Pseudo Kobayashi hyperbolicity of base spaces of families of minimal projective manifolds with maximal variation

Abstract

In this paper we prove that every quasi-projective base space $V$ of smooth family of minimal projective manifolds with maximal variation is pseudo Kobayashi hyperbolic, i.e. $V$ is Kobayashi hyperbolic modulo a proper subvariety $Z\subsetneq V$. In particular, $V$ is algebraically degenerate, that is, every nonconstant entire curve $f:\mathbb{C}\to V$ has image $f(\mathbb{C})$ which lies in that proper subvariety $Z\subsetneq V$. As a direct consequence, we prove the Brody hyperbolicity of moduli spaces of minimal projective manifolds, which answers a question by Viehweg-Zuo in 2003.

v2, 8 pages, a sketch of proof of Theorem 1.1 on the construction of Viehweg-Zuo Higgs bundles is added; the construction of Finsler metric is outlined; Remark 1.4 on the difficulty of Kobayashi hyperbolicity of moduli is added