Mathematical Research Letters

Volume 23 (2016)

Number 6

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

Pages: 1537 – 1564



Gwyn Bellamy (School of Mathematics and Statistics, University of Glasgow, Scotland, United Kingdom)

Travis Schedler (Department of Mathematics, Imperial College London, United Kingdom)


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.


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