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 |
Keywords:
quotient singularities; McKay correspondence; derived categories; group of automorphisms; three-dimensional complex variety; Hilbert scheme; crepant resolution; Fourier-Mukai transform; equivariant K-theoryReferences:
| [1] | M. F. Atiyah and F. Hirzebruch, The Riemann-Roch theorem for analytic embeddings, Topology 1 (1962), 151 – 166. · Zbl 0108.36402 · doi:10.1016/0040-9383(65)90023-6 |
| [2] | Michael Atiyah and Graeme Segal, On equivariant Euler characteristics, J. Geom. Phys. 6 (1989), no. 4, 671 – 677. · Zbl 0708.19004 · doi:10.1016/0393-0440(89)90032-6 |
| [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 · doi:10.1112/S0024609398004998 |
| [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 · doi:10.1016/0040-9383(95)00018-6 |
| [22] | Paul Roberts, Intersection theorems, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 417 – 436. · doi:10.1007/978-1-4612-3660-3_23 |
| [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 |
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