文档介绍：Invent. math. 73, 337-347 (1983) Inventiones
mathematicae
Springer-Verlag 1983
Arakelov's Theorem for Abelian Varieties
G. Faltings
Fachbereicb Mathematik, Universit~it-GesamthochschuleWuppertal,
GauBstr. 20, D-5600 Wuppertal 1, Federal Republic of Germany
w 1. Introduction
We start with an algebraic curve B over an algebraically closed field k of
characteristic zero, and let S~_B be a finite set of points. In [1] Arakelov has
shown that there are only finitely many families of algebraic curves of genus
g> 1 on B, with good reduction outside S, except for isotrivial families (iso-
trivial = es constant on a finite cover of B).
We want to consider the same question for principally polarized abelian
varieties. Here the answer is plicated:
There is a condition (*) such that the number of abelian varieties fulfilling
(,) is finite, while any variety not fulfilling it can be deformed. The condition
(,) says essentially that all endomorphisms of the cohomology of the abelian
variety are endomorphisms of the abelian variety itself.
The method of proof consists of bination of Arakelov's methods and
Deligne's description of abelian varieties via Hodge-structures. In the next
chapter we recall the necessary prerequisites, and after that we prove the
theorem in two steps as in [1].
We first derive a boundedness-theorem and then we show that families
fulfilling (,) cannot be deformed. From the form of the theorem it seems that it
is difficult to take it over to characteristic p>0 (see [7]). The author has
learned about this subject from L. Szpiro, who kindly printed out to him that
the following results are already known:
a) L. Moret-Bailly has proved a boundedness-theorem in any characteristic.
The result is contained in his thesis. (It will appear in the proceedings of the
"seminar on pencils of abelian varieties, Paris 1981/82", and in ptes-
Rendues). Unfortunately this theorem is we