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The McKay correspondence as an equivalence of derived categories


Authors: Tom Bridgeland, Alastair King and Miles Reid
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
Published electronically: March 22, 2001
MathSciNet review: 1824990
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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 $Y={\operatorname{\hbox{$G$ }-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.


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  • 1. M.F. Atiyah and F. Hirzebruch, The Riemann-Roch theorem for analytic embeddings, Topology 1 (1962) 151-166. MR 26:5593
  • 2. M.F. Atiyah and G. Segal, On equivariant Euler characteristics, J. Geom. Phys. 6 (1989) 671-677. MR 92c:19005
  • 3. W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces, Ergeb. Math. 4, Springer, 1984. MR 86c:32026
  • 4. A. Bondal, Representation of associative algebras and coherent sheaves, Math. USSR Izv. 34 (1990) 23-41. MR 90i:14017
  • 5. A. Bondal and M. Kapranov, Representable functors, Serre functors, and mutations, Math. USSR Izv. 35 (1990) 519-541. MR 91b:14013
  • 6. T. Bridgeland, Equivalences of triangulated categories and Fourier-Mukai transforms, Bull. London Math. Soc. 31 (1999) 25-34. MR 99k:18014
  • 7. T. Bridgeland and A. Maciocia, Fourier-Mukai transforms for K3 and elliptic fibrations, preprint, math.AG 9908022.
  • 8. A. Craw and M. Reid, How to calculate $\operatorname{\hbox{$A$ }-Hilb}\mathbb C^3$, preprint, math.AG 9909085, 29 pp.
  • 9. A. Grothendieck, Sur quelques points d'algèbre homologique, Tôhoku Math. J. 9 (1957) 119-221. MR 21:1328
  • 10. G. Gonzalez-Sprinberg and J.-L. Verdier, Construction géométrique de la correspondance de McKay, Ann. Sci. École Norm. Sup. 16 (1983) 409-449. MR 85k:14019
  • 11. R. Hartshorne, Residues and Duality, Lect. Notes Math. 20, Springer (1966). MR 36:5145
  • 12. Y. Ito and H. Nakajima, McKay correspondence and Hilbert schemes in dimension three, preprint, math.AG 9803120; Topology 39 (2000) 1155-1191. CMP 2001:01
  • 13. Y. Ito and M. Reid, The McKay correspondence for finite subgroups of $\operatorname{SL}(3,\mathbb C)$, in Higher dimensional complex varieties (Trento, 1994), M. Andreatta et al., Eds., de Gruyter, 1996, pp. 221-240. MR 98i:14018
  • 14. D. Kaledin, The McKay correspondence for symplectic quotient singularities, preprint, math.AG 9907087.
  • 15. M. Kapranov and E. Vasserot, Kleinian singularities, derived categories and Hall algebras, preprint, math.AG 9812016; Math. Ann. 316 (2000) 565-576. CMP 2000:11
  • 16. S. Mac Lane, Categories for the working mathematician, Graduate Texts in Math. 5, Springer, 1971. MR 50:7275
  • 17. S. Mukai, Duality between $\operatorname{D}(X)$ and $\operatorname{D}(\hat{X})$ with its application to Picard sheaves, Nagoya Math. J. 81 (1981) 153-175. MR 82f:14036
  • 18. I. Nakamura, Hilbert schemes of Abelian group orbits, to appear in J. Alg. Geom.
  • 19. A. Neeman, The Grothendieck duality theorem via Bousfield's techniques and Brown representability, J. Amer. Math. Soc. 9 (1996) 205-236. MR 96c:18006
  • 20. M. Reid, McKay correspondence, in Proc. of algebraic geometry symposium (Kinosaki, Nov 1996), T. Katsura (Ed.), 14-41, alg-geom 9702016.
  • 21. S.-S. Roan, Minimal resolutions of Gorenstein orbifolds in dimension 3, Topology 35 (1996) 489-508. MR 97c:14013
  • 22. P. Roberts, Intersection theorems, in Commutative algebra (Berkeley, 1987), MSRI Publ. 15, Springer, 1989, pp. 417-436. MR 90j:13024
  • 23. P. Roberts, Multiplicities and Chern classes in local algebra, CUP, 1998. MR 2001a:13029
  • 24. M. Verbitsky, Holomorphic symplectic geometry and orbifold singularities, preprint, math.AG 9903175; Asian J. Math. 4 (2000) 553-563. CMP 2001:05
  • 25. J.-L. Verdier, Des catégories dérivées des catégories abéliennes, Astérisque 239 (1996). MR 98c:18007

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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
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

DOI: https://doi.org/10.1090/S0894-0347-01-00368-X
Keywords: Quotient singularities, McKay correspondence, derived categories
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
Article copyright: © Copyright 2001 American Mathematical Society

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