# zbMATH — the first resource for mathematics

The McKay correspondence as an equivalence of derived categories. (English) Zbl 0966.14028
Summary: Let $$G$$ be a finite group of automorphisms of a non-singular three-dimensional complex variety $$M$$, whose canonical bundle $$\omega_M$$ is locally trivial as a $$G$$-sheaf. We prove that the Hilbert scheme $$Y=G \text{-Hilb }{M}$$ 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.

##### MSC:
 14J50 Automorphisms of surfaces and higher-dimensional varieties 18E30 Derived categories, triangulated categories (MSC2010) 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 19L47 Equivariant $$K$$-theory 14L30 Group actions on varieties or schemes (quotients) 14J30 $$3$$-folds
Full Text:
##### References:
 [1] M. F. Atiyah and F. Hirzebruch, The Riemann-Roch theorem for analytic embeddings, Topology 1 (1962), 151 – 166. · Zbl 0108.36402 [2] Michael Atiyah and Graeme Segal, On equivariant Euler characteristics, J. Geom. Phys. 6 (1989), no. 4, 671 – 677. · Zbl 0708.19004 [3] W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. · Zbl 0718.14023 [4] A. I. Bondal, Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 1, 25 – 44 (Russian); English transl., Math. USSR-Izv. 34 (1990), no. 1, 23 – 42. [5] A. I. Bondal and M. M. Kapranov, Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1183 – 1205, 1337 (Russian); English transl., Math. USSR-Izv. 35 (1990), no. 3, 519 – 541. · Zbl 0703.14011 [6] Tom Bridgeland, Equivalences of triangulated categories and Fourier-Mukai transforms, Bull. London Math. Soc. 31 (1999), no. 1, 25 – 34. · Zbl 0937.18012 [7] T. Bridgeland and A. Maciocia, Fourier-Mukai transforms for K3 and elliptic fibrations, preprint, math.AG 9908022. · Zbl 1066.14047 [8] A. Craw and M. Reid, How to calculate $$\operatorname{\hbox{}A}$$-Hilb [9] Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tôhoku Math. J. (2) 9 (1957), 119 – 221 (French). · Zbl 0118.26104 [10] G. Gonzalez-Sprinberg and J.-L. Verdier, Construction géométrique de la correspondance de McKay, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 3, 409 – 449 (1984) (French). · Zbl 0538.14033 [11] Robin Hartshorne, Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64. With an appendix by P. Deligne. Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. · Zbl 0212.26101 [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] Yukari Ito and Miles Reid, The McKay correspondence for finite subgroups of \?\?(3,\?), Higher-dimensional complex varieties (Trento, 1994) de Gruyter, Berlin, 1996, pp. 221-240. · Zbl 0894.14024 [14] D. Kaledin, The McKay correspondence for symplectic quotient singularities, preprint, math.AG 9907087. · Zbl 1060.14020 [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] Saunders MacLane, Categories for the working mathematician, Springer-Verlag, New York-Berlin, 1971. Graduate Texts in Mathematics, Vol. 5. · Zbl 0705.18001 [17] Shigeru Mukai, Duality between \?(\?) and \?(\?) with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153 – 175. · Zbl 0417.14036 [18] I. Nakamura, Hilbert schemes of Abelian group orbits, to appear in J. Alg. Geom. · Zbl 1104.14003 [19] Amnon Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), no. 1, 205 – 236. · Zbl 0864.14008 [20] M. Reid, McKay correspondence, in Proc. of algebraic geometry symposium (Kinosaki, Nov 1996), T. Katsura , 14-41, alg-geom 9702016. [21] Shi-Shyr Roan, Minimal resolutions of Gorenstein orbifolds in dimension three, Topology 35 (1996), no. 2, 489 – 508. · Zbl 0872.14034 [22] Paul Roberts, Intersection theorems, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 417 – 436. [23] Paul C. Roberts, Multiplicities and Chern classes in local algebra, Cambridge Tracts in Mathematics, vol. 133, Cambridge University Press, Cambridge, 1998. · Zbl 0917.13007 [24] M. Verbitsky, Holomorphic symplectic geometry and orbifold singularities, preprint, math.AG 9903175; Asian J. Math. 4 (2000) 553-563. CMP 2001:05 [25] Jean-Louis Verdier, Des catégories dérivées des catégories abéliennes, Astérisque 239 (1996), xii+253 pp. (1997) (French, with French summary). With a preface by Luc Illusie; Edited and with a note by Georges Maltsiniotis. · Zbl 0882.18010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.