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A generalized Bloch’s theorem and the hyperbolicity of the complement of an ample divisor in an abelian variety. (English) Zbl 0882.32009
Math. Ann. 306, No. 4, 743-758 (1996); erratum ibid. 326, No. 1, 205-207 (2003).
The purpose of this paper is to prove the following generalization of Bloch’s theorem: If the image of a holomorphic map \(f\) from \(\mathbb{C}\) to an abelian variety is Zariski dense, then the image of the differential of any order of the map \(f\) is invariant under any translation of the abelian variety.
As a corollary the authors prove the next statement: The image from \(\mathbb{C}\) to an abelian variety with Zariski dense image must intersect any hypersurface of the abelian variety.
This theorem together with Bloch’s theorem implies Lang’s conjecture [S. Lang: ‘Number theory’, III: Diophantine geometry. (1991; Zbl 0744.14012)].

MSC:
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
14K25 Theta functions and abelian varieties
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References:
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