Arithmetic Part of Faltings’s Proof

Abstract

In this chapter we will follow §5 of [22] quite closely. We start in the following situation:

k is a number field, A _k is an abelian variety over k, X _k ⊂ A _k is a subvariety such that $X_{\bar k}$ does not contain any translate of a positive dimensional abelian subvariety of $A_{\bar k}$, m is a sufficiently large integer as in Chapter IX, Lemma 1

Let R be the ring of integers in k. Since we want to apply Faltings’s version of Siegel’s Lemma (see Ch. X, Lemma 4) we need lattices in things like г(X ^m _k , line bundle). We obtain such lattices as:

г(proper model of X ^m _k over R, extension of line bundle).