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Counting resolutions of symplectic quotient singularities

Part of: Rings and algebras arising under various constructions Birational geometry Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  24 September 2015

Gwyn Bellamy
Gwyn Bellamy*
Affiliation:
School of Mathematics and Statistics, University Gardens, University of Glasgow, GlasgowG12 8QW, UK email gwyn.bellamy@glasgow.ac.uk
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Abstract

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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.

Keywords

symplectic resolutionssymplectic reflection algebrasOrlik–Solomon algebras

MSC classification

Primary: 14E15: Global theory and resolution of singularities
Secondary: 14E30: Minimal model program (Mori theory, extremal rays) 16S80: Deformations of rings 17B63: Poisson algebras
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 1 , January 2016 , pp. 99 - 114
Copyright
© The Author 2015 

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