Hyperbolicity of varieties with big linear representation of $\pi_1$

TL;DR

This paper proves that complex projective varieties with large linear representations of their fundamental group exhibit algebraic and hyperbolic properties, restricting the nature of holomorphic maps from curves.

Contribution

It establishes a new algebraicity result linking big fundamental group representations to the algebraic structure and hyperbolicity of the variety.

Findings

Existence of a proper subvariety Z controlling holomorphic maps from curves

Holomorphic maps from curves outside Z are induced by algebraic morphisms

Varieties with big reductive representations are pseudo-Brody hyperbolic

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 $\pi_1$.