Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Chern classes of the tangent bundle on the Hilbert scheme of points on the affine plane


Author: Samuel Boissière
Journal: J. Algebraic Geom. 14 (2005), 761-787
DOI: https://doi.org/10.1090/S1056-3911-05-00412-1
Published electronically: May 5, 2005
MathSciNet review: 2147349
Full-text PDF

Abstract | References | Additional Information

Abstract: The cohomology of the Hilbert schemes of points on smooth projective surfaces can be approached both with vertex algebra tools and equivariant tools. Using the first tool, we study the existence and the structure of universal formulas for the Chern classes of the tangent bundle over the Hilbert scheme of points on a projective surface. The second tool leads then to nice generating formulas in the particular case of the Hilbert scheme of points on the affine plane.


References [Enhancements On Off] (What's this?)

  • 1. D. Ben-Zvi and E. Frenkel, Vertex algebras and algebraic curves, Mathematical Surveys and Monographs, 88. American Mathematical Society, Providence, RI, 2001. xii+348 pp. MR 1849359 (2003f:17036)
  • 2. S. Boissière, On the McKay correspondences for the Hilbert scheme of points on the affine plane, arXiv:math.AG/0410281.
  • 3. -, Sur les correspondances de McKay pour le schéma de Hilbert de points sur le plan affine, Ph.D. thesis, Université de Nantes, 2004.
  • 4. A. Borel and J.-P. Serre, Le théorème de Riemann-Roch, Bull. Soc. Math. Fr. 86 (1958), 97-136. MR 0116022 (22:6817)
  • 5. Louis Comtet, Analyse combinatoire, Tomes I, II. (French) Collection SUP: ``Le Mathématicien", 4, 5 Presses Universitaires de France, Paris 1970 Vol. I: 192 pp.; Vol. II: 190 pp. MR 0262087 (41:6697)
  • 6. Gentiana Danila, Sur la cohomologie d'un fibré tautologique sur le schéma de Hilbert d'une surface, J. Alg. Geom. 10 (2001), 247-280, arXiv:math.AG/9904004. MR 1811556 (2001k:14003)
  • 7. G. Ellingsrud and L. Göttsche, Hilbert schemes of points and Heisenberg algebras, Moduli spaces in Algebraic Geometry, 1999.
  • 8. G. Ellingsrud, L. Göttsche, and M. Lehn, On the cobordism class of the Hilbert scheme of a surface, J. Alg. Geom. 10 (2001), 81-100, arXiv:math.AG/9904095. MR 1795551 (2001k:14005)
  • 9. G. Ellingsrud and S. A. Strømme, On the homology of the Hilbert scheme of points in the plane, Invent. Math. 87 (1987), 343-352. MR 0870732 (88c:14008)
  • 10. J. Fogarty, Algebraic families on an algebraic surface, Amer. J. Math. 10 (1968), 511-521. MR 0237496 (38:5778)
  • 11. W. Fulton and J. Harris, Representation theory, A first course. Graduate Texts in Mathematics, 129. Readings in Mathematics. Springer-Verlag, New York, 1991. xvi+551 pp. MR 1153249 (93a:20069)
  • 12. L. Göttsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann. 286 (1990), 193-207. MR 1032930 (91h:14007)
  • 13. I. Grojnowski, Instantons and affine algebras, I. The Hilbert scheme and vertex operators, Math. Res. Lett. 3 (1996), 275-291. MR 1386846 (97f:14041)
  • 14. A. Grothendieck, Techniques de construction et théorèmes d'existence en géométrie algébrique, IV : les schémas de Hilbert, Séminaire Bourbaki 221 (1960-1961).
  • 15. M. Haiman, $t,q$-Catalan numbers and the Hilbert scheme, Discrete Math. 193 (1998), 201-224. MR 1661369 (2000k:05264)
  • 16. R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977. MR 0463157 (57:3116)
  • 17. F. Hirzebruch, Topological methods in Algebraic Geometry, Springer, 1966.
  • 18. V. Kac, Vertex algebras for beginners, University Lecture Series, 10. American Mathematical Society, Providence, RI, 1997. MR 1417941 (99a:17027)
  • 19. M. Lehn, private communication.
  • 20. -, Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Invent. Math. 136 (1999), 157-207. MR 1681097 (2000h:14003)
  • 21. M. Lehn and C. Sorger, Symmetric groups and the cup product on the cohomology of Hilbert schemes, Duke Math. J. 110 (2001), 345-357. MR 1865244 (2002i:14004)
  • 22. W.-P. Li, Z. Qin, and W. Wang, Vertex algebras and the cohomology ring structure of Hilbert schemes of points on surfaces, Math. Ann. 324 (2002), 105-133. MR 1931760 (2003h:14009)
  • 23. -, Stability of the cohomology rings of Hilbert schemes of points on surfaces, J. reine angew. Math. 554 (2003), 217-234. MR 1952174 (2003m:14007)
  • 24. W.-Ping Li, Z. Qin, and W. Wang, Generators for the cohomology ring of Hilbert schemes of points on surfaces, Intern. Math. Res. Notices 20 (2001), 1057-1074. MR 1857595 (2002j:14003)
  • 25. I. G. Macdonald, Symmetric fonctions and hall polynomials, Oxford University Press (2nd edition 1995), 1979.
  • 26. -, Symmetric functions and orthogonal polynomials, AMS, 1991.
  • 27. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and theorems for the special functions in mathematical physics, third edition, Die Grundlehren der mathematischen Wissenschaften, Band 52 Springer-Verlag New York, Inc., New York 1966 viii+508 pp. MR 0232968 (38:1291)
  • 28. L. Manivel, Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence, (French) [Symmetric functions, Schubert polynomials and degeneracy loci] Cours Spécialisés [Specialized Courses], 3. Société Mathématique de France, Paris, 1998. vi+179 pp. MR 1638048 (99k:05159)
  • 29. H. Nakajima, Jack polynomials and Hilbert schemes of points on surfaces, 1996, arXiv:math.AG/9610021.
  • 30. -, Lectures on Hilbert schemes of points on surfaces, AMS, 1996.
  • 31. -, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Annals of math. 145 (1997), 379-388. MR 1441880 (98h:14006)
  • 32. E. Vasserot, Sur l'anneau de cohomologie du schéma de Hilbert de $\mathbf{C}^2$, C.-R. Acad. Sc. Paris 332 (2001), 7-12. MR 1805619 (2001k:14012)


Additional Information

Samuel Boissière
Affiliation: Fachbereich für Mathematik, Staudinger Weg 9, Johannes Gutenberg-Uni- versität Mainz, 55099 Mainz, Germany
Email: boissiere@mathematik.uni-mainz.de

DOI: https://doi.org/10.1090/S1056-3911-05-00412-1
Received by editor(s): November 5, 2004
Received by editor(s) in revised form: March 17, 2005
Published electronically: May 5, 2005

American Mathematical Society