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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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The McKay correspondence as an equivalence of derived categories
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by Tom Bridgeland, Alastair King and Miles Reid HTML | PDF
J. Amer. Math. Soc. 14 (2001), 535-554 Request permission

Abstract:

Let $G$ be a finite group of automorphisms of a nonsingular three-dimensional complex variety $M$, whose canonical bundle $\omega _M$ is locally trivial as a $G$-sheaf. We prove that the Hilbert scheme $Y = G$-$\operatorname {Hilb}M$ 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.

References
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Additional Information
  • Tom Bridgeland
  • Affiliation: Department of Mathematics and Statistics, University of Edinburgh, King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom
  • MR Author ID: 635821
  • ORCID: 0000-0001-5120-006X
  • Email: tab@maths.ed.ac.uk
  • Alastair King
  • Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
  • Email: a.d.king@maths.bath.ac.uk
  • Miles Reid
  • Affiliation: Math Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
  • Email: miles@maths.warwick.ac.uk
  • Received by editor(s): May 1, 2000
  • Received by editor(s) in revised form: November 1, 2000
  • Published electronically: March 22, 2001
  • Additional Notes: Earlier versions of this paper carried the additional title “Mukai implies McKay”

  • Dedicated: To Andrei Tyurin on his 60th birthday
  • © Copyright 2001 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 14 (2001), 535-554
  • MSC (2000): Primary 14E15, 14J30; Secondary 18E30, 19L47
  • DOI: https://doi.org/10.1090/S0894-0347-01-00368-X
  • MathSciNet review: 1824990