Skip to main content
Log in

Mixed Hodge polynomials of character varieties

With an appendix by Nicholas M. Katz

  • Published:
Inventiones mathematicae Aims and scope

Abstract

We calculate the E-polynomials of certain twisted GL(n,ℂ)-character varieties \(\mathcal{M}_{n}\) of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type \(\text{GL}(n,\mathbb{F}_{q})\) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n,ℂ)-character variety. The calculation also leads to several conjectures about the cohomology of \(\mathcal{M}_{n}\): an explicit conjecture for its mixed Hodge polynomial; a conjectured curious hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver. We prove these conjectures for n=2.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alvis, D.: The duality operation in the character ring of a finite Chevalley group. Bull. Am. Math. Soc., New Ser. 1(6), 907–911 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  2. Atiyah, M.F., Bott, R.: The Yang–Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond., Ser. A 308, 523–615 (1982)

    Article  MathSciNet  Google Scholar 

  3. Beilinson, A., Drinfeld V.: Quantization of Hitchins Integrable System and Hecke Eigensheaves. http://www.math.uchicago.edu/∼arinkin/langlands/ (ca. 1995) (preprint)

  4. Bott, R., Tolman, S., Weitsman, J.: Surjectivity for Hamiltonian loop group spaces. Invent. Math. 155(2), 225–251 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Crawley-Boevey, W., Van den Bergh, M.: Absolutely indecomposable representations and Kac–Moody Lie algebras. With an appendix by Hiraku Nakajima. Invent. Math. 155(3), 537–559 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Curtis, C.: Truncation and duality in the character ring of a finite group of Lie type. J. Algebra 62(2), 320–332 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  7. Danilov, V.I., Khovanskiĭ, A.G.: Newton polyhedra and an algorithm for calculating Hodge–Deligne numbers. Izv. Akad. Nauk SSSR, Ser. Mat. 50(5), 925–945 (1986)

    MathSciNet  Google Scholar 

  8. Deligne, P.: Théorie de Hodge II. Publ. Math., Inst. Hautes Étud. Sci. 40, 5–47 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  9. Deligne, P.: Théorie de Hodge III. Publ. Math., Inst. Hautes Étud. Sci. 44, 5–77 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  10. Deligne, P.: La conjecture de Weil. I. Publ. Math., Inst. Hautes Étud. Sci. 43, 273–307 (1974)

    Article  MathSciNet  Google Scholar 

  11. Deligne, P.: La conjecture de Weil II. Publ. Math., Inst. Hautes Étud. Sci. 52, 313–428 (1981)

    Google Scholar 

  12. Earl, R., Kirwan, F.: The Pontryagin rings of moduli spaces of arbitrary rank holomorphic bundles over a Riemann surface. J. Lond. Math. Soc., II. Ser. 60(3), 835–846 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Faber, C.: A conjectural description of the tautological ring of the moduli space of curves. In: Moduli of Curves and abelian Varieties. Aspects Math., vol. E33, pp. 109–129. Vieweg, Braunschweig (1999)

    Google Scholar 

  14. Faltings, G.: p-adic Hodge theory. J. Am. Math. Soc. 1(1), 255–299 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  15. Fontaine, J.-M., Messing, W.: p-adic periods and p-adic étale cohomology. In: Current Trends in Arithmetical Algebraic Geometry (Arcata, Calif., 1985). Contemp. Math., vol. 67, pp. 179–207. Am. Math. Soc., Providence, RI (1987)

    Google Scholar 

  16. Franz, M., Weber, A.: Weights in cohomology and the Eilenberg–Moore spectral sequence. Ann. Inst. Fourier (Grenoble) 55(2), 673–691 (2005)

    MATH  MathSciNet  Google Scholar 

  17. Freed, D., Quinn, F.: Chern–Simons theory with finite gauge group. Commun. Math. Phys. 156(3), 435–472 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  18. Frobenius, F.G.: Über Gruppencharaktere (1896). In: Gesammelte Abhandlungen III. Springer, Berlin–Heidelberg (1968)

    Google Scholar 

  19. Fulton, W.: Introduction to toric varieties. Ann. Math. Stud. The William H. Roever Lectures in Geometry, vol. 131. Princeton University Press, Princeton, NJ (1993)

    MATH  Google Scholar 

  20. Garsia, A.M., Haiman, M.: A remarkable q,t-Catalan sequence and q-Lagrange inversion. J. Algebr. Comb. 5(3), 191–244 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gessel, I., Viennot, G.: Binomial determinants, paths, and hook length formulae. Adv. Math. 58(3), 300–321 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  22. Green, J.A.: The characters of the finite general linear groups. Trans. Am. Math. Soc. 80, 402–447 (1955)

    Article  MATH  Google Scholar 

  23. Getzler, E.: Mixed Hodge Structures of Configuration Spaces. arXiv:alg-geom/9510018 (preprint)

  24. Gothen, P.B.: The Betti numbers of the moduli space of rank 3 Higgs bundles. Int. J. Math. 5, 861–875 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  25. Grothendieck, A.: Sur quelques points d’algèbre homologique. Tohoku Math. J. 9, 119–221 (1957)

    MATH  MathSciNet  Google Scholar 

  26. Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV. Publ. Math., Inst. Hautes Étud. Sci. 32, 5–361 (1967)

    Article  Google Scholar 

  27. Hanlon, P.: The fixed point partition lattices. Pac. J. Math. 96, 319–341 (1981)

    MATH  MathSciNet  Google Scholar 

  28. Harder, G., Narasimhan, M.S.: On the cohomology groups of moduli spaces of vector bundles over curves. Math. Ann. 212, 215–248 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  29. Hausel, T.: Vanishing of intersection numbers on the moduli space of Higgs bundles. Adv. Theor. Math. Phys. 2, 1011–1040 (1998) arXiv:math.AG/9805071

    MATH  MathSciNet  Google Scholar 

  30. Hausel, T. : Quaternionic geometry of matroids. Cent. Eur. J. Math. 3(1), 26–38 (2005) arXiv:math.AG/0308146

    Article  MATH  MathSciNet  Google Scholar 

  31. Hausel, T.: Mirror symmetry and Langlands duality in the non-abelian Hodge theory of a curve. In: Bogomolov, F., Tschinkel, Y. (eds.) Geometric Methods in Algebra and Number Theory. Progress in Mathematics, vol. 235. Birkhäuser, Boston (2005). arXiv:math.AG/0406380

    Chapter  Google Scholar 

  32. Hausel, T.: Betti numbers of holomorphic symplectic quotients, via arithmetic Fourier transform. Proc. Natl. Acad. Sci. USA 103(16), 6120–6124 (2006) arXiv: math.AG/0511163

    Article  MATH  MathSciNet  Google Scholar 

  33. Hausel, T.: S-duality in hyperkähler Hodge theory. arXiv:0709.0504v1 (to appear in: “The many facets of geometry: a tribute to Nigel Hitchin”, OUP)

  34. Hausel, T.: Arithmetic harmonic analysis, Macdonald polynomials and the topology of the Riemann–Hilbert monodromy map (in preparation)

  35. Hausel, T., Letellier, E., Rodriguez-Villegas, F.: Arithmetic harmonic analysis on character and quiver varieties (in preparation)

  36. Hausel, T., Sturmfels, B.: Toric hyperkähler varieties. Doc. Math. 7, 495–534 (2002) arXiv:math.AG/0203096

    MATH  MathSciNet  Google Scholar 

  37. Hausel, T., Thaddeus, M.: Relations in the cohomology ring of the moduli space of rank 2 Higgs bundles. J. Am. Math. Soc. 16, 303–329 (2003) arXiv:math.AG/0003094

    Article  MATH  MathSciNet  Google Scholar 

  38. Hausel, T., Thaddeus, M.: Mirror symmetry, Langlands duality and Hitchin systems. Invent. Math. 153(1), 197–229 (2003) arXiv:math.AG/0205236

    Article  MATH  MathSciNet  Google Scholar 

  39. Hausel, T., Thaddeus, M.: Generators for the cohomology ring of the moduli space of rank 2 Higgs bundles. Proc. Lond. Math. Soc. 88, 632–658 (2004) arXiv:math.AG/0003093

    Article  MATH  MathSciNet  Google Scholar 

  40. Hitchin, N.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc., III. Ser. 55, 59–126 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  41. Hua, J.: Counting representations of quivers over finite fields. J. Algebra 226(2), 1011–1033 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  42. Ito, T.: Stringy Hodge numbers and p-adic Hodge theory. Compos. Math. 140(6), 1499–1517 (2004)

    MATH  MathSciNet  Google Scholar 

  43. Jeffrey, L.: Group cohomology construction of the cohomology of moduli spaces of flat connections on 2-manifolds. Duke Math. J. 77, 407–429 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  44. Kac, V.: Infinite root systems, representations of graphs and invariant theory. Invent. Math. 56(1), 57–92 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  45. Kac, V.: Root systems, representations of quivers and invariant theory. In: Invariant Theory (Montecatini, 1982). Lect. Notes Math., vol. 996, pp. 74–108. Springer, Berlin (1983)

    Google Scholar 

  46. Kapustin, A., Witten, E.: Electric-magnetic duality and the geometric Langlands program. Commun. Number Theory Phys. 1(1), 1–236 (2007) arXiv:hep-th/0604151

    MathSciNet  Google Scholar 

  47. Katz, N.M.: Rigid Local Systems. Ann. Math. Studies, vol. 139. Princeton University Press, Princeton, NJ (1996)

    MATH  Google Scholar 

  48. Laumon, G.: Comparaison de caractéristiques d’Euler–Poincaré en cohomologie l-adique. C. R. Acad. Sci., Paris, Sér I, Math. 292(3), 209–212 (1981)

    MATH  MathSciNet  Google Scholar 

  49. Liebeck, M.W., Shalev, A.: Fuchsian groups, finite simple groups and representation varieties. Invent. Math. 159(2), 317–367 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  50. Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs, 2nd edn., Oxford Science Publications. The Clarendon Press Oxford University Press, New York (1995)

    Google Scholar 

  51. Markman, E.: Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces. J. Reine Angew. Math. 544, 61–82 (2002)

    MATH  MathSciNet  Google Scholar 

  52. McCleary, J.: A User’s Guide to Spectral Sequences, 2nd edn. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  53. Mednykh, A.D.: Determination of the number of nonequivalent coverings over a compact Riemann surface. Soviet Mathematics Doklady 19, 318–320 (1978)

    MATH  MathSciNet  Google Scholar 

  54. Meinrenken, A.: Witten’s formulas for intersection pairings on moduli spaces of flat G-bundles. Adv. Math. 197(1), 140–197 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  55. Mumford, D.: The Red Book of Varieties and Schemes. Springer, Berlin–Heidelberg (1999)

    MATH  Google Scholar 

  56. Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, 3rd edition. Springer, Berlin (1994)

    Google Scholar 

  57. Newstead, P.E.: Introduction to Moduli Problems and Orbit Spaces. Tata Inst. Bombay. Narosa Publishing House, New Delhi (1978)

    Google Scholar 

  58. Peters, C., Steenbrink, J.: Mixed Hodge Structures. Ergeb. Mathematik, vol. 52. Springer, Berlin (2008)

    MATH  Google Scholar 

  59. Racaniére, S.: Kirwan map and moduli space of flat connections. Math. Res. Lett. 11, 419–433 (2004)

    MATH  MathSciNet  Google Scholar 

  60. Reineke, M.: The Harder–Narasimhan system in quantum groups and cohomology of quiver moduli. Invent. Math. 152, 349–368 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  61. Rodriguez-Villegas, F.: Counting colorings on varieties. In: Proceedings of Primeras Jornadas de Teoría de Números (de Espanya), Villanova i la Geltrú, Spain, 2005. Publ. Mat., Barc., vol. extra 2007. Universitat Autònoma de Barcelona, Barcelona (2007)

    Google Scholar 

  62. Serre, J.-P.: Topics in Galois Theory. Jones and Bartlett Publishers, Boston (1992)

    MATH  Google Scholar 

  63. Seshadri, C.S.: Geometric reductivity over arbitrary base. Adv. Math. 26(3), 225–274 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  64. Simpson, C.T.: Nonabelian Hodge theory. In: Proceedings of the International Congress of Mathematicians, vols. I, II (Kyoto, 1990), pp. 747–756. Math. Soc. Jap., Tokyo (1991)

    Google Scholar 

  65. Van den Bogaart, T., Edixhoven, B.: Algebraic stacks whose number of points over finite fields is a polynomial. In: Number Fields and Function Fields – Two Parallel Worlds. Progr. Math., vol. 239, pp. 39–49. Birkhäuser, Boston, MA (2005)

    Chapter  Google Scholar 

  66. Wang, C.-L.: Cohomology theory in birational geometry. J. Differ. Geom. 60(2), 345–354 (2002)

    MATH  Google Scholar 

  67. Zagier, D.: Elementary aspects of the Verlinde formula and of the Harder–Narasimhan–Atiyah–Bott formula. In: Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993). Israel Math. Conf. Proc., vol. 9, pp. 445–462. Bar-Ilan Univ., Ramat Gan (1996)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Tamás Hausel.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hausel, T., Rodriguez-Villegas, F. Mixed Hodge polynomials of character varieties. Invent. math. 174, 555–624 (2008). https://doi.org/10.1007/s00222-008-0142-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-008-0142-x

Keywords

Navigation