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
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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.
- Edward Frenkel and David Ben-Zvi, Vertex algebras and algebraic curves, Mathematical Surveys and Monographs, vol. 88, American Mathematical Society, Providence, RI, 2001. MR 1849359
SB1 S. Boissière, On the McKay correspondences for the Hilbert scheme of points on the affine plane, arXiv:math.AG/0410281.
SB2 ---, Sur les correspondances de McKay pour le schéma de Hilbert de points sur le plan affine, Ph.D. thesis, Université de Nantes, 2004.
- Armand Borel and Jean-Pierre Serre, Le théorème de Riemann-Roch, Bull. Soc. Math. France 86 (1958), 97–136 (French). MR 116022
- Louis Comtet, Analyse combinatoire. Tomes I, II, Collection SUP: “Le Mathématicien”, 4, vol. 5, Presses Universitaires de France, Paris, 1970 (French). MR 0262087
- Gentiana Danila, Sur la cohomologie d’un fibré tautologique sur le schéma de Hilbert d’une surface, J. Algebraic Geom. 10 (2001), no. 2, 247–280 (French, with English summary). MR 1811556
EG G. Ellingsrud and L. Göttsche, Hilbert schemes of points and Heisenberg algebras, Moduli spaces in Algebraic Geometry, 1999.
- Geir Ellingsrud, Lothar Göttsche, and Manfred Lehn, On the cobordism class of the Hilbert scheme of a surface, J. Algebraic Geom. 10 (2001), no. 1, 81–100. MR 1795551
- Geir Ellingsrud and Stein Arild Strømme, On the homology of the Hilbert scheme of points in the plane, Invent. Math. 87 (1987), no. 2, 343–352. MR 870732, DOI https://doi.org/10.1007/BF01389419
- John Fogarty, Algebraic families on an algebraic surface, Amer. J. Math. 90 (1968), 511–521. MR 237496, DOI https://doi.org/10.2307/2373541
- William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249
- Lothar Göttsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann. 286 (1990), no. 1-3, 193–207. MR 1032930, DOI https://doi.org/10.1007/BF01453572
- I. Grojnowski, Instantons and affine algebras. I. The Hilbert scheme and vertex operators, Math. Res. Lett. 3 (1996), no. 2, 275–291. MR 1386846, DOI https://doi.org/10.4310/MRL.1996.v3.n2.a12
G A. Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique, IV : les schémas de Hilbert, Séminaire Bourbaki 221 (1960-1961).
- Mark Haiman, $t,q$-Catalan numbers and the Hilbert scheme, Discrete Math. 193 (1998), no. 1-3, 201–224. Selected papers in honor of Adriano Garsia (Taormina, 1994). MR 1661369, DOI https://doi.org/10.1016/S0012-365X%2898%2900141-1
- Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
Hi F. Hirzebruch, Topological methods in Algebraic Geometry, Springer, 1966.
- Victor Kac, Vertex algebras for beginners, University Lecture Series, vol. 10, American Mathematical Society, Providence, RI, 1997. MR 1417941
L2 M. Lehn, private communication.
- Manfred Lehn, Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Invent. Math. 136 (1999), no. 1, 157–207. MR 1681097, DOI https://doi.org/10.1007/s002220050307
- Manfred Lehn and Christoph Sorger, Symmetric groups and the cup product on the cohomology of Hilbert schemes, Duke Math. J. 110 (2001), no. 2, 345–357. MR 1865244, DOI https://doi.org/10.1215/S0012-7094-01-11026-0
- Wei-ping Li, Zhenbo Qin, and Weiqiang Wang, Vertex algebras and the cohomology ring structure of Hilbert schemes of points on surfaces, Math. Ann. 324 (2002), no. 1, 105–133. MR 1931760, DOI https://doi.org/10.1007/s002080200330
- Wei-Ping Li, Zhenbo Qin, and Weiqiang Wang, Stability of the cohomology rings of Hilbert schemes of points on surfaces, J. Reine Angew. Math. 554 (2003), 217–234. MR 1952174, DOI https://doi.org/10.1515/crll.2003.007
- Wei-ping Li, Zhenbo Qin, and Weiqiang Wang, Generators for the cohomology ring of Hilbert schemes of points on surfaces, Internat. Math. Res. Notices 20 (2001), 1057–1074. MR 1857595, DOI https://doi.org/10.1155/S1073792801000502
McDo1 I. G. Macdonald, Symmetric fonctions and hall polynomials, Oxford University Press (2nd edition 1995), 1979.
McDo2 ---, Symmetric functions and orthogonal polynomials, AMS, 1991.
- Wilhelm Magnus, Fritz Oberhettinger, and Raj Pal Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966. MR 0232968
- Laurent Manivel, Fonctions symétriques, polynômes de Schubert et lieux de dégénérescence, Cours Spécialisés [Specialized Courses], vol. 3, Société Mathématique de France, Paris, 1998 (French, with English and French summaries). MR 1638048
N2 H. Nakajima, Jack polynomials and Hilbert schemes of points on surfaces, 1996, arXiv:math.AG/9610021.
N1 ---, Lectures on Hilbert schemes of points on surfaces, AMS, 1996.
- Hiraku Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. of Math. (2) 145 (1997), no. 2, 379–388. MR 1441880, DOI https://doi.org/10.2307/2951818
- Eric Vasserot, Sur l’anneau de cohomologie du schéma de Hilbert de $\mathbf C^2$, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 1, 7–12 (French, with English and French summaries). MR 1805619, DOI https://doi.org/10.1016/S0764-4442%2800%2901766-3
BF 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.
SB1 S. Boissière, On the McKay correspondences for the Hilbert scheme of points on the affine plane, arXiv:math.AG/0410281.
SB2 ---, Sur les correspondances de McKay pour le schéma de Hilbert de points sur le plan affine, Ph.D. thesis, Université de Nantes, 2004.
BS A. Borel and J.-P. Serre, Le théorème de Riemann-Roch, Bull. Soc. Math. Fr. 86 (1958), 97–136.
Com 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.
D 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.
EG G. Ellingsrud and L. Göttsche, Hilbert schemes of points and Heisenberg algebras, Moduli spaces in Algebraic Geometry, 1999.
EGL 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.
ES G. Ellingsrud and S. A. Strømme, On the homology of the Hilbert scheme of points in the plane, Invent. Math. 87 (1987), 343–352.
Fo J. Fogarty, Algebraic families on an algebraic surface, Amer. J. Math. 10 (1968), 511–521.
FH 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.
Goe L. Göttsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann. 286 (1990), 193–207.
Gro I. Grojnowski, Instantons and affine algebras, I. The Hilbert scheme and vertex operators, Math. Res. Lett. 3 (1996), 275–291.
G A. Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique, IV : les schémas de Hilbert, Séminaire Bourbaki 221 (1960-1961).
H M. Haiman, $t,q$-Catalan numbers and the Hilbert scheme, Discrete Math. 193 (1998), 201–224.
Ha R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.
Hi F. Hirzebruch, Topological methods in Algebraic Geometry, Springer, 1966.
Kac V. Kac, Vertex algebras for beginners, University Lecture Series, 10. American Mathematical Society, Providence, RI, 1997.
L2 M. Lehn, private communication.
L ---, Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Invent. Math. 136 (1999), 157–207.
LS M. Lehn and C. Sorger, Symmetric groups and the cup product on the cohomology of Hilbert schemes, Duke Math. J. 110 (2001), 345–357.
LQW3 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.
LQW2 ---, Stability of the cohomology rings of Hilbert schemes of points on surfaces, J. reine angew. Math. 554 (2003), 217–234.
LQW4 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.
McDo1 I. G. Macdonald, Symmetric fonctions and hall polynomials, Oxford University Press (2nd edition 1995), 1979.
McDo2 ---, Symmetric functions and orthogonal polynomials, AMS, 1991.
MOS 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.
M 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.
N2 H. Nakajima, Jack polynomials and Hilbert schemes of points on surfaces, 1996, arXiv:math.AG/9610021.
N1 ---, Lectures on Hilbert schemes of points on surfaces, AMS, 1996.
N3 ---, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Annals of math. 145 (1997), 379–388.
V E. Vasserot, Sur l’anneau de cohomologie du schéma de Hilbert de $\mathbf {C}^2$, C.-R. Acad. Sc. Paris 332 (2001), 7–12.
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
Received by editor(s):
November 5, 2004
Received by editor(s) in revised form:
March 17, 2005
Published electronically:
May 5, 2005
