Cv ) satisfying the conditions of the lemma. In particular,
divide γ by the section in condition (iii). This gives a power series f on Dr
which satisfies
X
(11.3.1) − log kf ksup ≥ − log kγksup,v − cv d si2 .
For each j , the Cartier divisor φ∗ Dj is some multiple µk of some coordinate

µ
hyperplane zk = 0; thus zk k is a local generator of φ∗ O (−Dj ). Recall that gD,j
is defined as minus the logarithm of the metric of the section 1 ∈ O ,→ O (Dj ).
Therefore the metric k · kj of zk ∈ O (−φ∗ Dj ) equals −µk log kzk k − φ∗ gD,j . By
condition (ii) of the lemma it follows that

(11.3.2) − log kzk kj ≤ gj (ξ) ,
172 P. Vojta
where gj is a generalized Weil function on C1 × . . . × Cn . Moreover, let i be such

that Dj is the pull-back of a divisor on Xei ; then gj is the pull-back of a generalized
Weil function on Ci . We may decrease r so that r = J ; since P ∈ D1 ∩ . . . ∩ DJ ,
it follows that φ−1 (P ) = {0}. We also assume that k = j above.
The inequality (11.3.2) holds in particular at φ−1 (P ). Writing
X
f (z) = a(i ) z(i )
(i )≥0
with a(i ) ∈ Cv , it follows that the derivative D` γ(D1 ∩ . . . ∩ DJ ) corresponds to

the term
a` z` .
By (11.3.1) and (11.3.2) it therefore suffices to show that
ka` k ≤ sup kf k .
Dr
This inequality holds by Cauchy’s inequalities ([V 4], Lemma 15.1).
Summing this inequality over all places v of k and applying Lemma 8.4 (f)
then gives the following bound of heights.
Q
Theorem 11.4. There exist height functions h1 , . . . , hJ on Ci and a constant
c such that, for any tuple `, we may clear denominators in the map γ 7→ D` γ to
Q f
obtain an R-module map D`0 from the appropriate subset of Γ Xi , O (dM−,s )
to the R-module
n Y \
γ∈Γ X fi ∩ Dj , O (dM−,s − `1 D1 − . . . − `J DJ )T D
j
o

kγksup,v ≤ 1 for all v ∈/ S∞ .
For archimedean places v we metrize this module

via the supremum norm on the
line sheaf O (dM−,s − `1 D1 − . . . − `J DJ )T D ; then
j
Y Y X
(11.4.1) kD`0 γkv ≤ kγkv · exp `1 h1 (ξ) + . . . + `J hJ (ξ) + cd si2 .
v|∞ v|∞
Moreover, if each Dj is the pull-back of a metrized divisor on Γi (j ) , then each hj

can be taken as the pull-back of a height function on Ci (j ) . Here c depends only
on the families Γ1 , . . . , Γn over C1 , . . . , Cn , the divisors D1 , . . . , DJ , their Weil
functions gD,1 , . . . , gD,J , the model V , and the extensions of various line sheaves
to V .
In the next section, the above theorem will be combined with the following
proposition to show that existence of a small section of Γ (W , dM−,s ) implies
the existence of a small section of Γ (W , dL−,s ).
Proposition 11.5. Let C1 , . . . , Cn be projective arithmetic schemes over Spec R
with generic fibers C1 , . . . , Cn respectively, over Spec k . For i = 1, . . . , n let
πi : Γi → Ci be surjective projective morphisms. Let L1 , . . . , Ln and L01 , . . . , L0n be

metrized divisor classes on Γ1 , . . . , Γn , respectively. Let A0 be an abelian variety
defined over k , and for i = 1, . . . , n let ρi : Γi ×R k → A0 be a morphism. Let P
be a Poincaré divisor class on A0 × A0 ; for i < j let Mij be a metrized divisor
class on Γi ×R Γj which equals (ρi × ρj )∗ P on the generic fiber (over k ). Let h()
denote logarithmic height functions on C1 , . . . , Cn relative to ample divisors.
Then there exist nonempty Zariski-open Ui ⊆ Ci , i = 1, . . . , n, with the fol-
lowing property. For ξi ∈ Ui (k p ) let Xi denote the variety πi−1 (ξi ). Let d1 , . . . , dn
be positive integers such that di dj ∈ Z for all pairs (i , j ), let `1 , . . . , `n be
positive integers with ì ≤ rdi for all i , and let γ be a nonzero global section in
Y X X Xp
Γ Xi , di pr∗i Li + ì pr∗i L0i + di dj (pri × prj )∗ Mij .
i <j
Then
(11.5.1) Ã !
Y X
n Xp X
n
0 00
kγksup,v ≥ exp −c di h(ξi ) − c di dj h(ξi )h(ξj ) − c di .
v i =1 i <j i =1
0 00
Here the constants c, c , and c depend only on π1 , . . . , πn , r, and the metrized
divisor classes.
Proof . For all i let mi = dim Γi − dim Ci and let ψi : Γi → PCii be a generically
m
finite rational map which is finite over the generic point of Ci . After expanding
Γi , we may assume that ψi is a morphism. The proof of ([V 4], Corollary 18.3)
shows that there exist generically finite surjective morphisms φi : Γi] → Γi such
that the metrized divisor classes Li and L0i define norms L]i and (L0i )] on Γi] which
agree with suitable multiples of φ∗ ψ ∗ O (1) up to divisors supported on fiber
components over Spec R and, correspondingly, changes of metrics at archimedean
places. This proof also constructs norms Mij] associated to Mij on Γi] ×R Γj] ; the
following lemma characterizes these norms.
Lemma 11.5.2. For i = 1, 2 let πi] : Γi] → Ci be a surjective morphism of
complex projective varieties, let Ai be abelian varieties, and let ρ]i : Γi] → Ai be
morphisms. Let P be a divisor class on A1 ×A2 whose restrictions to {0}×A2 and
A1 × {0} are algebraically equivalent to zero, and let M = (ρ]1 × ρ]2 )∗ P . Assume
that C1 and C2 are normal, and that the restriction of M to the generic fiber of
π1] × π2] is trivial. Then some positive integral multiple of M equals (π1] × π2] )∗ M 0
for some divisor class M 0 on C1 ×C2 . Moreover, the restrictions of M 0 to {ξ1 }×C2
and C1 × {ξ2 } are trivial for ξ1 ∈ C1 and ξ2 ∈ C2 .
Proof . For i = 1, 2 let Bi be the smallest translated abelian subvariety of Ai

containing ρ]i (Γi] ). Also for all ξi ∈ Ci let Di (ξi ) denote the smallest abelian
subvariety of Ai such that ρ]i ((πi] )−1 (ξi )) is contained in a translate of Di (ξi ). For
generic ξi , D(ξi ) equals its maximal value; let this be denoted Di . Since M = 0
on fibers of π1] × π2] , it follows that P = 0 on ρ]1 ((π1] )−1 (ξ1 )) × ρ]2 ((π2] )−1 (ξ2 )). For
174 P. Vojta
generic (ξ1 , ξ2 ), ρ]1 ((π1] )−1 (ξ1 )) × ρ]2 ((π2] )−1 (ξ2 )) spans the corresponding translate
of D1 × D2 as an abelian variety (for any choice of origin); hence some positive
multiple mP is trivial on this translate. Moreover, there exists one m which works
for all (ξ1 , ξ2 ). For i = 1, 2 let αi : Bi → Ei be the quotient under the action of Di ;
then by ([H 2], III Ex. 12.4) there exists a divisor class Q on E1 × E2 such that
mP = (α1 × α2 )∗ Q. Since fibers of πi] are collapsed by αi , there exist morphisms
βi : Ci → Ei making the following diagram commute:
]
ρ
Γi] −→i
Bi ⊆ Ai
 ] α
yπi y i
βi
Ci −→ Ei
Let M 0 = (β1 × β2 )∗ Q. It then follows that mM = (π1 × π2 )∗ M 0 .
Finally, the restrictions of M to {η1 } × Γ2] and Γ1] × {η2 } (for η1 ∈ Γ1] and
η2 ∈ Γ2] ) are numerically equivalent to zero; hence a similar statement is true
for M 0 . The last assertion of the lemma then follows from [Ma].
The statement of the proposition is unchanged if we multiply all divisor
classes by a positive integer and raise γ to that same power. Therefore we may
assume that Mij] is a pull-back of a metrized divisor class on Ci × Cj , as in the
lemma, up to changes at fiber components over Spec R and changes in metrics.
deg ψ
When applying the lemma, note that Ai = A0 i .
Corresponding to the section γ there is a nonzero global section
Ã !
Y ] X X Xp
] ∗ ] ∗ 0] ∗ ]
γ ∈Γ Γi , di pri Li + ì pri Li + di dj (pri × prj ) Mij .
i <j
The above divisor class coincides with a product of multiples of pr∗i φ∗i ψi∗ O (1)
up to fiber components over Spec R, changes of metrics, and a divisor class
Xp
di dj (pri × prj )∗ (πi] × πj] )∗ Mij0 ,
i <j
where Mij0 is as in the lemma. However, we note that Poincaré-like divisors Mij0
satisfy
p
(11.5.3) hM 0 (ξi , ξj ) ≤ c 0 h(ξi )h(ξj ) .
ij
The proof then

Q mconcludes as in ([V 4], Lemma 13.9) by pulling up a suitable
point from PCii and applying (11.5.3).
12. Construction of a global section
Let Γ be a metrized, torsion free finitely generated module of rank δ over the
ring of integers R of k . For all archimedean places v of k , let the completion
Γv := Γ ⊗R kv of Γ at v be given a Haar measure such that the unit ball has
Q
measure 1, and let covol(Γ Q ) denote the covolume of Γ in v|∞ Γv . Define a
length function `(γ) = v kγkv , and for i = 1, . . . , δ define successive minima
λi to be the minimum λ such that there exist R-linearly independent elements
γ1 , . . . , γi ∈ Γ such that `(γj ) ≤ λ for all j = 1, . . . , i .
Lemma 12.1. In this situation, there exist constants c5 and c6 depending only on
k such that
λ1 . . . λ δ
≤ c6δ δ [k :Q]δ/2 .
1
(12.1.1) ≤
c5δ δ [k :Q]δ/2 covol(Γ )
Proof . This follows from ([V 1], Theorem 6.1.11 and Remark 6.1.12). (When
k = Q this is Minkowski’s “second theorem.”) The factors δ ±[k :Q]δ/2 come from
the factorials in loc. cit. and from the volume of the unit ball in Euclidean space.

Lemma 12.2. Let β : Γ1 → Γ2 be a homomorphism of metrized R-modules. Let
δ0 and δ2 be the ranks (over R) of the kernel and image of β, respectively. For
all v | ∞ assume that Cv is a constant such that
(12.2.1) kβ(γ)kv ≤ Cv kγkv for all γ ∈ Γ1 ,
Q
and let C = v|∞ Cv . Then
(12.2.2) covol(Γ1 ) ≥ 2−[k :Q]δ0 covol(Ker β)C −δ2 covol(Image β) .
Proof . See ([Ko], Lemma 5).

These lemmas, together with the results of Sect. 10, provide us with the
main tools needed to construct a small section. At this point we introduce the
assumption that
1
(12.3) si ≈ √ .
hL (Pi )
Theorem 12.4. Let be as in Proposition 10.7, let M−,s be as in (3.6), and let
k · k0 denote a metric as in (10.5). Then for all tuples s = (s1 , . . . , sn ) of positive
rational numbers satisfying (12.3) and for all sufficiently large (and divisible)
d (depending on s), there exists a section γ ∈ Γ (V , dM−,s ) such that kγk0 is
bounded and such that the inequality
Ã !
Y X n
(12.4.1) kγksup,v ≤ exp cd si2
v|∞ i =1
holds. Here the constant c is independent of s and d .
Proof . This proof is a matter of obtaining bounds for covolumes of various

modules in the diagram
176 P. Vojta
0 −→ Γ (V , dM−,s )
P a β P b
α
−→ Γ V , d si2 pr∗i L0 −→ Γ V , d si2 pr∗i L00
∪ P∪ a
Γ 0 (V , dM−,s ) −→ Γ 0 V , d si2 pr∗i L0 ,
where the top row is the Faltings complex (10.4) and the symbols Γ 0 in the
bottom row denote the submodules of sections γQfor which kγk0Q is bounded.
First, note that if F is a divisor class on X i , then Γ X i , dF ) ,→
Γ (V , dF ) and the cokernel is annihilated by an integer independent of d . There-
fore we may pass between the two modules a few times without affecting the
estimates.
To shorten notation, let Γ0 , Γ1 , and Γ2 denote the modules in the top row,
and Γ00 and Γ10 the modules in the bottom row. For all except Γ2 , we shall use
metrics induced by the injections into Γ1 ; on Γ1 and Γ2 the metrics shall be
those induced by the largest of the sup norms on the direct summands. Also let
δ1 = rank Γ1 .
First, by ([V 4], Lemma 13.8),
X [k :Q]δ /2
covol(Γ1 ) ≤ exp δ1 · cd si2 · δ1 1
X
≤ exp δ1 · cd si2 ,
where from now on c is a constant which is independent of d and s, but whose

value may change from line to line. The extra factor in the first step appears
because we are using the first half of (12.1.1); it disappears because δ1 grows
only polynomially in d , so this factor can be absorbed into the other factor if d
is sufficiently large. This argument will be used implicitly several times in this
proof.
Next we show a similar bound for covol(Γ10 ), as Q
follows.P ei be a resolution
Let X
as in Sect. 11, and regard Γ1 as a submodule of Γ e 2 ∗ 0
Xi , d si pri L . We have
Γ10 = Γ1,m ⊆ Γ1,m−1 ⊆ . . . ⊆ Γ1,0 = Γ1 ,
where each Γ1,f is the kernel of a map βf from Γ1,f −1 to some module determined
by certain partial derivatives, as in Lemma 11.2. We will apply Lemma 12.2 to
βf . First consider (11.4.1). Since each Dj in Theorem 11.4 is the pull-back of a
metrized divisor on some X ei , the corresponding height satisfies h(ξj ) ≤ ch(Xi (j ) )+
c 0 . Moreover by Lemma 11.2 the factors `j are bounded by cdsi2 , so P by2 (4.5.4)
and the condition (12.3) on s, theP bound
(11.4.1) is at most exp cd si . Thus
(12.2.1) holds with C = exp cd si2 . Similarly, (4.5.4) and (12.3) (with Pdi =
dsi2 ) imply that the right-hand side of (11.5.1) is not smaller than exp −cd si2 .
This gives
X
covol(βf (Γ1,f −1 )) ≥ exp − rank βf (Γ1,f −1 ) · cd si2 .
By (12.2.2), therefore,
covol(Γ1,f ) X
≤ exp δ1 · cd si2
covol(Γ1,f −1 )
and thus, by descending induction on f ,
X
covol(Γ10 ) ≤ exp δ1 · cd si2 .
Here we note that some steps can be done in Pparallel: in fact, the above bounds
on the `j imply that we can take m ≤ cd si2 , so the power of 2 in (12.2.2)
does not affect the shape of this estimate.
Next, it is an easy matter to bound covol(Γ00 ); this argument is the same as
in [F 1]. Indeed, by ([V 4], Lemma 13.9),
X
covol(β(Γ10 )) ≥ exp − rank β(Γ10 ) · cd si2 ,
P
and by construction (12.2.1) holds with C ≤ exp cd si2 . Thus by (12.2.2)
X
(12.4.2) covol(Γ00 ) ≤ exp δ1 · cd si2 .
Now let δ00 = rank Γ00 . By Proposition 10.7, δ1 /δ00 is bounded, so (12.4.2)
holds (after adjusting c) with δ1 replaced by δ00 . Then, by the second half of
(12.1.1), there exists a nonzero section γ ∈ Γ00 with
Y X
kγkv ≤ exp cd si2 .
v|∞
This formula still refers to the norm on Γ1 ; however, by ([V 4], Lemma 13.2b
and Corollary 13.7), the same bound holds using the norm on Γ (V , dM−,s ).

Applying Proposition 10.10 then gives the following result:
Q
Proposition 12.5. Let P ∈ Xi (k ) and let γ be a global section as in Theorem
12.4. Then we have
(12.5.1)
Ã ! 2
ds 2 2
ds
Y X n YY µ Y αvmii + αvmjj
kγ(P )kv ≤ exp cd si2 · ,
ds 2 2 dsj2 2
v∈S i =1 v∈S m=1 i <j 1 + α i 1 + α
vmi vmj
where αvmi = pr∗i αm as defined in (2.7), relative to kv .
Proof . This is immediate from Proposition 10.10 and (10.6).

Remark 12.6. Theorem 12.4 can also be used to find a section whose norm is
bounded with respect to a given finite set of singular metrics. Indeed, one only
needs to repeat the fifth paragraph of the proof for each of the metrics. This fact
will be used in a subsequent paper.
178 P. Vojta
13. End of the proof
The next few steps of the proof of Theorem 0.2 are much the same as have
appeared previously; therefore they can be described very quickly.
To begin, for places v ∈ S let λm,v be the Weil function defined in Proposition
2.6. The set A(RS ) can be embedded into the finite dimensional vector space

V := A0 (k ) ⊗Z R ⊕ Rµ·#S
via the map

P 7→ (ρ(P ), (λm,v (P ))m,v ) .
On the first factor, the canonical height defines a length function
q
`0 (P ) = ĥL0 (ρ(P )) ;
on the second factor, let

µ
1 XX
`1 (P ) = |λm,v (P )| .
[k : Q]
v∈S m=1
Both length functions are nondegenerate. We note that `1 (P ) is related to the

height function hL1 (P ). Indeed, the functions max(λm,v , 0) and max(−λm,v , 0)
are Weil functions for the divisors [0]m and [∞]m , respectively, at the place v,
although they may differ by a bounded amount from the metrics (2.8). However,
this fact still implies that
(13.1) hL1 (P ) = `1 (P ) + O(1) .
(This equality requires the fact that P ∈ A(RS ). It will be needed for (13.7).)
This argument uses the same sphere packing argument as in [V 4], except
that now it must take place simultaneously on two spheres, due to the fact that
hL0 behaves quadratically in the group law, whereas hL1 behaves linearly. But
note that both hL0 and hL1 are bounded from below, and hL = hL0 + hL1 . Then,
for points P ∈ A(RS ), the points
· ρ(P ) ∈ A0 (k ) ⊗Z R
1
√
hL (P )
and
∈ R#S ·µ ;
1
· λm,v (P )
hL (P ) v∈S , 1≤m≤µ
both lie in the unit balls relative to the length functions `0 and `1 , respectively.
Now we assume that for some predetermined 1 > 0, the points P1 , . . . , Pn
have been chosen such that
Ã !
1 1
(13.2) `0 √ · ρ(Pi ) − p · ρ(Pj ) ≤ 1
hL (Pi ) hL (Pj )
and

1 1
(13.3) `1 · Pi − · Pj ≤ 1
hL (Pi ) hL (Pj )
for all i < j . For such
√a tuple (P1 , . . . , Pn ) choose rational numbers s1 , . . . , sn
sufficiently close to 1 hL (Pi ), i = 1, . . . , n, such that

(13.4) `0 si · ρ(Pi ) − sj · ρ(Pj ) ≤ 1
and

(13.5) `1 si2 · Pi − sj2 · Pj ≤ 1
for all i < j .
Let E be the arithmetic curve on V corresponding to (P1 , . . . , Pn ). Applying
(13.4) to the definition (3.3) of M−,s gives
(13.6)
1 n(n − 1) X
n X

deg M−,s E ≤ 1 − n + (n − 1) si2 hL1 (Pi ) + O si2 .
[k : Q] 2
i =1
This follows as in ([V 4], 17.2). But also (13.5) implies that `1 (dsi2 ·Pi −dsj2 ·Pj ) ≤
d 1 ; combining this with the inequality

(α + β)2 1 1
min α, min β, e | log(α/β)| , α > 0, β > 0
(1 + α)2 (1 + β)2 α β
implies that
(13.7)
dsi2 hL1 (Pi ) + dsj2 hL1 (Pj )

ds 2 2
ds 2
1 XX
µ αvmii + αvmjj X .
≤ − log si2
[k : Q] + d 1 + O d
dsi2 2 dsj2 2
v∈S m=1 1 + αvmi 1 + αvmj
Therefore

ds 2 2
ds 2
X
n
1 XXX
µ αvmii + αvmjj
(n − 1) dsi2 hL1 (Pi ) ≤ − log
[k : Q]
ds 2 2 ds 2 2
i =1 i <j
v∈S m=1 1 + αvmii 1 + αvmjj
n(n − 1)d X
+ 1 + O d si2 .
2
Combining this with (13.6) and (12.5.1) then gives
1 X X
− log γ E v ≤ dn((n − 1)1 − ) + O d si2 .
[k : Q]
v ∈S
/
If 1 and 2 > 0 are chosen sufficiently small and if the heights hL (Pi ) are
sufficiently large, then this bound becomes
180 P. Vojta
1 X
− log γ E v ≤ −d 2 .
[k : Q]
v ∈S
/
These bounds depend only on X ⊆ A. As in [F 1] and ([V 4], Proposition 16.1),

we obtain a positive lower bound for the index of γ along E :
t(γ, (P1 , . . . , Pn ), ds12 , . . . , dsn2 ) ≥ 3 ,
where 3 = 3 (2 , X ⊆ A).

The last step of the proof consists of applying Faltings’ product theorem, as
was done in ([F 1], Sect. 6) or [V 4]. This implies that at least one of P1 , . . . , Pn
lies in a strictly smaller Xi , still satisfying (4.5). The inductive step of Sect. 4
may then be carried out.
14. Proof of Corollary 0.3
Let g = h 1 (X , OX ). Then the Albanese variety Alb(X ) has dimension g. By the

condition on the number of components of Supp D, there are at least dim X −g +1
linearly independent divisors Ei such that Supp Ei ⊆ Supp D, and such that each
Ei is algebraically equivalent to zero. The line sheaves O (Ei ) can be used to
define a semiabelian variety A, which X \D maps into. But since dim A > dim X ,
the image of X \ D is a proper subvariety, to which we can apply Theorem 0.2.
This implies Corollary 0.3.
References
[F 1] Faltings, G.: Diophantine approximation on abelian varieties. Ann. Math., II. Ser. 133, 549–
576 (1991)
[F 2] Faltings, G.: The general case of S. Lang’s conjecture. In: Christante, V. and Messing, W.
(eds.) Barsotti symposium in algebraic geometry. (Perspectives in mathematics, vol. 15) San
Diego, Calif.: Academic Press (1994) pp. 175–182
[H 1] Hartshorne, R.: Ample subvarieties of algebraic varieties. (Lect. Notes Math., vol. 156) Berlin
Heidelberg New York: Springer (1970)
[H 2] Hartshorne, R.: Algebraic geometry (Grad. Texts Math. 52). New York: Springer (1977)
[I 1] Iitaka, S.: Logarithmic forms of algebraic varieties. J. Fac. Sci., Univ. Tokyo, Sect. I A 23,
525–544 (1976)
[I 2] Iitaka, S.: On logarithmic Kodaira dimension of algebraic varieties. In: Baily, Jr., W. L. and
Shioda, T. (eds.) Complex analysis and algebraic geometry: a collection of papers dedicated
to K. Kodaira. Cambridge: Cambridge University Press (1977) pp. 175–189
[Kl] Kleiman, S. L.: Toward a numerical theory of ampleness. Ann. Math., II. Ser. 84, 293–344
(1966)
[Ko] Kooman, R. J.: Faltings’s version of Siegel’s lemma. In: Edixhoven, B. and Evertse, J.-H.
(eds.) Diophantine approximation and abelian varieties. (Lect. Notes Math., vol. 1566) Berlin
Heidelberg: Springer (1993) pp. 93–96
[L 1] Lang, S.: Integral points on curves. Publ. Math., Inst. Hautes Étud. Sci. 6, 27–43 (1960)
[L 2] Lang, S.: Fundamentals of diophantine geometry. New York: Springer (1983)
[L 3] Lang, S.: Hyperbolic and diophantine analysis. Bull. Am. Math. Soc., New Ser. 14, 159–205
(1986)
[L 4] Lang, S.: Number theory III: diophantine geometry. (Encyclopaedia of Mathematical Sci-
ences, vol. 60) Berlin Heidelberg: Springer (1991)
[Ma] Matsusaka, T.: The criterion for algebraic equivalence and the torsion group. Amer. J. Math.
79, 53–66 (1957)
[McQ] McQuillan, M.: Division points on semi-abelian varieties. Invent. Math. 120, 143–159 (1995)
[N 1] Noguchi, J.: Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties.
Nagoya Math. J. 83, 213–233 (1981)
[N 2] Noguchi, J.: A higher dimensional analogue of Mordell’s conjecture over function fields.
Math. Ann. 258, 207–212 (1981)
[S 1] Serre, J.-P.: Groupes algébriques et corps de classes. (Actualites scientifiques et industrielles,
vol. 1264) Paris: Hermann (1975)
[S 2] Serre, J.-P.: Quelques propriétés des groupes algébriques commutatifs. In: Waldshmidt, M.
(ed.) Nombres transcendants et groupes algébriques. Astérisque 69–70, 191–203 (1979)
[V 1] Vojta, P: Diophantine approximations and value distribution theory. (Lect. Notes Math., vol.
1239) Berlin Heidelberg New York: Springer (1987)
[V 2] Vojta, P: Siegel’s theorem in the compact case. Ann. Math., II. Ser. 133, 509–548 (1991)
[V 3] Vojta, P: A generalization of theorems of Faltings and Thue-Siegel-Roth-Wirsing. J. Am.
Math. Soc. 5, 763–804 (1992)
[V 4] Vojta, P: Applications of arithmetic algebraic geometry to diophantine approximations. In:
Ballico, E. (ed.) Arithmetic Algebraic Geometry, Trento, 1991. (Lect. Notes Math., vol. 1553)
Berlin Heidelberg: Springer (1993) pp. 164–208
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Invent. math.

126, 133–181 (1996)

Integral points on subvarieties of semiabelian

Oblatum 7-VII-1993 & 18-XII-1995

Dedicated to Professor Wolfgang M. Schmidt on the occasion of his sixtieth birthday

? Partially supported by NSF grants DMS-9001372 and DMS-9304899.

Γ be the division group of Γ ; i.e., the group of all x ∈ A such that nx ∈ Γ

2. Structure of semiabelian varieties

(2.3) Ā := P(OA0 ⊕ M1 ) ×A0 . . . ×A0 P(OA0 ⊕ Mµ ) .

The morphism ρ : Ā → A0 extends in the obvious manner. This completion was

(2.4) 0 → Pic(A0 ) → Pic(Ā) → Zµ → 0 ,

(2.5) OA0 ⊕ Mm → OA0 and OA0 ⊕ Mm → Mm .

so that in particular we have the numerical equivalence

Proposition 2.6. Let Ā be the completion of a semiabelian variety A defined over

for all P , Q ∈ A(Cv ) and all places v of k . Moreover, if v is archimedean, then

Proof . We may assume µ = m = 1; the general case follows by pulling back to

(2.6.2) (z ) = [0] − [∞] − ρ∗ (s) .

(2.6.3) λ(P ) = (− log kz kv + λ(s) ◦ ρ) − (− log kz (0)kv + λ(s) (0)) ,

provided s is chosen so as to generate M at 0 ∈ A0 . By (1) and (2) and (2.6.2),

(f , gs) 7→ (z1 z2 /z0 u(0))f − g .

Indeed, this defines a rational map A × A 99K A. Replacing s by s 0 = hs changes

(2.7) αm (P + Q) = αm (P )αm (Q)

for all P , Q ∈ A(Cv ) and all places v of k .

Proof . This is a straightforward application of Seshadri’s criterion for ampleness

as a Q-divisor class on Ws . Also let

Let ` be a positive integer. Multiplication by ` extends to a morphism

as a divisor on Ws . This is homogeneous of degree 1 in s (up to pulling back, as

Proposition 3.5. The Q-divisor Qs,m

Proof . We may assume that s is a tuple of integers. To shorten notation, let

pr∗i [0]m ∩ pr∗j [0]m and pr∗i [∞]m ∩ pr∗j [∞]m .

First of all, we may assume that X is relatively closed in A and geometrically

in A. Then the restriction of the quotient map A → A/B (X ) to X exhibits X as a

and choose points P1 , . . . , Pn in X (RS )\Z (X ) satisfying conditions CP (c1 , c2 , 1 ):

Here and throughout the proof, constants c and ci depend only on X , A, n, k ,

Proof . If X is a closed subvariety of A and P ∈ Xreg , then the tangent space TX ,P

Pij := (pri + prj )∗ L0 − pr∗i L0 − pr∗j L0

is represented over A0 (C)n by a form in

Proof . See ([V 4], Lemma 11.3).

it follows that if µ = 0 then the highest self-intersection number of Lδ,s is homo-

where fmij ∈ Z and ρm,i is a nonzero smooth function on the closure of Ω. We

(smooth function) · dzi1 ∧ . . . ∧ dziN −p ∧ d z̄j1 ∧ . . . ∧ d z̄jN −p .

Let F = (fij ) be a p × N matrix with entries in Z, and for i = 1, . . . , p let

Then for each e the integral

converges to a value bounded uniformly in e. Moreover, let F 0 be the matrix

it follows that (5.5.3.1) equals

Hence if F 0 is singular then the integral vanishes.

to bound the absolute value of the integral by

The first assertion of the lemma is then clear, since

Proof . We use the identity

we find that the factor

Proof . We may regard Ws as a closed subscheme of Ān+n(n−1)/2 in an obvious

for all sufficiently large d (depending on s).

Proof . By Lemma 6.1, Lδ,s is ample. Riemann-Roch therefore implies that, as

7. Generalized Weil functions

Proposition 7.2. Generalized Weil functions on a scheme X form an abelian

Proof . Since min(g1 , g2 ) = g2 + min(g1 − g2 , 0), we may assume that g2 = 0. By

Proof . Replacing X with a suitable nonsingular blowing-up, we may assume that

for some Mk -constant (cv ), where φ1 , . . . , φn , ψ1 , . . . , ψm are nonzero rational

Definition 7.6. Let g be a generalized Weil function on a variety X and let F ⊆ X

Proposition 7.7. Min-min generalized Weil functions on a variety X form a

and choose points P1 , . . . , Pn in X (RS )\Z (X ) satisfying conditions CP (c1 , c2 , 1 ):

(10.1) γ ∈ Γ (V , dM−,s ) ∩ Γ (Ws , dL−,s ) .

L0 = 2(n − 1)ρ∗ L0 + (n − 1)L1 + (n − 1 − )L