Transformation Groups

, Volume 23, Issue 3, pp 671–705 | Cite as


Open Access


We study properties of the Hirzebruch class of quotient singularities ℂ n /G, where G is a finite matrix group. The main result states that the Hirzebruch class coincides with the Molien series of G under suitable substitution of variables. The Hirzebruch class of a crepant resolution can be described specializing the orbifold elliptic genus constructed by Borisov and Libgober. It is equal to the combination of Molien series of centralizers of elements of G. This is an incarnation of the McKay correspondence. The results are illustrated with several examples, in particular of 4-dimensional symplectic quotient singularities.


  1. [AB84]
    M. F. Atiyah, R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), 1–28.MathSciNetCrossRefMATHGoogle Scholar
  2. [Alu13]
    P. Aluffi, Grothendieck classes and Chern classes of hyperplane arrangements, Int. Math. Res. Not. IMRN 2013 (2013), no. 8, 1873–1900.MathSciNetCrossRefMATHGoogle Scholar
  3. [AM09]
    P. Aluffi, L. C. Mihalcea, Chern classes of Schubert cells and varieties, J. Algebraic Geom. 18 (2009), no. 1, 63–100.MathSciNetCrossRefMATHGoogle Scholar
  4. [AW14]
    M. Andreatta, J. A. Wiśniewski, 4-dimensional symplectic contractions, Geom. Dedicata 168 (2014), 311–337.MathSciNetCrossRefMATHGoogle Scholar
  5. [Bat99]
    V. V. Batyrev. Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs, J. Eur. Math. Soc. (JEMS) 1 (1999), no. 1, 5–33.MathSciNetCrossRefMATHGoogle Scholar
  6. [Bau82]
    P. Baum, Fixed point formula for singular varieties, in: Current Trends in Algebraic Topology, Part 2 (London, Ont., 1981), CMS Conf. Proc., Vol. 2, Amer. Math. Soc., Providence, RI, 1982, pp. pages 3–22.Google Scholar
  7. [BC88]
    P. Baum, A. Connes, Chern character for discrete groups, in: A fête of Topology, Academic Press, Boston, MA, 1988, pp. 163–232.Google Scholar
  8. [BFQ79]
    P. Baum, W. Fulton, G. Quart, Lefschetz-Riemann-Roch for singular varieties, Acta Math. 143 (1979), no. 3–4, 193–211.MathSciNetCrossRefMATHGoogle Scholar
  9. [BKR01]
    T. Bridgeland, A. King, M. Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), no. 3, 535–554 (electronic).Google Scholar
  10. [BL00]
    L. A. Borisov, A. Libgober, Elliptic genera of toric varieties and applications to mirror symmetry, Invent. Math. 140 (2000), no. 2, 453–485.MathSciNetCrossRefMATHGoogle Scholar
  11. [BL03]
    L. Borisov, A. Libgober, Elliptic genera of singular varieties, Duke Math. J. 116 (2002), no. 2, 319–351.MathSciNetMATHGoogle Scholar
  12. [BL05]
    L. Borisov, A. Libgober, McKay correspondence for elliptic genera, Ann. of Math. (2) 161 (2005), no. 3, 1521–1569.Google Scholar
  13. [BS13]
    G. Bellamy, T. Schedler, On the (non)existence of symplectic resolutions for imprimitive symplectic reection groups, ArXiv e-prints, September 2013.Google Scholar
  14. [BSY10]
    J.-P. Brasselet, J. Schürmann, S. Yokura, Hirzebruch classes and motivic Chern classes for singular spaces, J. Topol. Anal. 2 (2010), no. 1, 1–55.MathSciNetCrossRefMATHGoogle Scholar
  15. [BV82]
    N. Berline, M. Vergne. Classes caractéristiques équivariantes. Formule de localisation en cohomologie équivariante, C. R. Acad. Sci. Paris Sér. I Math. 295 (1982), no. 9, 539–541.Google Scholar
  16. [BV97]
    M. Brion, M. Vergne. An equivariant Riemann-Roch theorem for complete, simplicial toric varieties, J. Reine Angew. Math. 482 (1997), 67–92.MathSciNetMATHGoogle Scholar
  17. [CDGZ04]
    A. Campillo, F. Delgado, S. M. Gusein-Zade. Poincaré series of a rational surface singularity, Invent. Math. 155 (2004), no. 1, 41–53.MathSciNetCrossRefMATHGoogle Scholar
  18. [Cha85]
    K. Chandrasekharan, Elliptic Functions, Grundlehren der Mathematischen Wissenschaften, Vol. 281, Springer-Verlag, Berlin, 1985.Google Scholar
  19. [CMSS12]
    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), no. 12, 1722–1769.MathSciNetCrossRefMATHGoogle Scholar
  20. [Coh80]
    A. M. Cohen, Finite quaternionic reflection groups, J. Algebra 64 (1980), no. 2, 293–324.MathSciNetCrossRefMATHGoogle Scholar
  21. [DBW14]
    M. Donten-Bury, J. A. Wiśniewski, On 81 symplectic resolutions of a 4-dimensional quotient by a group of order 32, arXiv:1409.4204 (2014).Google Scholar
  22. [Don69]
    P. Donovan, The Lefschetz-Riemann-Roch formula, Bull. Soc. Math. France 97 (1969), 257–273.MathSciNetCrossRefMATHGoogle Scholar
  23. [Ful98]
    W. Fulton, Intersection Theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Vol. 2 Springer-Verlag, Berlin, 1998.Google Scholar
  24. [Hir56]
    F. Hirzebruch, Neue Topologische Methoden in der Algebraischen Geometrie, Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Heft 9. Springer-Verlag, Berlin, 1956.Google Scholar
  25. [Huh16]
    J. Huh, Positivity of Chern classes of Schubert cells and varieties, J. Algebraic Geom. 25 (2016), no. 1, 177–199.MathSciNetCrossRefMATHGoogle Scholar
  26. [Huy05]
    D. Huybrechts, Complex Geometry, An introduction, Universitext. Springer-Verlag, Berlin, 2005.Google Scholar
  27. [IR96]
    Y. Ito, M. Reid, The McKay correspondence for finite subgroups of SL(3, C), in: Higher-dimensional Complex Varieties (Trento, 1994), de Gruyter, Berlin, 1996., pp. 221–240.Google Scholar
  28. [Kal02]
    D. Kaledin, McKay correspondence for symplectic quotient singularities, Invent. Math. 148 (2002), no 1, 151–175.MathSciNetCrossRefMATHGoogle Scholar
  29. [Lib15]
    A. Libgober, Elliptic genus of phases of N = 2 theories, Comm. Math. Phys. 340 (2015), no. 3, 939–958.MathSciNetCrossRefMATHGoogle Scholar
  30. [LS12]
    M. Lehn, C. Sorger, A symplectic resolution for the binary tetrahedral group, in: Geometric Methods in Representation Theory. II, Sémin. Congr., Vol. 24, Soc. Math. France, Paris, 2012, pp. 429–435.Google Scholar
  31. [McK80]
    J. McKay, Graphs, singularities, and finite groups, in: The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), Proc. Sympos. Pure Math., Vol. 37, Amer. Math. Soc., Providence, RI, 1980, pp. 183–186.Google Scholar
  32. [MW15]
    M. Mikosz, A. Weber, Equivariant Hirzebruch class for quadratic cones via degenerations, J. Singul. 12 (2015), 131–140.MathSciNetMATHGoogle Scholar
  33. [Ohm06]
    T. Ohmoto, Equivariant Chern classes of singular algebraic varieties with group actions, Math. Proc. Cambridge Philos. Soc. 140 (2006), no. 1, 115–134.MathSciNetCrossRefMATHGoogle Scholar
  34. [PT07]
    A. Pedroza, L. W. Tu, On the localization formula in equivariant cohomology, Topology Appl. 154 (2007), no. 7, 1493–1501.MathSciNetCrossRefMATHGoogle Scholar
  35. [Rei97]
    M. Reid, McKay correspondence, arXiv:alg-geom/9702016 (1997).Google Scholar
  36. [Sai00]
    M. Saito, Mixed Hodge complexes on algebraic varieties, Math. Ann. 316 (2000), no. 2, 283–331.MathSciNetCrossRefMATHGoogle Scholar
  37. [Ver00]
    M. Verbitsky, Holomorphic symplectic geometry and orbifold singularities, Asian J. Math. 4 (2000), no. 3, 553–563.MathSciNetCrossRefMATHGoogle Scholar
  38. [Vey03]
    W. Veys, Stringy zeta functions for Q-Gorenstein varieties, Duke Math. J. 120 (2003), no. 3, 469–514.MathSciNetCrossRefMATHGoogle Scholar
  39. [Wae08a]
    R. Waelder, Equivariant elliptic genera, Pacific J. Math. 235 (2008), no. 2, 345–377.MathSciNetCrossRefMATHGoogle Scholar
  40. [Wae08b]
    R. Waelder, Equivariant elliptic genera and local McKay correspondences, Asian J. Math. 12 (2008), no. 2, 251–284.MathSciNetCrossRefMATHGoogle Scholar
  41. [Web12]
    A. Weber, Equivariant Chern classes and localization theorem, J. Singul. 5 (2012), 153–176.MathSciNetMATHGoogle Scholar
  42. [Web16a]
    A. Weber, Equivariant Hirzebruch class for singular varieties, Selecta Math. (N.S.) 22 (2016), no. 3, 1413–1454.Google Scholar
  43. [Web16b]
    A. Weber, Hirzebruch class and Bia lynicki-Birula decomposition, Transform. Groups 22 (2017), no. 2, 537–557.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of WarsawWarszawaPoland
  2. 2.Institute of MathematicsFreie Universität BerlinBerlinGermany
  3. 3.Institute of Mathematics, Polish Academy of SciencesWarszawaPoland

Personalised recommendations