Advertisement

Geometriae Dedicata

, Volume 172, Issue 1, pp 245–291 | Cite as

Tensor products of tautological bundles under the Bridgeland–King–Reid–Haiman equivalence

  • Andreas Krug
Original Paper

Abstract

We give a description of the image of tensor products of tautological bundles on Hilbert schemes of points on surfaces under the Bridgeland–King–Reid–Haiman equivalence. Using this, some new formulas for cohomological invariants of these bundles are obtained. In particular, we give formulas for the Euler characteristic of arbitrary tensor products on the Hilbert scheme of two points and of triple tensor products in general.

Keywords

Algebraic geometry Hilbert schemes of points Tautological bundles 

Mathematics Subject Classification

14C05 14F05 14J60 

Notes

Acknowledgments

Most of the content of this article is also part of the author’s PhD thesis. The author wants to thank his adviser Marc Nieper-Wißkirchen for his support. The article was finished during the authors stay at the SFB Transregio 45 in Bonn.

References

  1. 1.
    Bridgeland, T., King, K., Reid, M.: The McKay correspondence as an equivalence of derived categories. J. Am. Math. Soc 14(3), 535–554 (2001). (electronic)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Boissière, S., Nieper-Wißkirchen, M.A.: Universal formulas for characteristic classes on the Hilbert schemes of points on surfaces. J. Algebra 315(2), 924–953 (2007)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Boissière, S.: Chern classes of the tangent bundle on the Hilbert scheme of points on the affine plane. J. Algebraic Geom. 14(4), 761–787 (2005)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Danila, Gentiana: Sections du fibré déterminant sur l’espace de modules des faisceaux semi-stables de rang 2 sur le plan projectif. Ann. Inst. Fourier (Grenoble) 50(5), 1323–1374 (2000)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Danila, G.: Sur la cohomologie d’un fibré tautologique sur le schéma de Hilbert d’une surface. J. Algebraic Geom. 10(2), 247–280 (2001)MATHMathSciNetGoogle Scholar
  6. 6.
    Danila, G.: Sections de la puissance tensorielle du fibré tautologique sur le schéma de Hilbert des points d’une surface. Bull. Lond. Math. Soc. 39(2), 311–316 (2007)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Drezet, J.-M., Narasimhan, M.S.: Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques. Invent. Math. 97(1), 53–94 (1989)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Fulton, W., Harris, J.: Representation Theory, Volume 129 of Graduate Texts in Mathematics. Springer, New York (1991) (A first course, Readings in Mathematics)Google Scholar
  9. 9.
    Fogarty, J.: Algebraic families on an algebraic surface. Am. J. Math 90, 511–521 (1968)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Haiman, M.: Hilbert schemes, polygraphs and the Macdonald positivity conjecture. J. Am. Math. Soc. 14(4), 941–1006 (2001). (electronic)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Hartshorne, R.: Algebraic geometry. In: Graduate Texts in Mathematics, No. 52. Springer, New York (1977)Google Scholar
  12. 12.
    Huybrechts, D., Thomas, R.: \({\mathbb{P}}\)-objects and autoequivalences of derived categories. Math. Res. Lett. 13(1), 87–98 (2006)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Krug, A.: Extension groups of tautological sheaves on Hilbert schemes of points on surfaces. arXiv:1111.4263 (2011)Google Scholar
  14. 14.
    Lehn, M.: Chern classes of tautological sheaves on Hilbert schemes of points on surfaces. Invent. Math. 136(1), 157–207 (1999)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Lipman, J., Hashimoto, M.: Foundations of Grothendieck Duality for Diagrams of Schemes, Volume 1960 of Lecture Notes in Mathematics. Springer, Berlin (2009)Google Scholar
  16. 16.
    Nieper-Wißkirchen, M.: Chern Numbers and Rozansky-Witten Invariants of Compact Hyper-Kähler Manifolds. World Scientific Publishing Co. Inc., River Edge, NJ (2004)CrossRefMATHGoogle Scholar
  17. 17.
    Scala, L.: Cohomology of the Hilbert scheme of points on a surface with values in representations of tautological bundles. Duke Math. J. 150(2), 211–267 (2009a)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Scala, L.: Some remarks on tautological sheaves on Hilbert schemes of points on a surface. Geom. Dedicata 139, 313–329 (2009b)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Universität BonnBonnGermany

Personalised recommendations