Smooth Families of Tori and Linear K\"ahler Groups

TL;DR

This paper proves that linear Kähler groups are fundamental groups of smooth complex projective varieties by studying the deformation of smooth torus families in an equivariant setting.

Contribution

It establishes that linear Kähler groups can be realized as fundamental groups of smooth projective varieties, extending previous results with new deformation techniques.

Findings

Linear Kähler groups are fundamental groups of smooth projective varieties.

Deformation of smooth torus families is key to the proof.

Enhances previous results in [CCE14].

Abstract

That short note, meant as an addendum to [CCE14], enhances the results contained in loc. cit. In particular it is proven here that a linear K{\"a}hler group is already the fundamental group of a smooth complex projective variety. This is achieved by studying the relative deformation of the total space of a smooth family of tori in an equivariant context.