Holomorphic symplectic geometry and orbifold singularities

TL;DR

This paper proves that if a finite group acting on a symplectic vector space admits a holomorphic symplectic resolution, then the group is generated by symplectic reflections, extending the understanding of orbifold singularities and their resolutions.

Contribution

It establishes that groups with symplectic resolutions are generated by symplectic reflections, providing a symplectic analogue of classical conjectures and deepening the link between group actions and singularity resolutions.

Findings

G is generated by symplectic reflections when V/G admits a symplectic resolution

Symplectic resolutions are always semismall

A crepant resolution of V/G is always symplectic

Abstract

Let G be a finite group acting on a symplectic complex vector space V. Assume that the quotient V/G has a holomorphic symplectic resolution. We prove that G is generated by "symplectic reflectionsd"', i.e. symplectomorphisms with fixed space of codimension 2 in V. Symplectic resolutions are always semismall. A crepant resolution of V/G is always symplectic. We give a symplectic version of Nakamura conjectures.