Mukai implies McKay: the McKay correspondence as an equivalence of derived categories
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.
Keywords
Cite
@article{arxiv.math/9908027,
title = {Mukai implies McKay: the McKay correspondence as an equivalence of derived categories},
author = {Tom Bridgeland and Alastair King and Miles Reid},
journal= {arXiv preprint arXiv:math/9908027},
year = {2007}
}
Comments
Dedicated to Andrei Tyurin's 60th birthday. This draft is completely rewritten; it contains in particular a complete proof of Nakamura's conjecture that the Hilbert scheme of G-clusters is a crepant resolution for G in SL(3,C). 27 pp. submitted to J. Amer. Math. Soc