Singularities of character varieties

TL;DR

The paper provides a comprehensive study of singularities in character varieties Ch(Π,G) for complex reductive G over genus g surfaces. It develops a uniform framework unifying almost simple and general reductive groups, using dimension theory and endoscopic techniques to control orbifold and endoscopic singularities for g>1, while leveraging Borel–Friedman–Morgan for g=1. The main achievements are establishing that all components are normal, $\mathbb{Q}$-factorial, and have symplectic singularities, and giving a complete classification of components that admit symplectic resolutions; in particular, it recovers and extends Bellamy–Schedler’s results. The work also clarifies how central isogenies and nonconnected target groups interact with symplectic structures, and it situates these results within related constructions such as twisted representation varieties and quasi-Hamiltonian reduction, with potential implications for mirror symmetry and nonabelian Hodge theory.

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.

Theorems & Definitions (99)

• Theorem A
• Corollary 1.1
• Theorem B
• Remark 1.2
• Remark 1.3
• Theorem C
• Proposition 3.1
• proof
• Corollary 3.2
• proof