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Counting resolutions of symplectic quotient singularities
Published online by Cambridge University Press: 24 September 2015
Abstract
Let ${\rm\Gamma}$ be a finite subgroup of $\text{Sp}(V)$. In this article we count the number of symplectic resolutions admitted by the quotient singularity $V/{\rm\Gamma}$. 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 ${\rm\Gamma}$ for which it is known that $V/{\rm\Gamma}$ admits a symplectic resolution. As a consequence of our results, we confirm a conjecture of Ginzburg and Kaledin.
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- Research Article
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- © The Author 2015
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