Hyperbolicity of varieties with big linear representation of ${\pi }_1$

Abstract

We show the following algebraicity result for a complex projective variety $X$ with big representation of ${\pi }_1$ into a semi-simple algebraic group: There exists a proper subvariety $Z\subset X$ such that for any algebraic curve $C$, any holomorphic map $\gamma :C\to X$ with $\gamma (C)\not\subset Z$ is induced from an algebraic morphism. As an application, we prove pseudo-Brody hyperbolicity of certain varieties with big reductive representations of ${\pi }_1$.

13 pages. Comments welcome. Section 3 simplified