McKay equivalence for symplectic resolutions of singularities

TL;DR

This paper proves an equivalence of derived categories between crepant resolutions of symplectic quotient singularities and G-equivariant sheaves on the original space, extending McKay correspondence to symplectic resolutions.

Contribution

It establishes a derived category equivalence for symplectic resolutions of quotient singularities, generalizing the McKay correspondence.

Findings

Derived categories of crepant resolutions are equivalent to G-equivariant sheaves.

The result applies to any crepant resolution of V/G.

Extends McKay correspondence to symplectic cases.

Abstract

Let $V$ be a finite-dimensional symplectic vector space over a field of characteristic 0, and let $G \subset Sp(V)$ be a finite subgroup. We prove that for any crepant resolution $X \to V/G$, the bounded derived category $D^b(Coh(X))$ of coherent sheaves on $X$ is equivalent to the bounded derived category $D^b_G(Coh(V))$ of $G$-equivariant coherent sheaves on $V$.