On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties

TL;DR

This paper demonstrates that any affine variety over Q can be related to the character variety of certain Artin groups, leading to new examples of groups not realizable as fundamental groups of smooth complex algebraic varieties.

Contribution

It establishes a connection between affine varieties and character varieties of Artin groups, and constructs finitely-presented groups that are not fundamental groups of smooth complex algebraic varieties.

Findings

Any affine variety over Q has a Zariski open subset isomorphic to a subset of a character variety of an Artin group.

Constructs examples of finitely-presented groups not realizable as fundamental groups of smooth complex algebraic varieties.

Shows the relationship between algebraic varieties, Artin groups, and fundamental groups of algebraic varieties.

Abstract

We prove that for any affine variety S defined over Q there exist Shephard and Artin groups G such that a Zariski open subset U of S is biregular isomorphic to a Zariski open subset of the character variety Hom(G, PO(3))//PO(3). The subset U contains all real points of S . As an application we construct new examples of finitely-presented groups which are not fundamental groups of smooth complex algebraic varieties.