Counting resolutions of symplectic quotient singularities

Abstract

Let $\Gamma$ be a finite subgroup of $\mathrm{Sp}(V)$. In this article we count the number of symplectic resolutions admitted by the quotient singularity $V/\Gamma$. Our approach is to compare the universal Poisson deformation of the symplectic quotient singularity with the deformation given by the Calogero-Moser space. In this way, we give a simple formula for the number of $\mathbb{Q}$-factorial terminalizations admitted by the symplectic quotient singularity in terms of the dimension of a certain Orlik-Solomon algebra naturally associated to the Calogero-Moser deformation. This dimension is explicitly calculated for all groups $\Gamma$ for which it is known that $V/\Gamma$ admits a symplectic resolution. As a consequence of our results, we confirm a conjecture of Ginzburg and Kaledin.

13 pages, final version

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