Pseudo Kobayashi hyperbolicity of subvarieties of general type on abelian varieties

Abstract

We prove that the Kobayashi pseudo distance of a closed subvariety $X$ of an abelian variety $A$ is a true distance outside the special set $\operatorname{Sp}(X)$ of $X$, where $\operatorname{Sp}(X)$ is the union of all positive dimensional translated abelian subvarieties of $A$ which are contained in $X$. More strongly, we prove that a closed subvariety $X$ of an abelian variety is taut modulo $\operatorname{Sp}(X)$; Every sequence $f_n:{\mathbb{D}}\to X$ of holomorphic mappings from the unit disc ${\mathbb{D}}$ admits a subsequence which converges locally uniformly, unless the image $f_n(K)$ of a fixed compact set $K$ of ${\mathbb{D}}$ eventually gets arbitrarily close to $\operatorname{Sp}(X)$ as $n$ gets larger. These generalize a classical theorem on algebraic degeneracy of entire curves in irregular varieties.