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.

arXiv commentDedicated 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

References in corpus (4)

• McKay correspondence
• How to calculate A-Hilb C^3
• Fourier-Mukai transforms for K3 and elliptic fibrations
• Holomorphic symplectic geometry and orbifold singularities

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