Hyperbolicity and fundamental groups of complex quasi-projective varieties (III): applications
Benoit Cadorel, Ya Deng, Katsutoshi Yamanoi
TL;DR
This work advances the understanding of the interplay between hyperbolicity, fundamental groups, and birational geometry for complex quasi-projective varieties in Campana’s program. By combining Nevanlinna theory, Deligne’s mixed Hodge theory, and hyperbolicity techniques, it proves that for smooth quasi-projective $X$ which are special or $h$-special, any linear representation $\rho:\pi_1(X)\to GL_N(\mathbb{C})$ has an image whose identity component splits as $U\times T$ with $U$ unipotent and $T$ a torus, hence $\rho(\pi_1(X))$ is virtually nilpotent (virtually abelian if $\rho$ is reductive). It establishes a structure theorem for quasi-projective varieties with big, reductive representations, linking nonnegative logarithmic Kodaira dimension to a birational model with a log Iitaka fibration over a semi-abelian base, and it provides a refined view of Kollár’s 1995 conjecture in the quasi-projective setting. The paper also furnishes explicit examples illustrating sharpness of the results and the distinct behavior in non-compact cases, including a counterexample to Campana’s original abelianity conjecture and a characterization of varieties birational to semi-abelian varieties under reductive hypotheses. Overall, the results deepen the connections between fundamental group representations, hyperbolicity, and birational structure in the quasi-projective world, with potential implications for arithmetic geometry and the classification of complex varieties.
Abstract
This paper is Part III of a series of three. We begin by introducing the notion of $h$-special varieties, which can be seen as varieties "chain-connected by the Zariski closures of entire curves." We prove that if $X$ is either a special complex quasi-projective variety in the sense of Campana or an $h$-special variety, then for any linear representation $\varrho:π_1(X)\to \mathrm{GL}_N(\mathbb{C})$, the image $\varrho(π_1(X))$ is virtually nilpotent. We also provide examples showing that this result is sharp, leading to a revised form of Campana's abelianity conjecture for smooth quasi-projective varieties. In addition, we prove a structure theorem for quasi-projective varieties with big and semisimple representations of the fundamental groups, thereby addressing a conjecture by Kollár in 1995. We also construct several examples of quasi-projective varieties that are special and $h$-special, highlighting certain atypical properties of the non-compact case in contrast with the projective setting.