Repr\'esentations lin\'eaires des groupes k\"ahl\'eriens : Factorisations et conjecture de Shafarevich lin\'eaire

TL;DR

This paper extends classical results on linear representations of fundamental groups from complex projective to compact Kähler manifolds, refining existing theorems and exploring holomorphic convexity and Shafarevich conjecture in this broader context.

Contribution

It generalizes fundamental group representation results to Kähler manifolds and refines the understanding of holomorphic convexity and Shafarevich conjecture in this setting.

Findings

Refined classical results on fundamental group representations for Kähler manifolds.

Extended holomorphic convexity results and Shafarevich conjecture to the Kähler case.

Removed incorrect proof regarding linear Kähler groups being virtually complex-projective.

Abstract

We extend to compact K\"ahler manifolds some classical results on linear representation of fundamental groups of complex projective manifolds. Our approach based on an interversion lemma for fibrations with tori versus general type manifolds as fibers gives a refinement of the classical work of Zuo. We extend to the kahler case some general results on holomorphic convexity of coverings such as the linear shafarevich conjecture. In the first version, the proof of the statement that a linear Kahler group is virtually complex-projective was wrong. We removed it from this new version. The proof will be given in a forthcoming work.