Symplectic resolutions of character varieties

TL;DR

This paper studies the geometric structure of character varieties of compact Riemann surfaces, showing they are symplectic singularities and classifying when they admit symplectic resolutions based on genus and group type.

Contribution

It provides a classification of symplectic resolutions for character varieties of genus g > 0 for certain complex reductive groups, reducing the problem to the semi-simple case.

Findings

Character varieties are symplectic singularities.

Resolutions exist only for specific genus and group combinations.

Classification reduces to semi-simple cases with explicit conditions.

Abstract

In this article we consider the connected component of the identity of $G$-character varieties of compact Riemann surfaces of genus $g > 0$, for connected complex reductive groups $G$ of type $A$ (e.g., $SL_n$ and $GL_n$). We show that these varieties are symplectic singularities and classify which admit symplectic resolutions. The classification reduces to the semi-simple case, where we show that a resolution exists if and only if either $g=1$ and $G$ is a product of special linear groups of any rank and copies of the group $PGL_2$, or if $g=2$ and $G = (SL_2)^m$ for some $m$.