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.
Similar content being viewed by others
References
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)
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)
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)
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)
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)
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)
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)
Fulton, W., Harris, J.: Representation Theory, Volume 129 of Graduate Texts in Mathematics. Springer, New York (1991) (A first course, Readings in Mathematics)
Fogarty, J.: Algebraic families on an algebraic surface. Am. J. Math 90, 511–521 (1968)
Haiman, M.: Hilbert schemes, polygraphs and the Macdonald positivity conjecture. J. Am. Math. Soc. 14(4), 941–1006 (2001). (electronic)
Hartshorne, R.: Algebraic geometry. In: Graduate Texts in Mathematics, No. 52. Springer, New York (1977)
Huybrechts, D., Thomas, R.: \({\mathbb{P}}\)-objects and autoequivalences of derived categories. Math. Res. Lett. 13(1), 87–98 (2006)
Krug, A.: Extension groups of tautological sheaves on Hilbert schemes of points on surfaces. arXiv:1111.4263 (2011)
Lehn, M.: Chern classes of tautological sheaves on Hilbert schemes of points on surfaces. Invent. Math. 136(1), 157–207 (1999)
Lipman, J., Hashimoto, M.: Foundations of Grothendieck Duality for Diagrams of Schemes, Volume 1960 of Lecture Notes in Mathematics. Springer, Berlin (2009)
Nieper-Wißkirchen, M.: Chern Numbers and Rozansky-Witten Invariants of Compact Hyper-Kähler Manifolds. World Scientific Publishing Co. Inc., River Edge, NJ (2004)
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)
Scala, L.: Some remarks on tautological sheaves on Hilbert schemes of points on a surface. Geom. Dedicata 139, 313–329 (2009b)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Krug, A. Tensor products of tautological bundles under the Bridgeland–King–Reid–Haiman equivalence. Geom Dedicata 172, 245–291 (2014). https://doi.org/10.1007/s10711-013-9919-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-013-9919-1