Advertisement

Geometriae Dedicata

, Volume 136, Issue 1, pp 17–46 | Cite as

Hodge polynomials of the moduli spaces of rank 3 pairs

  • Vicente MuñozEmail author
Original Paper

Abstract

Let X be a smooth projective curve of genus g ≥ 2 over the complex numbers. A holomorphic triple \({(E_1, E_2, \phi)}\) on X consists of two holomorphic vector bundles E 1 and E 2 over X and a holomorphic map \({\phi \colon E_{2}\to E_{1}}\) . There is a concept of stability for triples which depends on a real parameter σ. In this paper, we determine the Hodge polynomials of the moduli spaces of σ-stable triples with rk(E 1) = 3, rk(E 2) = 1, using the theory of mixed Hodge structures. This gives in particular the Poincaré polynomials of these moduli spaces. As a byproduct, we recover the Hodge polynomial of the moduli space of odd degree rank 3 stable vector bundles.

Keywords

Moduli space Complex curve Stable triple Hodge polynomial 

Mathematics Subject Classification (2000)

14F45 14D20 14H60 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bradlow S.B., García-Prada O.: Stable triples, equivariant bundles and dimensional reduction. Math. Ann. 304, 225–252 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bradlow S.B., García-Prada O., Gothen P.B.: Moduli spaces of holomorphic triples over compact Riemann surfaces. Math. Ann. 328, 299–351 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Burillo J.: El polinomio de Poincaré-Hodge de un producto simétrico de variedades kählerianas compactas. Collect. Math. 41, 59–69 (1990)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Del Baño S.: On the motive of moduli spaces of rank two vector bundles over a curve. Compos. Math. 131, 1–30 (2002)zbMATHCrossRefGoogle Scholar
  5. 5.
    Deligne, P.: Théorie de Hodge I,II,III. In: Proc. I.C.M., vol. 1, 1970, pp. 425–430; in Publ. Math. I.H.E.S. 40, 5–58 (1971); ibid. 44, 5–77 (1974)Google Scholar
  6. 6.
    Durfee, A.H.: Algebraic varieties which are a disjoint union of subvarieties. Lect. Notes Pure Appl. Math. 105, 99–102. Marcel Dekker (1987)Google Scholar
  7. 7.
    Danivol V.I., Khovanskiǐ A.G.: Newton polyhedra and an algorithm for computing Hodge-Deligne numbers. Math. U.S.S.R. Izv. 29, 279–298 (1987)Google Scholar
  8. 8.
    Earl R., Kirwan F.: The Hodge numbers of the moduli spaces of vector bundles over a Riemann surface. Q. J. Math. 51, 465–483 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    García-Prada O.: Dimensional reduction of stable bundles, vortices and stable pairs. Int. J. Math. 5, 1–52 (1994)zbMATHCrossRefGoogle Scholar
  10. 10.
    García–Prada, O., Gothen, P.B., Muñoz, V.: Betti numbers of the moduli space of rank 3 parabolic Higgs bundles. Mem. Am. Math. Soc. 187, VIII+80pp (2007)Google Scholar
  11. 11.
    Muñoz V., Ortega D., Vázquez-Gallo M.-J.: Hodge polynomials of the moduli spaces of pairs. Int. J. Math. 18, 695–721 (2007)zbMATHCrossRefGoogle Scholar
  12. 12.
    Muñoz, V., Ortega, D., Vázquez-Gallo, M.-J.: Hodge polynomials of the moduli spaces of triples of rank (2, 2). Q. J. Math. (in press). doi: 10.1093/qmath/han007
  13. 13.
    Schmitt A.: A universal construction for the moduli spaces of decorated vector bundles. Transform. Groups 9, 167–209 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Thaddeus M.: Stable pairs, linear systems and the Verlinde formula. Invent. Math. 117, 317–353 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    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. 9, 445–462 (1996)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Instituto de Ciencias Matemáticas CSIC-UAM-UCM-UC3MConsejo Superior de Investigaciones CientíficasMadridSpain

Personalised recommendations