Towards the classification of symplectic linear quotient singularities admitting a symplectic resolution

TL;DR

This paper advances the classification of symplectic linear quotient singularities by proving non-existence of symplectic resolutions for most cases, reducing the open problem to finitely many remaining instances.

Contribution

It completes the classification for infinite series in dimension 4 and partially addresses exceptional groups, narrowing down the open cases to 48.

Findings

Most infinite series cases do not admit symplectic resolutions.

One exceptional group is proven not to admit a symplectic resolution.

Remaining open cases are reduced to 48, with no expectation of resolutions.

Abstract

Over the past two decades, there has been much progress on the classification of symplectic linear quotient singularities V/G admitting a symplectic (equivalently, crepant) resolution of singularities. The classification is almost complete but there is an infinite series of groups in dimension 4 - the symplectically primitive but complex imprimitive groups - and 10 exceptional groups up to dimension 10, for which it is still open. In this paper, we treat the remaining infinite series and prove that for all but possibly 39 cases there is no symplectic resolution. We thereby reduce the classification problem to finitely many open cases. We furthermore prove non-existence of a symplectic resolution for one exceptional group, leaving 39+9=48 open cases in total. We do not expect any of the remaining cases to admit a symplectic resolution.