A generalized Bloch's theorem and the hyperbolicity of the complement of an ample divisor in an Abelian variety

Abstract

Bloch's theorem on the hyperbolicity of nonlinear subvarieties of abelian varieties states that the Zariski closure of the image of a holomorphic map from C to an abelian variety is precisely the translate of an abelian subvariety [B26,KS0,O77,W80,GG79] (for a survey on the proofs of Bloch's theorem see e.g., [Si95,Chapter 7]). The purpose of this note is to prove the following generalization of Bloch's theorem. If the image of a holomorphic map f from C to an abelian variety is Zariski dense, then the Zariski closure of the image of the differential of any order of the map f is invariant under any translation of the abelian variety (Theorem(2.2) below). This generalization has as a corollary the conjecture of Lang [La91] that the complement of an ample divisor in an abelian variety is hyperbolic. The following slightly more general statement, which together with Bloch's theorem implies Lang 's conjecture (see Corollary(3.2)), is a corollary of our generalized Bloch's theorem. The image from C to an abelian variety with Zariski dense image must intersect any complex hypersurface of the abelian variety (Theorem(3. !)).