Equivariant characteristic classes of external and symmetric products of varieties

Maxim, Laurenţiu¹ ; Schürmann, Jörg²

¹ Department of Mathematics, University of Wisconsin-Madison, Madison, WI, United States
² Mathematische Institut, Universität Münster, Münster, Germany

Geometry & topology, Tome 22 (2018) no. 1, pp. 471-515.

Résumé

We obtain refined generating series formulae for equivariant characteristic classes of external and symmetric products of singular complex quasiprojective varieties. More concretely, we study equivariant versions of Todd, Chern and Hirzebruch classes for singular spaces, with values in delocalized Borel–Moore homology of external and symmetric products. As a byproduct, we recover our previous characteristic class formulae for symmetric products and obtain new equivariant generalizations of these results, in particular also in the context of twisting by representations of the symmetric group.

Détail

DOI : 10.2140/gt.2018.22.471

Classification : 55S15, 57R20, 20C30
Keywords: characteristic classes, orbifold classes, Hirzebruch– and Lefschetz–Riemann–Roch, external and symmetric products of varieties, generating series, representations of symmetric groups

Bibliographie

[1] M F Atiyah, Power operations in K–theory, Quart. J. Math. Oxford Ser. 17 (1966) 165 | DOI

[2] M F Atiyah, I M Singer, The index of elliptic operators, III, Ann. of Math. 87 (1968) 546 | DOI

[3] P Baum, J L Brylinski, R Macpherson, Cohomologie équivariante délocalisée, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985) 605

[4] P Baum, A Connes, Chern character for discrete groups, from: "A fête of topology" (editors Y Matsumoto, T Mizutani, S Morita), Academic Press (1988) 163 | DOI

[5] P Baum, W Fulton, R Macpherson, Riemann–Roch for singular varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975) 101 | DOI

[6] P Baum, W Fulton, G Quart, Lefschetz–Riemann–Roch for singular varieties, Acta Math. 143 (1979) 193 | DOI

[7] D Bergh, The binomial theorem and motivic classes of universal quasi-split tori, preprint (2014)

[8] L Borisov, A Libgober, McKay correspondence for elliptic genera, Ann. of Math. 161 (2005) 1521 | DOI

[9] J P Brasselet, J Schürmann, S Yokura, Hirzebruch classes and motivic Chern classes for singular spaces, J. Topol. Anal. 2 (2010) 1 | DOI

[10] S Cappell, L Maxim, T Ohmoto, J Schürmann, S Yokura, Characteristic classes of Hilbert schemes of points via symmetric products, Geom. Topol. 17 (2013) 1165 | DOI

[11] S E Cappell, L G Maxim, J Schürmann, J L Shaneson, Equivariant characteristic classes of singular complex algebraic varieties, Comm. Pure Appl. Math. 65 (2012) 1722 | DOI

[12] S E Cappell, L Maxim, J Schürmann, J L Shaneson, S Yokura, Characteristic classes of symmetric products of complex quasi-projective varieties, J. Reine Angew. Math. 728 (2017) 35 | DOI

[13] P Deligne, Catégories tensorielles, Mosc. Math. J. 2 (2002) 227

[14] D Edidin, Riemann–Roch for Deligne–Mumford stacks, from: "A celebration of algebraic geometry" (editors B Hassett, J McKernan, J Starr, R Vakil), Clay Math. Proc. 18, Amer. Math. Soc. (2013) 241

[15] D Edidin, W Graham, Riemann–Roch for equivariant Chow groups, Duke Math. J. 102 (2000) 567 | DOI

[16] C Farsi, M J Pflaum, C Seaton, Stratifications of inertia spaces of compact Lie group actions, J. Singul. 13 (2015) 107 | DOI

[17] E Getzler, Mixed Hodge structures of configuration spaces, preprint (1995)

[18] E Gorsky, Adams operations and power structures, Mosc. Math. J. 9 (2009) 305

[19] F Heinloth, A note on functional equations for zeta functions with values in Chow motives, Ann. Inst. Fourier Grenoble 57 (2007) 1927 | DOI

[20] F Hirzebruch, D Zagier, The Atiyah–Singer theorem and elementary number theory, 3, Publish or Perish (1974)

[21] T Kawasaki, The index of elliptic operators over V –manifolds, Nagoya Math. J. 84 (1981) 135 | DOI

[22] A Krug, Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological bundles, preprint (2016)

[23] I G Macdonald, The Poincaré polynomial of a symmetric product, Proc. Cambridge Philos. Soc. 58 (1962) 563 | DOI

[24] I G Macdonald, Symmetric functions and Hall polynomials, Clarendon (1979)

[25] R D Macpherson, Chern classes for singular algebraic varieties, Ann. of Math. 100 (1974) 423 | DOI

[26] L Maxim, M Saito, J Schürmann, Symmetric products of mixed Hodge modules, J. Math. Pures Appl. 96 (2011) 462 | DOI

[27] L Maxim, J Schürmann, Twisted genera of symmetric products, Selecta Math. 18 (2012) 283 | DOI

[28] L Maxim, J Schürmann, Plethysm and cohomology representations of external and symmetric products, preprint (2016)

[29] S Meinhardt, M Reineke, Donaldson–Thomas invariants versus intersection cohomology of quiver moduli, J. Reine Angew. Math. (2017) | DOI

[30] B Moonen, Das Lefschetz–Riemann–Roch–Theorem für singuläre Varietäten, Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn (1978)

[31] M V Nori, The Hirzebruch–Riemann–Roch theorem, Michigan Math. J. 48 (2000) 473 | DOI

[32] T Ohmoto, Equivariant Chern classes of singular algebraic varieties with group actions, Math. Proc. Cambridge Philos. Soc. 140 (2006) 115 | DOI

[33] T Ohmoto, Generating functions of orbifold Chern classes, I : Symmetric products, Math. Proc. Cambridge Philos. Soc. 144 (2008) 423 | DOI

[34] Z Qin, W Wang, Hilbert schemes and symmetric products: a dictionary, from: "Orbifolds in mathematics and physics" (editors A Adem, J Morava, Y Ruan), Contemp. Math. 310, Amer. Math. Soc. (2002) 233 | DOI

[35] L Scala, Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles, Duke Math. J. 150 (2009) 211 | DOI

[36] L Scala, Some remarks on tautological sheaves on Hilbert schemes of points on a surface, Geom. Dedicata 139 (2009) 313 | DOI

[37] J Schürmann, A general construction of partial Grothendieck transformations, preprint (2002)

[38] J Schürmann, Characteristic classes of mixed Hodge modules, from: "Topology of stratified spaces" (editors G Friedman, E Hunsicker, A Libgober, L Maxim), Math. Sci. Res. Inst. Publ. 58, Cambridge Univ. Press (2011) 419

[39] B Toen, Théorèmes de Riemann–Roch pour les champs de Deligne–Mumford, –Theory 18 (1999) 33 | DOI

[40] W Wang, Equivariant K–theory, wreath products, and Heisenberg algebra, Duke Math. J. 103 (2000) 1 | DOI

[41] W Wang, J Zhou, Orbifold Hodge numbers of the wreath product orbifolds, J. Geom. Phys. 38 (2001) 152 | DOI

[42] Z Wang, J Zhou, Tautological sheaves on Hilbert schemes of points, J. Algebraic Geom. 23 (2014) 669 | DOI

[43] J Zhou, Delocalized equivariant coholomogy of symmetric products, preprint (1999)