On the (non)existence of symplectic resolutions for imprimitive symplectic reflection groups

TL;DR

This paper classifies when symplectic quotient singularities from imprimitive symplectic reflection groups admit projective symplectic resolutions, identifying specific cases where such resolutions exist or remain unresolved, especially in dimensions other than four.

Contribution

It provides a classification of symplectically irreducible and imprimitive groups for which the quotient singularities admit symplectic resolutions, excluding certain cases, and highlights unresolved cases in higher dimensions.

Findings

Classified all such quotient singularities with symplectic resolutions in most dimensions.

Identified four explicit unresolved singularities in dimensions up to 10.

Excluded certain group types from admitting resolutions.

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 $K < \SL_2(\C)$, for which the corresponding quotient singularity admits a projective symplectic resolution. As a consequence, for $\dim V \neq 4$, we classify all quotient singularities $V/G$ admitting a projective symplectic resolution which do not decompose as a product of smaller-dimensional quotient singularities, except for at most four explicit singularities, that occur in dimensions at most 10, for whom the question of existence remains open.