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Generating functions of orbifold Chern classes I: symmetric products

  • TORU OHMOTO (a1)


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 GSn. 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)|.



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[1]Aluffi, P.. Modification systems and integration in their Chow groups. Selecta Math. 11 (2005), 155202.
[2]Batyrev, V.. Non-archimedean integrals and stringy Euler numbers of log-terminal pairs. J. Eur. Math. Soc. 1 (1999), 533.
[3]Batyrev, V.. Canonical abelianization of finite group action. Preprint, math.AG/0009043.
[4]Boissière, S.. Chern classes of the tangent bundle on the Hilbert schemes of points on the affine plane. J. Alg. Geom. 14} (2005), 761787.
[5]Borisov, L. and Libgober, A.. Elliptic genera of singular varieties. Duke Math. J. 116 (2003), 319351.
[6]Brasselet, J. P., Schürmann, J. and Yokura, S.. Hirzebruch classes and motivic Chern classes for singular spaces. Preprint, math.AG/0503492.
[7]Bryan, J. and Fulman, J.. Orbifold Euler characteristics and the number of commuting m-tuples in the symmetric groups. Ann. Comb. 2 (1998), 16.
[8]Edidin, D. and Graham, W.. Equivariant intersection theory. Invent. Math. 131 (1998), 595634.
[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.
[10]Hirzebruch, F. and Höfer, T.. On the Euler number of an orbifold. Math. Ann. 286 (1990), 255260.
[11]Kennedy, G.. MacPherson's Chern classes of singular algebraic varieties. Comm. Algebra 18 (1990), 28212839.
[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), 625628.
[13]Lehn, M.. Chern classes of tautological sheaves on Hilbert schemes of points on surfaces. Invent. Math. 136 (1999), 157207.
[14]Macdonald, I. G.. The Poincaré polynomial of a symmetric product. Proc. Camb. Phil. Soc. 58 (1962), 563568.
[15]MacPherson, R.. Chern classes for singular algebraic varieties. Annals of Math. 100 (1974), 421432.
[16]Müller, T.. Enumerating representations in finite wreath product. Adv. Math. 153 (2000), 118154.
[17]Ohmoto, T.. Equivariant Chern classes of singular algebraic varieties with group actions. Math. Proc. Camb. Phil. Soc. 140 (2006), 115134.
[18]Schwartz, M. H.. Classes caractéristiques définies par une stratification d'une variété analytique complexe. C. R. Acad. Sci. Paris 260 (1965), 3262-3264, 35353537.
[19]Schürmann, J. and Yokura, S.. A survey on characteristic classes of singular spaces. Preprint, math.AG/0511175.
[20]Stanley, R.. Enumerative Combinatorics, Vol. 2. Cambridge Studies in Adv. Math., 62 (Cambridge University Press, 1999).
[21]Tamanoi, H.. Generalized orbifold Euler characteristic of symmetric products and equivariant Morava K-Theory. Alg. Geom. Topology 1 (2001), 115141.
[22]Totaro, B.. The Chow ring of a classifying space. Proc. Symposia in Pure Math., AMS 67 (1999), 249281.
[23]Wohlfahrt, K.. Über einen satz von dey und die modulgruppe. Arch. Math. (Basel) 29 (1977), 455457.
[24]Yoshida, T.. Classical problems in group theory (I): enumerating subgroups and homomorphisms. Sugaku Expositions, AMS, (9) 2 (1996), 169188.
[25]Yoshida, T.. Categorical aspects of generating functions (I): exponential formulas and Krull–Schmidt category. J. Algebra 240 (2001), 4082.


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