Selecta Mathematica

, Volume 24, Issue 5, pp 3889–3926 | Cite as

Intersection cohomology of moduli spaces of sheaves on surfaces

  • Jan ManschotEmail author
  • Sergey Mozgovoy


We study intersection cohomology of moduli spaces of semistable vector bundles on a complex algebraic surface. Our main result relates intersection Poincaré polynomials of the moduli spaces to Donaldson–Thomas invariants of the surface. In support of this result, we compute explicitly intersection Poincaré polynomials for sheaves with rank two and three on ruled surfaces.

Mathematics Subject Classification

14F05 14J60 14N35 55N33 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Beilinson, A., Ginzburg, V., Soergel, W.: Koszul duality patterns in representation theory. J. Am. Math. Soc. 9(2), 473–527 (1996)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Biglari, S.: On rings and categories of general representations (2010). arXiv:1002.2801
  3. 3.
    Bridgeland, T.: An introduction to motivic Hall algebras. Adv. Math. 229(1), 102–138 (2012). arXiv:1002.4372 MathSciNetCrossRefGoogle Scholar
  4. 4.
    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)MathSciNetGoogle Scholar
  5. 5.
    Deligne, P.: Théorie de Hodge. II. Inst. Hautes Études Sci. Publ. Math. 40, 5–57 (1971)CrossRefGoogle Scholar
  6. 6.
    Donaldson, S.K.: Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc. (3) 50(1), 1–26 (1985)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Getzler, E.: Mixed Hodge structures of configuration spaces, Preprint 96-61, Max Planck Institute for Mathematics, Bonn (1996). arXiv:alg-geom/9510018
  8. 8.
    Gothen, P.B., King, A.D.: Homological algebra of twisted quiver bundles. J. Lond. Math. Soc. (2) 71 (1), 85–99 (2005). arXiv:math/0202033 MathSciNetCrossRefGoogle Scholar
  9. 9.
    Göttsche, L.: The Betti numbers of the hilbert scheme of points on a smooth projective surface. Mathematische Annalen 286(1), 193–207 (1990)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Göttsche, L.: Theta functions and Hodge numbers of moduli spaces of sheaves on rational surfaces. Commun. Math. Phys. 206(1), 105–136 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Göttsche, L., Zagier, D.: Jacobi forms and the structure of Donaldson invariants for \(4\)-manifolds with \(b_+=1\). Selecta Math. (N.S.) 4(1), 69–115 (1998)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Haghighat, B.: From strings in 6d to strings in 5d. J. High Energy Phys. 1, 062 (2016). arXiv:1502.06645 MathSciNetCrossRefGoogle Scholar
  13. 13.
    Heinloth, F.: A note on functional equations for zeta functions with values in Chow motives. Ann. Inst. Fourier (Grenoble) 57(6), 1927–1945 (2007). arXiv:math/0512237 MathSciNetCrossRefGoogle Scholar
  14. 14.
    Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves, 2nd ed. Cambridge University Press, Cambridge (2010)Google Scholar
  15. 15.
    Huybrechts, D., Lehn, M.: Stable pairs on curves and surfaces. J. Algebraic Geom. 4(1), 67–104 (1995). arXiv:alg-geom/9211001 MathSciNetzbMATHGoogle Scholar
  16. 16.
    Joyce, D.: Configurations in abelian categories. IV. Invariants and changing stability conditions. Adv. Math. 217(1), 124–204 (2008). arXiv:math/0410268 MathSciNetCrossRefGoogle Scholar
  17. 17.
    Joyce, D.: Configurations in abelian categories. II. Ringel–Hall algebras. Adv. Math. 210(2), 635–706 (2007). arXiv:math.AG/0503029 MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kirwan, F.: On the homology of compactifications of moduli spaces of vector bundles over a Riemann surface. Proc. Lond. Math. Soc. (3) 53(2), 237–266 (1986)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kresch, A.: Cycle groups for Artin stacks. Invent. Math. 138(3), 495–536 (1999)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lübke, M., Teleman, A.: The Kobayashi–Hitchin Correspondence. World Scientific Publishing Co., Inc., River Edge (1995)CrossRefGoogle Scholar
  21. 21.
    Manschot, J.: The Betti numbers of the moduli space of stable sheaves of rank \(3\) on \(\mathbb{P}^2\). Lett. Math. Phys. 98(1), 65–78 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Manschot, J.: BPS invariants of semi-stable sheaves on rational surfaces. Lett. Math. Phys. 103(8), 895–918 (2013). arXiv:1109.4861 MathSciNetCrossRefGoogle Scholar
  23. 23.
    Manschot, J.: Sheaves on \(\mathbb{P}^2\) and generalized Appell functions, 2014. Adv. Theor. Math. Phys. 21(3), 655–681 (2017). arXiv:1407.7785 MathSciNetCrossRefGoogle Scholar
  24. 24.
    Maxim, L., Saito, M., Schürmann, J.: Symmetric products of mixed Hodge modules. J. Math. Pures Appl. (9) 96(5), 462–483 (2011). arXiv:1008.5345 MathSciNetCrossRefGoogle Scholar
  25. 25.
    Maxim, L., Schürmann, J.: Twisted genera of symmetric products. Selecta Math. (N.S.) 18(1), 283–317 (2012). arXiv:0906.1264 MathSciNetCrossRefGoogle Scholar
  26. 26.
    Meinhardt, S.: Donaldson–Thomas invariants vs. intersection cohomology for categories of homological dimension one (2015). arXiv:1512.03343
  27. 27.
    Meinhardt, S., Reineke, M.: Donaldson–Thomas invariants versus intersection cohomology of quiver moduli (2014). arXiv:1411.4062
  28. 28.
    Mozgovoy, S.: A computational criterion for the Kac conjecture. J. Algebra 318(2), 669–679 (2007). arXiv:math/0608321 MathSciNetCrossRefGoogle Scholar
  29. 29.
    Mozgovoy, S.: Invariants of moduli spaces of stable sheaves on ruled surfaces (2013). arXiv:1302.4134
  30. 30.
    Mozgovoy, S., Reineke, M.: Intersection cohomology of moduli spaces of vector bundles over curves (2015). arXiv:1512.04076
  31. 31.
    Peters, C.A.M., Joseph, H.M.: Steenbrink, Mixed Hodge structures, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 52. Springer, Berlin (2008)Google Scholar
  32. 32.
    Reineke, M.: Counting rational points of quiver moduli. Int. Math. Res. Not. 17, ID 70456 (2006). arXiv:math/0505389
  33. 33.
    Saito, M.: Introduction to mixed Hodge modules, Astérisque 179–180, 145–162 (1989). Actes du Colloque de Théorie de Hodge (Luminy, 1987)Google Scholar
  34. 34.
    Uhlenbeck, K., Yau, S.-T.: On the existence of Hermitian–Yang–Mills connections in stable vector bundles. Commun. Pure Appl. Math. 39(S, suppl.), 257–293 (1986). Frontiers of the Mathematical Sciences: 1985 (New York, 1985)Google Scholar
  35. 35.
    Vafa, C., Witten, E.: A strong coupling test of \(S\)-duality. Nucl. Phys. B 431, 3–77 (1994). arXiv:hep-th/9408074 MathSciNetCrossRefGoogle Scholar
  36. 36.
    Yoshioka, K.: The Betti numbers of the moduli space of stable sheaves of rank \(2\) on \(\mathbb{P}^2\). J. Reine Angew. Math. 453, 193–220 (1994)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Yoshioka, K.: The Betti numbers of the moduli space of stable sheaves of rank \(2\) on a ruled surface. Math. Ann. 302(3), 519–540 (1995)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Zwegers, S.: Mock theta functions, Ph.D. thesis, Utrecht University (2002)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of MathematicsTrinity College DublinDublin 2Ireland
  2. 2.Hamilton Mathematics InstituteDublin 2Ireland

Personalised recommendations