Pseudo-quotients of algebraic actions and their application to character varieties

TL;DR

This paper introduces pseudo-quotients as a topological alternative to classical GIT quotients, facilitating easier geometric analysis and computation of character varieties in algebraic geometry.

Contribution

It defines pseudo-quotients, explores their properties, and applies them to compute virtual classes of SL(2) character varieties for various groups.

Findings

Pseudo-quotients are unique up to virtual class in characteristic zero.

They simplify the identification of quotients in geometric constructions.

Virtual classes of SL(2) character varieties are explicitly computed.

Abstract

In this paper, we propose a weak version of quotient for the algebraic action of a group on a variety, which we shall call a pseudo-quotient. They arise when we focus on the purely topological properties of good GIT quotients regardless of their algebraic properties. The flexibility granted by their topological nature enables an easier identification in geometric constructions than classical GIT quotients. We obtain several results about the interplay between pseudo-quotients and good quotients. Additionally, we show that in characteristic zero pseudo-quotients are unique up to virtual class in the Grothendieck ring of algebraic varieties. As an application, we compute the virtual class of $\mathrm{SL}_{2}(k)$-character varieties for free groups and surface groups as well as their parabolic counterparts with punctures of Jordan type.