Symplectic resolutions of character varieties

Bellamy, Gwyn; Schedler, Travis

doi:10.2140/gt.2023.27.51

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Symplectic resolutions of character varieties

Gwyn Bellamy and Travis Schedler

Geometry & Topology 27 (2023) 51–86

DOI: 10.2140/gt.2023.27.51

Abstract

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 $A$ (eg ${SL}_{n}$ and ${GL}_{n}$ ). We show that these varieties are $ℚ$ –factorial symplectic singularities and classify which admit symplectic resolutions. The classification reduces to the semisimple 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 $g = 2$ and $G = {({SL}_{2})}^{m}$ for some $m$ .

Keywords

symplectic resolution, character variety, Poisson variety

Mathematical Subject Classification 2010

Primary: 14D20, 16D20

Secondary: 16S80, 17B63

References

Publication

Received: 26 September 2019

Revised: 10 July 2021

Accepted: 15 August 2021

Published: 1 May 2023

Proposed: Ciprian Manolescu

Seconded: Frances Kirwan, Richard P Thomas

Authors

Gwyn Bellamy
School of Mathematics and Statistics University of Glasgow Glasgow United Kingdom

Travis Schedler
Department of Mathematics Imperial College London London United Kingdom

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Volume 27, issue 1 (2023)