Mukai implies McKay: the McKay correspondence as an equivalence of derived categories

TL;DR

This paper proves that for a finite group acting on a complex threefold with trivial canonical bundle, the Hilbert scheme provides a crepant resolution and establishes a derived category equivalence, extending the McKay correspondence.

Contribution

It generalizes the McKay correspondence to threefolds by showing the Hilbert scheme yields a crepant resolution and a derived equivalence with G-sheaves.

Findings

Hilbert scheme Y is a crepant resolution of X=M/G

Derived equivalence (Fourier-Mukai transform) between Y and G-sheaves on M

K-theory of Y matches equivariant K-theory of M

Abstract

Let G be a finite group of automorphisms of a nonsingular complex threefold M such that the canonical bundle omega_M is locally trivial as a G-sheaf. We prove that the Hilbert scheme Y=GHilb M parametrising G-clusters in M is a crepant resolution of X=M/G and that there is a derived equivalence (Fourier- Mukai transform) between coherent sheaves on Y and coherent G-sheaves on M. This identifies the K theory of Y with the equivariant K theory of M, and thus generalises the classical McKay correspondence. Some higher dimensional extensions are possible.