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# Mathematical Research Letters

## Volume 23 (2016)

### Number 6

### On the (non)existence of symplectic resolutions of linear quotients

Pages: 1537 – 1564

DOI: https://dx.doi.org/10.4310/MRL.2016.v23.n6.a1

#### Authors

#### Abstract

We study the existence of symplectic resolutions of quotient singularities $V/G$, where $V$ is a symplectic vector space and $G$ acts symplectically. Namely, we classify the symplectically irreducible and imprimitive groups, excluding those of the form $K \rtimes S_2$ where $K \lt \mathsf{SL}_2 (\mathbf{C})$, for which the corresponding quotient singularity admits a projective symplectic resolution. As a consequence, for $\dim V \neq 4$, we classify all symplectically irreducible quotient singularities $V/G$ admitting a projective symplectic resolution, except for at most four explicit singularities, that occur in dimensions at most $10$, for which the question of existence remains open.

#### Keywords

symplectic resolution, symplectic smoothing, symplectic reflection algebra, Poisson variety, quotient singularity, McKay correspondence

#### 2010 Mathematics Subject Classification

16S80, 17B63

Accepted 17 September 2015

Published 21 February 2017