Hyperbolicity and fundamental groups of complex quasi-projective varieties (II): via non-abelian Hodge theories

Abstract

This is Part II of a series of three papers. We studies the hyperbolicity of complex quasi-projective varieties $X$ in the presence of a big and reductive representation $\varrho: \pi_1(X)\to {\rm GL}_N(\mathbb{C})$. For any Galois conjugate variety $X^\sigma$ with $\sigma \in {\rm Aut}(\mathbb{C}/\mathbb{Q})$, we prove the generalized Green-Griffiths-Lang conjecture. When $\varrho$ is furthermore large, we show that the special subsets of $X^\sigma$ describing the non-hyperbolicity locus coincide, and that this locus is proper exactly when $X$ is of log general type. Moreover, if the Zariski closure of $\rho(\pi_1(X))$ is semisimple, we prove that there exists a proper Zariski closed subset $Z \subsetneq X^\sigma$ such that every subvariety not contained in $Z$ is of log general type and all entire curves in $X^\sigma$ are contained in $Z$. This result extends the theorems of the third author (2010) and of Campana-Claudon-Eyssidieux (2015) from projective to quasi-projective varieties, and yields stronger conclusions even in the projective case.