THE CANONICAL ARITHMETIC HEIGHT OF SUBVARIETIES OF AN ABELIAN VARIETY OVER A FINITELY GENER.pdf
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THE CANONICAL ARITHMETIC HEIGHT OF SUBVARIETIES OF AN ABELIAN VARIETY OVER A FINITELY GENER
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THE CANONICAL ARITHMETIC HEIGHT OF SUBVARIETIES
OF AN ABELIAN VARIETY
OVER A FINITELY GENERATED FIELD
ATSUSHI MORIWAKI
Introduction
This paper is the sequel of [2]. In [4], S. Zhang defined the canonical height of subvarieties
of an abelian variety over a number field in terms of adelic metrics. In this paper, we
generalize it to an abelian variety defined over a finitely generated field over Q. Our way is
slightly different from his method. Instead of using adelic metrics directly, we introduce an
adelic sequence and an adelic structure (cf. §§3.1).
Let K be a finitely generated field over Q with d = tr. degQ(K), and B = (B;H1, . . . , Hd)
a polarization of K, i.e., B is a projective arithmetic variety whose function field is K, and
H1, . . . , Hd are nef C∞-hermitian line bundles on B. Let A be an abelian variety over K,
and L a symmetric ample line bundle on A. Fix a projective arithmetic variety A over B
and a nef C∞-hermitian Q-line bundle L on A such that A is the generic fiber of A→ B and
L is isomorphic to L on A. Then we can assign the naive height hB
(A,L)(X) to a subvariety
X of AK . Indeed, if X is defined over K, h
B
(A,L)(X) is given by
d?eg
(
c?1(L
∣∣
X )
·dimX+1 · c?1(π?X (H1)) · · · c?1(π?X (Hd))
)
(dimX + 1) deg(L|dimXX )
,
where X is the Zariski closure of X in A and πX : X → B is the canonical morphism.
The canonical height h?BL (X) of X with respect to L and B is characterized by the following
properties:
(a) h?BL (X) ≥ 0 for all subvarieties X of AK .
(b) There is a constant C such that∣∣∣h?BL (X) ? hB(A,L)(X)∣∣∣ ≤ C
for all subvarieties X of AK .
(c) h?BL ([N ](X)) = N
2h?BL (X) for all subvarieties X of AK and all non-zero integers N .
The main result of this paper is the following theorem, which is a generalization of [5].
Theorem (cf. Theorem 5.1). If the polarization B is big (i.e., H1, . . . , Hd are nef and big),
then, for a subvariety X of AK, the following are equivalent.
Date: 19/Oc
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