Hyperbolicity and fundamental groups of complex quasi-projective varieties (II): via non-abelian Hodge theories
Benoit Cadorel, Ya Deng, Katsutoshi Yamanoi
TL;DR
We address hyperbolicity for complex quasi-projective varieties $X$ endowed with a big reductive representation $\varrho: \pi_1(X)\to GL_N(\mathbb{C})$, proving that all Galois conjugates $X^\sigma$ are of log general type away from a proper locus and satisfy pseudo Picard/Brody hyperbolicity. The approach leverages non-abelian Hodge theory, including a KZ-type reduction, spectral coverings, and harmonic maps into Bruhat–Tits buildings, to translate representation-theoretic data into geometric positivity. A key outcome is a stability-under-conjugation theorem: the log-general-type property and hyperbolicity loci defined via Sp-sets are equivalent for $X^\sigma$ for any $\sigma$, with a robust factorization framework for linear representations through fibrations. The results extend prior projective cases to quasi-projective varieties, provide a unified non-abelian toolbox, and lay the groundwork for Part III applications to Campana’s special varieties and related structure theorems.
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 $\varrho: π_1(X)\to {\rm GL}_N(\mathbb{C})$. For any Galois conjugate variety $X^σ$ with $σ\in {\rm Aut}(\mathbb{C}/\mathbb{Q})$, we prove the generalized Green-Griffiths-Lang conjecture. When $\varrho$ is furthermore large, we show that the special subsets of $X^σ$ 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 $ρ(π_1(X))$ is semisimple, we prove that there exists a proper Zariski closed subset $Z \subsetneq X^σ$ such that every subvariety not contained in $Z$ is of log general type and all entire curves in $X^σ$ 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.