Hyperbolicity and fundamental groups of complex quasi-projective varieties

This paper investigates the relationship between the hyperbolicity of complex quasi-projective varieties X and the (topological) fundamental group π_1(X) in the presence of a linear representation ϱ: π_1(X) → GL_N(ℂ). We present our main results in three parts. Firstly, we show that if ϱ is bigand the Zariski closure of ϱ(π_1(X)) semisimple, then for any X^σ:=X×_σℂ where σ∈ Aut(ℂ/ℚ), there exists a proper Zariski closed subset Z ⫋ X^σ such that any closed irreducible subvariety V of X^σ not contained in Z is of log general type, and any holomorphic map from the punctured disk 𝔻^* to X^σ with image not contained in Z does not have an essential singularity at the origin. In particular, all entire curves in X^σ lie on Z. We provide examples to illustrate the optimality of this condition. Secondly, assuming that ϱ is big and reductive, we prove the generalized Green-Griffiths-Lang conjecture for X^σ. Furthermore, if ϱ is large, we show that the special subsets of X^σ that capture the non-hyperbolicity locus of X^σ from different perspectives are equal, and this subset is proper if and only if X is of log general type. Lastly, we prove that if X is a special quasi-projective manifold in the sense of Campana or h-special, then ϱ(π_1(X)) is virtually nilpotent. We provides examples to demonstrate that this result is sharp and thus revise Campana's abelianity conjecture for smooth quasi-projective varieties. To prove these theorems, we develop new features in non-abelian Hodge theory, geometric group theory, and Nevanlinna theory. Some byproducts are obtained.