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Generating functions of orbifold Chern classes I: symmetric products

Published online by Cambridge University Press:  01 March 2008

TORU OHMOTO
TORU OHMOTO*
Affiliation:
Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan.
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Abstract

In this paper, for a possibly singular complex variety X, generating functions of total orbifold Chern homology classes of the symmetric products SnX are given. These are very natural “class versions” of known generating function formulae of (generalized) orbifold Euler characteristics of SnX. Our Chern classes work covariantly for proper morphisms. We state the result more generally. Let G be a finite group and Gn the wreath product G ∼ Sn. For a G-variety X and a group A, we show a “Dey–Wohlfahrt type formula“ for equivariant Chern–Schwartz–MacPherson classes associated to Gn-representations of A (Theorem 1ċ1 and 1ċ2). When X is a point, our formula is just the classical one in group theory generating numbers |Hom(A, Gn)|.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 144 , Issue 2 , March 2008 , pp. 423 - 438
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]Aluffi, P.. Modification systems and integration in their Chow groups. Selecta Math. 11 (2005), 155–202.CrossRefGoogle Scholar
[2]Batyrev, V.. Non-archimedean integrals and stringy Euler numbers of log-terminal pairs. J. Eur. Math. Soc. 1 (1999), 5–33.Google Scholar
[3]Batyrev, V.. Canonical abelianization of finite group action. Preprint, math.AG/0009043.Google Scholar
[4]Boissière, S.. Chern classes of the tangent bundle on the Hilbert schemes of points on the affine plane. J. Alg. Geom. 14} (2005), 761–787.Google Scholar
[5]Borisov, L. and Libgober, A.. Elliptic genera of singular varieties. Duke Math. J. 116 (2003), 319–351.Google Scholar
[6]Brasselet, J. P., Schürmann, J. and Yokura, S.. Hirzebruch classes and motivic Chern classes for singular spaces. Preprint, math.AG/0503492.Google Scholar
[7]Bryan, J. and Fulman, J.. Orbifold Euler characteristics and the number of commuting m-tuples in the symmetric groups. Ann. Comb. 2 (1998), 1–6.CrossRefGoogle Scholar
[8]Edidin, D. and Graham, W.. Equivariant intersection theory. Invent. Math. 131 (1998), 595–634.CrossRefGoogle Scholar
[9]de Fernex, T., Lupercio, E., Nevins, T. and Uribe, B.. Stringy Chern classes of singular varieties. math.AG/0407314, to appear in Adv. Math.Google Scholar
[10]Hirzebruch, F. and Höfer, T.. On the Euler number of an orbifold. Math. Ann. 286 (1990), 255–260.CrossRefGoogle Scholar
[11]Kennedy, G.. MacPherson's Chern classes of singular algebraic varieties. Comm. Algebra 18 (1990), 2821–2839.CrossRefGoogle Scholar
[12]Kwieciński, M.. Formule du produit pour les classes caract'eristiques de Chern-Schwartz-MacPherson et homologie d'intersection. C. R. Acad. Sci. Paris 314 (1992), 625–628.Google Scholar
[13]Lehn, M.. Chern classes of tautological sheaves on Hilbert schemes of points on surfaces. Invent. Math. 136 (1999), 157–207.CrossRefGoogle Scholar
[14]Macdonald, I. G.. The Poincaré polynomial of a symmetric product. Proc. Camb. Phil. Soc. 58 (1962), 563–568.CrossRefGoogle Scholar
[15]MacPherson, R.. Chern classes for singular algebraic varieties. Annals of Math. 100 (1974), 421–432.Google Scholar
[16]Müller, T.. Enumerating representations in finite wreath product. Adv. Math. 153 (2000), 118–154.CrossRefGoogle Scholar
[17]Ohmoto, T.. Equivariant Chern classes of singular algebraic varieties with group actions. Math. Proc. Camb. Phil. Soc. 140 (2006), 115–134.CrossRefGoogle Scholar
[18]Schwartz, M. H.. Classes caractéristiques définies par une stratification d'une variété analytique complexe. C. R. Acad. Sci. Paris 260 (1965), 3262-3264, 3535–3537.Google Scholar
[19]Schürmann, J. and Yokura, S.. A survey on characteristic classes of singular spaces. Preprint, math.AG/0511175.Google Scholar
[20]Stanley, R.. Enumerative Combinatorics, Vol. 2. Cambridge Studies in Adv. Math., 62 (Cambridge University Press, 1999).Google Scholar
[21]Tamanoi, H.. Generalized orbifold Euler characteristic of symmetric products and equivariant Morava K-Theory. Alg. Geom. Topology 1 (2001), 115–141.CrossRefGoogle Scholar
[22]Totaro, B.. The Chow ring of a classifying space. Proc. Symposia in Pure Math., AMS 67 (1999), 249–281.Google Scholar
[23]Wohlfahrt, K.. Über einen satz von dey und die modulgruppe. Arch. Math. (Basel) 29 (1977), 455–457.Google Scholar
[24]Yoshida, T.. Classical problems in group theory (I): enumerating subgroups and homomorphisms. Sugaku Expositions, AMS, (9) 2 (1996), 169–188.Google Scholar
[25]Yoshida, T.. Categorical aspects of generating functions (I): exponential formulas and Krull–Schmidt category. J. Algebra 240 (2001), 40–82.CrossRefGoogle Scholar