Symplectic resolutions of character varieties

Bellamy, Gwyn¹ ; Schedler, Travis²

¹ School of Mathematics and Statistics, University of Glasgow, Glasgow, United Kingdom
² Department of Mathematics, Imperial College London, London, United Kingdom

Geometry & topology, Tome 27 (2023) no. 1, pp. 51-86.

Résumé

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 ${\mathrm{SL}}_n$ and ${\mathrm{GL}}_n$). We show that these varieties are $\mathbb{Q}$–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 ${\mathrm{PGL}}_2$, or $g=2$ and $G={({\mathrm{SL}}_2)}^m$ for some $m$.

Détail

DOI : 10.2140/gt.2023.27.51

Classification : 14D20, 16D20, 16S80, 17B63
Keywords: symplectic resolution, character variety, Poisson variety

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