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 Sp(𝑋) of 𝑋, where 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 Sp(𝑋); Every sequence 𝑓_𝑛 : 𝔻 → 𝑋 of holomorphic mappings from the unit disc 𝔻 admits a subsequence which converges locally uniformly, unless the image 𝑓_𝑛(𝐾) of a fixed compact set 𝐾 of 𝔻 eventually gets arbitrarily close to Sp(𝑋) as 𝑛 gets larger. These generalize a classical theorem on algebraic degeneracy of entire curves in irregular varieties.