Counting resolutions of symplectic quotient singularities

TL;DR

This paper provides a formula to count symplectic resolutions of quotient singularities by finite symplectic groups, linking deformation theory with algebraic invariants, and confirms a conjecture by Ginzburg and Kaledin.

Contribution

It introduces a new method to count symplectic resolutions using Poisson deformations and Orlik-Solomon algebra calculations, confirming a key conjecture.

Findings

Derived a formula for the number of symplectic resolutions.

Explicitly computed the relevant algebraic dimension for known cases.

Confirmed the Ginzburg-Kaledin conjecture for these cases.

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.