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In this paper we prove that every quasi-projective base space $V$ of smoothfamily of minimal projective manifolds with maximal variation is pseudoKobayashi hyperbolic, i.e. $V$ is Kobayashi hyperbolic modulo a propersubvariety $Z\subsetneq V$. 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 adirect consequence, we prove the Brody hyperbolicity of moduli spaces ofminimal projective manifolds, which answers a question by Viehweg-Zuo in 2003.

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