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Counting resolutions of symplectic quotient singularities
- Part of
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- Journal:
- Compositio Mathematica / Volume 152 / Issue 1 / January 2016
- Published online by Cambridge University Press:
- 24 September 2015, pp. 99-114
- Print publication:
- January 2016
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- Article
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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.
