Two-step nilpotent monodromy of local systems on special varieties

TL;DR

This paper proves that monodromy groups of local systems on special quasi-projective varieties are virtually nilpotent of class at most 2, refining previous results and developing a deformation theory for such systems.

Contribution

It introduces a deformation theory for local systems on quasi-compact K"ahler manifolds and extends properties of the quasi-Albanese map to the quasi-projective setting.

Findings

Monodromy groups are virtually nilpotent of class at most 2.

Developed a universal deformation construction for local systems.

Extended the understanding of the quasi-Albanese map to special quasi-projective varieties.

Abstract

Let $X$ be a smooth complex quasi-projective variety that is special in the sense of Campana. We prove that the monodromy group of any complex local system on $X$ is virtually nilpotent of class at most $2$. This result sharply refines a theorem of Cadorel, Yamanoi, and the second author. To establish this result, we develop a deformation theory for certain local systems on quasi-compact K\"ahler manifolds by constructing universal deformations for such local systems. As a byproduct of our argument, we also show that a general fiber of the quasi-Albanese map of $X$ is special, extending a result of Campana and Claudon from the projective to the quasi-projective setting.