Orbifold cohomology for global quotients. (English) Zbl 1086.14046

Fantechi, Barbara; Göttsche, Lothar

Duke Math. J. 117, No. 2, 197-227 (2003).

Ruan’s “Cohomological Hyper-Kähler Resolution Conjecture” aims to identify the orbifold cohomology of a complex orbifold \(X\) admitting a hyper-Kähler resolution with the ordinary cohomology of the resolution. The orbifold cohomology theory which appears in the conjecture was introduced by Chen and Ruan, and it is closely related to the degree \(0\) Gromov-Witten invariants of the orbifold \(X\).
The current paper deals with the case of global quotients \([Y/G]\), where \(Y\) is a smooth complex manifold with the action of a finite group \(G\). First, the authors define a (larger and non-commutative) ring \(H^{\star}(Y,G)\) with a \(G\) action, whose \(G\) invariant part equals the Chen-Ruan orbifold cohomology. As a vector space \(H^{\star}(Y,G)\) is isomorphic to the sum of cohomologies of the fixed loci \(\bigoplus_{g\in G} H^{\star}(Y^{g})\). The definition of the multiplitive structure is trickier and involves the Euler classes of certain “obstruction bundles”, carefully defined in the first section of the paper.
Secondly, the authors check a special case of Ruan’s conjecture. If \(S\) is a smooth surface with trivial canonical bundle, the Hilbert scheme \(S^{[n]}\) of \(n\) points on \(S\) provides a hyper-Kähler resolution for the symmetric product \(\text{Sym}^{n}S\). The authors compute the orbifold cohomology of the symmetric product \(\text{Sym}^{n}S\) and compare their answer to M. Lehn and C. Sorger’s calculation of the cohomology ring of the Hilbert scheme [Invent. Math. 152, No. 2, 305–329 (2003; Zbl 1035.14001)]. A closely related result can be found in B. Uribe’s paper [Commun. Anal. Geom. 13, No. 1, 113–128 (2005; Zbl 1087.32012)]. In addition, the authors discuss the orbifold cohomology of Beauville’s generalized Kummer varieties.

Reviewer: Dragos Oprea (Palo Alto)

MSC:

14N35	Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14C05	Parametrization (Chow and Hilbert schemes)
14A20	Generalizations (algebraic spaces, stacks)
14E15	Global theory and resolution of singularities (algebro-geometric aspects)

Keywords:

orbifold cohomology; Hilbert scheme of points

Citations:

Zbl 1035.14001; Zbl 1087.32012

References:

[1]	D. Abramovich and A. Vistoli, Compactifying the space of stable maps , J. Amer. Math. Soc. 15 (2002), 27–75. JSTOR: · Zbl 0991.14007
[2]	A Adem and Y Ruan, Twisted orbifold K-theory ,
[3]	V. V Batyrev and L. A Borisov, Mirror duality and string-theoretic Hodge numbers , Invent. Math. 126 (1996), 183–203. · Zbl 0872.14035
[4]	A Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle , J. Differential Geom. 18 (1983), 755–782. · Zbl 0537.53056
[5]	M. Britze, On the cohomology of generalized Kummer varieties , dissertation, Universität Köln, 2003.
[6]	W Chen and Y Ruan, A new cohomology theory for orbifold , · Zbl 1063.53091
[7]	L Dixon, J A Harvey, C Vafa, and E Witten, Strings on orbifolds , Nuclear Phys. B 261 (1985), 678–686.
[8]	–. –. –. –., Strings on orbifolds, II , Nuclear Phys. B 274 (1986), 285–314.
[9]	P Etingof and V Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism , Invent Math. 147 (2002), 243–348. \CMP1 881 922 · Zbl 1061.16032
[10]	B Fantechi, The orbifold Chow ring , in preparation.
[11]	W Fulton and J Harris, Representation Theory: A First Course , Grad. Texts in Math. 129 , Springer, New York, 1991. · Zbl 0744.22001
[12]	L Göttsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface , Math. Ann. 286 (1990), 193–207. · Zbl 0679.14007
[13]	L Göttsche and W Soergel, Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces , Math. Ann. 296 (1993), 235–245. · Zbl 0789.14002
[14]	I Grojnowski, Instantons and affine algebras, I: The Hilbert scheme and vertex operators , Math. Res. Lett. 3 (1996), 275–291. · Zbl 0879.17011
[15]	A Grothendieck, “Sur le mémoire de Weil: généralisation des fonctions abéliennes” in Séminaire Bourbaki, Vol. 4 , Soc. Math. France, Montrouge, 1995, exp. 141, 57–71. \CMP1 610 945
[16]	J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves , appendix by W. Fulton, Invent. Math. 67 (1982), 23–88. · Zbl 0506.14016
[17]	F Hirzebruch and T Höfer, On the Euler number of an orbifold , Math. Ann. 286 (1990), 255–260. · Zbl 0679.14006
[18]	R. M. Kaufmann, Orbifolding Frobenius algebras , · Zbl 1083.57037
[19]	M Lehn, Chern classes of tautological sheaves on Hilbert schemes of points on surfaces , Invent. Math. 136 (1999), 157–207. · Zbl 0919.14001
[20]	M Lehn and C Sorger, Symmetric groups and the cup product on the cohomology of Hilbert schemes , Duke Math. J. 110 (2001), 345–357. · Zbl 1093.14008
[21]	——–, The cup product of the Hilbert scheme for \(K3\) surfaces , to appear in Invent. Math., · Zbl 1035.14001
[22]	W.-P Li, Z Qin, and W Wang, Vertex algebras and the cohomology ring structure of Hilbert schemes of points on surfaces , Math. Ann. 324 (2002), 105–133. \CMP1 931 760 · Zbl 1024.14001
[23]	H Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces , Ann. of Math. (2) 145 (1997), 379–388. JSTOR: · Zbl 0915.14001
[24]	——–, Lectures on Hilbert Schemes of Points on Surfaces , Univ. Lecture Ser. 18 , Amer. Math. Soc., Providence, 1999. · Zbl 0949.14001
[25]	D Quillen, Elementary proofs of some results of cobordism using Steenrod operations , Adv. Math. 7 (1971), 29–56. · Zbl 0214.50502
[26]	M Reid, La correspondance de McKay , Astérisque 276 (2002), 53–72., Séminaire Bourbaki 1999/2000, exp. 867. \CMP1 886 756
[27]	Y Ruan, “Stringy geometry and topology of orbifolds” in Symposium in Honor of C H Clemens (Salt Lake City, Utah, 2000) , Contemp. Math. 312 , Amer. Math. Soc., Providence, 2002, 187–233. \CMP1 942 135 · Zbl 1060.14080
[28]	S Tufféry, Déformations de courbes avec action de groupe , Forum Math. 5 (1993), 243–259. · Zbl 0809.14005
[29]	B. Uribe, Orbifold cohomology of the symmetric product , · Zbl 1202.57027
[30]	E Vasserot, Sur l’anneau de cohomologie du schéma de Hilbert de \(\mathbb C^2\) , C. R. Acad. Sci. Paris Sér I Math. 332 (2001), 7–12. · Zbl 0991.14001
[31]	E Zaslow, Topological orbifold models and quantum cohomology rings , Comm. Math. Phys. 156 (1993), 301–331. · Zbl 0795.53074