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Stringy K-theory and the Chern character. (English) Zbl 1132.14047

In the paper the authors construct new \(G\)-equivariant rings: \({\mathcal K}(X,G)\) and \({\mathcal H}(X,G)\), which they call the stringy \(K\)-theory and stringy cohomology, respectively. Here \(G\) is a finite group and \(X\) is a smooth projective \(G\)-variety. The stringy cohomology \({\mathcal H}(X,G)\) defined here is equivalent to B. Fantechi’s and L. Göttsche’s construction of stringy cohomology [Duke Math. J. 117 No. 2, 197–227 (2003; Zbl 1086.14046)]. However, the construction given here is much simpler than that of Fantechi and Göttsche and moreover it is functorial therefore an analogous construction yields the stringy Chow ring \({\mathcal A}(X,G).\) The other main result of the paper is the consruction of a new stringy Chern character \({\mathcal C}h : {\mathcal K}(X,G)\rightarrow {\mathcal H}(X,G)\). This stringy Chern character is proved to be an isomorphism of rings preserving all properties of a pre-\(G\)-Frobenius algebra except those which involve the metric.
The authors also define two new orbifold \(K\)-theories. The first one \({\text{K}}_{orb}(\mathcal X)\) is defined for a smooth Deligne-Mumford stack or general almost complex orbifold. The second is defined for \({\mathcal X} = [X/G]\) as the algebra of invariants \(K_{orb}(\mathcal X)={\mathcal K}(X,G)^{G}\) of the stringy \(K\)-theory of \(X\) and it is called the small orbifold \(K\)-theory of \(\mathcal X\).

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14C40 Riemann-Roch theorems
19E08 \(K\)-theory of schemes
19L10 Riemann-Roch theorems, Chern characters
19L47 Equivariant \(K\)-theory
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds

Citations:

Zbl 1086.14046
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References:

[1] Abramovich, D., Graber, T., Vistoli, A.: Algebraic orbifold quantum products. In: Adem, A., Morava, J., Ruan, Y. (eds.) Orbifolds in Mathematics and Physics. Contemp. Math., vol. 310, pp. 1–25. (2002). Am. Math. Soc., Providence, RI (2002). math.AG/0112004 · Zbl 1067.14055
[2] Adem, A., Ruan, Y.: Twisted orbifold K-theory. Commun. Math. Phys. 273(3), 533–56 (2003). math.AT/0107168 · Zbl 1051.57022
[3] Adem, A., Ruan, Y., Zhan, B.: A stringy product on twisted orbifold K-theory. Preprint. math.AT/0605534
[4] Atiyah, M.F., Hirzebruch, F.: The Riemann–Roch theorem for analytic embeddings. Topology 1, 151–166 (1962) · Zbl 0108.36402 · doi:10.1016/0040-9383(65)90023-6
[5] Atiyah, M.F., Segal, G.: On equivariant Euler characteristics. J. Geom. Phys. 6, 671–677 (1989) · Zbl 0708.19004 · doi:10.1016/0393-0440(89)90032-6
[6] Chen, W., Ruan, Y.: A new cohomology theory for orbifold. Commun. Math. Phys. 248(1), 1–31 (2004). math.AG/0004129
[7] Chen, W., Ruan, Y.: Orbifold Gromov–Witten theory. In: Adem, A., Morava, J., Ruan, Y., (eds.) Orbifolds in Mathematics and Physics. Contemp. Math., vol. 310, pp. 25–85. Am. Math. Soc., Providence, RI (2002). math.AG/0103156 · Zbl 1091.53058
[8] Chen, B., Hu, S.: A deRham model for Chen–Ruan cohomology ring of Abelian orbifolds. Math. Ann. 336(1), 51–71 (2006). math.SG/0408265 · Zbl 1122.14018 · doi:10.1007/s00208-006-0774-3
[9] Dixon, L., Harvey, J.A., Vafa, C., Witten, E.: Strings on Orbifolds. Nucl. Phys. B261, 678 (1985) · doi:10.1016/0550-3213(85)90593-0
[10] Dolgushev, V., Etingof, P.: Hochschild cohomology of quantized symplectic orbifolds and the Chen–Ruan cohomology. Int. Math. Res. Not. 2005(27), 1657–1688 (2005). math.QA/0410562 · Zbl 1088.53061 · doi:10.1155/IMRN.2005.1657
[11] Edidin, D., Graham, W.: NonAbelian localization in equivariant K-theory and Riemann–Roch for quotients. Adv. Math. 198(2), 547–582 (2005). math.AG/0411213 · Zbl 1093.19004
[12] Fantechi, B., Göttsche, L.: Orbifold cohomology for global quotients. Duke Math. J. 117(2), 197–227 (2003). math.AG/0104207 · Zbl 1086.14046
[13] Farkas, H., Kra, I.: Riemann Surfaces, 2nd edn. Springer, New York (1991) · Zbl 0475.30001
[14] Frenkel, E., Szczesny, M.: Chiral de Rham complex and orbifolds. Preprint. math.AG/0307181 · Zbl 1184.14033
[15] Fulton, W.: Intersection Theory. Springer, New York (1998) · Zbl 0885.14002
[16] Fulton, W., Harris, J.: Representation Theory: a First Course. Springer, New York (1991) · Zbl 0744.22001
[17] Fulton, W., Lang, S.: Riemann–Roch Algebra. Springer, New York (1985)
[18] Goldin, R., Holm, T.S., Knutson, A.: Orbifold cohomology of torus quotients. Preprint. math.SG/0502429 · Zbl 1159.14029
[19] Ginzburg, V., Guillemin, V., Karshon, Y.: Moment maps, cobordisms, and Hamiltonian group actions. Am. Math. Soc., Providence, RI (2002) · Zbl 1197.53002
[20] Givental, A.: On the WDVV-equation in quantum K-theory. Mich. Math. J. 48, 295–304 (2000). math.AG/0003158 · Zbl 1081.14523
[21] Jarvis, T., Kaufmann, R., Kimura, T.: Pointed admissible G-covers and G-equivariant cohomological Field Theories. Compos. Math. 141(4), 926–978 (2005). math.AG/0302316 · Zbl 1091.14014
[22] Joshua, R.: Higher Intersection theory for algebraic stacks: I. K-Theory, 27(2), 134–195 (2002) · Zbl 1058.14004
[23] Joshua, R.: K-Theory and absolute cohomology for algebraic stacks. K-theory archive #0732. Preprint (2005)
[24] Kaledin, D.: Multiplicative McKay correspondence in the symplectic case. Preprint. math.AG/0311409
[25] Kani, E.: The Galois-module structure of the space of holomorphic differentials of a curve. J. Reine Angew. Math. 367, 187–206 (1986) · Zbl 0606.14027 · doi:10.1515/crll.1986.367.187
[26] Karoubi, M.: K-Theory, An Introduction. Springer, Berlin–New York (1978) · Zbl 0382.55002
[27] Kaufmann, R.: Orbifold Frobenius algebras, cobordisms, and monodromies. In: Adem, A., Morava, J., Ruan, Y. (eds.) Orbifolds in Mathematics and Physics. Contemp. Math., vol. 310, pp. 135–162 (2002) · Zbl 1084.57027
[28] Kaufmann, R.: Orbifolding Frobenius algebras. Int. J. Math. 14(6), 573–617 (2003). math.AG/0107163 · Zbl 1083.57037
[29] Kaufmann, R.: The algebra of discrete torsion. J. Algebra 282(1), 232–259 (2004). math.AG/0208081 · Zbl 1106.16040
[30] Kaufmann, R.: Discrete torsion, symmetric products and the Hilbert scheme. In: Hertling, C., Marcolli, M. (eds.) Frobenius Manifolds, Quantum Cohomology and Singularities. Aspects Math., vol. E36, Vieweg, Wiesbaden (2004)
[31] Kleiman, S.: Algebraic Cycles and the Weil Conjectures. In: Grothendieck, A., Kuiper, N. (eds.) Dix exposés sur la cohomologie des schémas. North-Holland, Amsterdam (1968)
[32] Lee, Y.P.: Quantum K-theory I: Foundations. Duke Math. J. 121(3), 389–424 (2004). math.AG/0105014 · Zbl 1051.14064
[33] Quillen, D.: Elementary proofs of some results of cobordism theory using Steenrod operations. Adv. Math. 7, 29–56 (1971) · Zbl 0214.50502 · doi:10.1016/0001-8708(71)90041-7
[34] Ruan, Y.: Stringy orbifolds. In: Adem, A., Morava, J., Ruan, Y. (eds.) Orbifolds in Mathematics and Physics. Contemp. Math., vol. 310, pp. 259–299 (2002) · Zbl 1080.14500
[35] Berthelot, P., Grothendieck, A., Illusie, L.: Théorie des intersections et théorème de Riemann–Roch. Lect. Notes Math., vol. 225. Springer, Berlin (1971) · Zbl 0218.14001
[36] Shanahan, P.: The Atiyah–Singer Index Theorem. Springer, New York (1978) · Zbl 0369.58021
[37] Toen, B.: Théorèmes de Riemann–Roch pour les champs de Deligne–Mumford. K-Theory 18(1), 33–76 (1999) · Zbl 0946.14004 · doi:10.1023/A:1007791200714
[38] Totaro, B.: The resolution property for schemes and stacks. J. Reine Angew. Math. 577, 1–22 (2004). math.AG/0207210 · Zbl 1077.14004 · doi:10.1515/crll.2004.2004.577.1
[39] Turaev, V.: Homotopy field theory in dimension 2 and group-algebras. Preprint. math.QA/9910010
[40] Uribe, B.: Orbifold cohomology of the symmetric product. Commun. Anal. Geom. 13(1), 113–128 (2005). math.AT/0109125 · Zbl 1087.32012
[41] Vistoli, A.: Higher equivariant K-theory for finite group actions. Duke Math. J. 63(2), 399–419 (1991) · Zbl 0738.55002 · doi:10.1215/S0012-7094-91-06317-9
[42] Vezzosi, G., Vistoli, A.: Higher algebraic K-theory for actions of diagonalizable groups. Invent. Math. 153(1), 1–44 (2003). math.AG/0107174 · Zbl 1032.19001
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