Counting resolutions of symplectic quotient singularities (Q2787621)

Let \(V/\Gamma\) be a symplectic quotient singularity, i.e. a quotient of a finite dimensional vector space \(V\) by a linear action of a finite subgroup \(\Gamma \subset \mathrm {Sp}(V)\). The article under review concerns \(\mathbb{Q}\)-factorial terminalizations of \(V/\Gamma\), i.e. projective, crepant, birational morphisms \(\rho : Y \rightarrow V/\Gamma\) such that \(Y\) has only \(\mathbb{Q}\)-factorial terminal singularities, and symplectic resolutions of \(V/\Gamma\), i.e. smooth \(\mathbb{Q}\)-factorial terminalizations. By results of Namikawa, a symplectic quotient singularity admits finitely many \(\mathbb{Q}\)-factorial terminalizations, and if one of them is smooth then all are smooth.NEWLINENEWLINEThe main result is a formula for the number of non-isomorphic \(\mathbb{Q}\)-factorial terminalizations of \(V/\Gamma\), expressed in terms of the Calogero-Moser deformation of \(V/\Gamma\) and the Namikawa Weyl group associated to \(V/\Gamma\). From this theorem the author derives a more explicit formula or obtains a number of symplectic resolutions for all groups \(\Gamma\) such that \(V/\Gamma\) is known to admit a symplectic resolution. These are: the infinite series of wreath products \(\mathcal{S}_n \wr G\), where \(G \subset \mathrm {SL}(2,\mathbb{C})\), acting on \(V = \mathbb{C}^{2n}\), a 4-dimensional representation \(G_4\) of the binary tetrahedral group and a 4-dimensional representation of \(Q_8 \times_{\mathbb{Z}_2} D_8\). In the case of \(\mathcal{S}_n \wr G\) the result is a formula involving the description of the Weyl group associated to \(G\) via the McKay correspondence, and for the remaining two cases the author obtains 2 and 81 symplectic resolutions respectively.