Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-29T05:23:43.819Z Has data issue: false hasContentIssue false
  • English
  • Français

Generating Series in the Cohomology of Hilbert Schemes of Points on Surfaces

Published online by Cambridge University Press:  01 February 2010

Samuel Boissière  and
Marc A. Nieper-Wisskirchen
Samuel Boissière
Affiliation:
Laboratoire J. A. Dieudonné, UMR CNRS 6621, Université de Nice Sophia-AntipolisParc Valrose, 06108 Nice, France, [email protected], http://math.unice.fr/~sb
Marc A. Nieper-Wisskirchen
Affiliation:
Institut für Mathematik,, Johannes-Gutenberg Universität, 55099 Mainz, Germany, [email protected], http://www.mathematik.uni-mainz.de/Members/nieper

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the study of the rational cohomology of Hilbert schemes of points on a smooth surface, it is particularly interesting to understand the characteristic classes of the tautological bundles and the tangent bundle. In this note we pursue this study. We first collect all results appearing separately in the literature and prove some new formulas using Ohmoto's results on orbifold Chern classes on Hilbert schemes. We also explain the algorithmic counterpart of the topic: the cohomology space is governed by a vertex algebra that can be used to compute characteristic classes. We present an implementation of the vertex operators in the rewriting logic system MAUDE, and address observations and conjectures obtained after symbolic computations.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 10 , 2007 , pp. 254 - 270
Copyright
Copyright © London Mathematical Society 2007

References

reference

1.Boissière, Samuel, ‘Chern classes of the tangent bundle on the Hilbert scheme of points on the affine plane’, J. Algebraic Geom. 14 (2005) 761–787.CrossRefGoogle Scholar
2.Boissiére, Samuel, ‘On the McKay correspondences for the Hilbert scheme of points on the affine plane‘, Math. Ann. 334 (2006) 419–438.CrossRefGoogle Scholar
3.Boissiére, Samuel and Nieper-Wisskirchen, Marc A., ‘Universal formulas for characteristic classes on the Hilbert schemes of points on surfaces’, J. Algebra, to appear.Google Scholar
4.Clavel, Manuel, Durán, Francisco, Eker, Steven, Lincoln, Patrick, Martí-Oliet, Narciso, Meseguer, José and Quesda, José F., ‘Maude: specification and programming in rewriting logic’, Theoret. comput. Sci. 285 (2002) 187–243.CrossRefGoogle Scholar
5.Göttsche, Lothar, ‘The Betti numbers of the Hilbert scheme of points on a smooth projective surface’, Math. Ann. 286 (1990) 193–207.CrossRefGoogle Scholar
6.Lehn, Manfred, ‘Chern classes of tautological sheaves on Hilbert schemes of points on surfaces‘, Invent. Math. 136 (1999) 157–207.CrossRefGoogle Scholar
7.Lehn, Manfred, ‘Lectures on Hilbert schemes’, Algebraic structures and moduli spaces, CRM Proc. Lecture Notes 38 (Amer. Math. Soc, Providence, RI, 2004) 1–30.CrossRefGoogle Scholar
8.Li, Wei-Ping, Qin, Zhenbo and Wang, Weiqiang, ‘Hilbert schemes and W algebras’, Int. Math. Res. Not. (2002) 1427–1456.CrossRefGoogle Scholar
9.Nakajima, Hiraku, ‘Heisenberg algebra and Hilbert schemes of points on projective surfaces‘, Ann. of Math. (2) 145 (1997) 379–388.CrossRefGoogle Scholar
10.Nieper-Wisskirchen, Marc, ‘Equivariant cohomology, symmetric functions and the Hilbert scheme of points on the total space of the invertible sheaf oPl(-2) over the projective line’, arXiv:math.AG/0610834.Google Scholar
11.Ohmoto, Toru, ‘Generating functions of orbifold Chern classes I: symmetric products’, arXiv:math.AG/0604583.Google Scholar
Supplementary material: File

JCM 10 Boissiere and Nieper-Wisskirchen Appendix A

Boissiere and Nieper-Wisskirchen Appendix A

Download JCM 10 Boissiere and Nieper-Wisskirchen Appendix A(File)
File 13.2 KB