Singularities of character varieties

TL;DR

This paper studies the geometric and singularity properties of character varieties associated with complex reductive groups on Riemann surfaces, classifying when they admit symplectic resolutions and analyzing their singularities.

Contribution

It proves that all connected components of these character varieties are $Q$-factorial with symplectic singularities and classifies those with symplectic resolutions, using novel techniques for different genus cases.

Findings

Connected components are $Q$-factorial and have symplectic singularities.

Classification of components admitting symplectic resolutions.

Control of singularities via elliptic endoscopic groups for genus > 1.

Abstract

For any complex reductive group $G$ and any compact Riemann surface with genus $g>0$, we show that every connected component of the associated character variety is $\mathbb{Q}$-factorial and has symplectic singularities, and classify the connected components that admit symplectic resolutions. When $g>1$, we use elliptic endoscopic groups to control the singularities caused by irreducible local systems with automorphism groups larger than the centre of $G$; when $g=1$, our analysis is based on some results of Borel-Friedman-Morgan. The main results for $g>1$ were obtained by Herbig-Schwarz-Seaton via a different approach.