Ideals of the cohomology rings of Hilbert schemes and their applications

Authors:
Wei-Ping Li, Zhenbo Qin and Weiqiang Wang

Journal:
Trans. Amer. Math. Soc. **356** (2004), 245-265

MSC (2000):
Primary 14C05; Secondary 14F25, 17B69

Published electronically:
August 26, 2003

MathSciNet review:
2020031

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the ideals of the rational cohomology ring of the Hilbert scheme of points on a smooth projective surface . As an application, for a large class of smooth quasi-projective surfaces , we show that every cup product structure constant of is independent of ; moreover, we obtain two sets of ring generators for the cohomology ring .

Similar results are established for the Chen-Ruan orbifold cohomology ring of the symmetric product. In particular, we prove a ring isomorphism between and for a large class of smooth quasi-projective surfaces with numerically trivial canonical class.

**[CR]**W. Chen, Y. Ruan,*A new cohomology theory for orbifold*, arXiv:math.AG/0004129.**[ES]**Geir Ellingsrud and Stein Arild Strømme,*Towards the Chow ring of the Hilbert scheme of 𝑃²*, J. Reine Angew. Math.**441**(1993), 33–44. MR**1228610****[FG]**B. Fantechi, L. Göttsche,*Orbifold cohomology for global quotients*, Duke Math. J.**117**(2003), 197-227.**[FH]**H. K. Farahat and G. Higman,*The centres of symmetric group rings*, Proc. Roy. Soc. London Ser. A**250**(1959), 212–221. MR**0103935****[Got]**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**, 10.1007/BF01453572**[Gro]**I. Grojnowski,*Instantons and affine algebras. I. The Hilbert scheme and vertex operators*, Math. Res. Lett.**3**(1996), no. 2, 275–291. MR**1386846**, 10.4310/MRL.1996.v3.n2.a12**[Lehn]**Manfred Lehn,*Chern classes of tautological sheaves on Hilbert schemes of points on surfaces*, Invent. Math.**136**(1999), no. 1, 157–207. MR**1681097**, 10.1007/s002220050307**[LS1]**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**, 10.1215/S0012-7094-01-11026-0**[LS2]**M. Lehn, C. Sorger,*The cup product of the Hilbert scheme for**surfaces*, Invent. Math.**152**(2003) 305-329.**[LQW1]**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**, 10.1007/s002080200330**[LQW2]**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**, 10.1155/S1073792801000502**[LQW3]**W.-P. Li, Z. Qin, W. Wang,*Stability of the cohomology rings of Hilbert schemes of points on surfaces*, J. Reine Angew. Math.**554**(2003), 217-234.**[LQW4]**Wei-Ping Li, Zhenbo Qin, and Weiqiang Wang,*Hilbert schemes and 𝒲 algebras*, Int. Math. Res. Not.**27**(2002), 1427–1456. MR**1908477**, 10.1155/S1073792802110129**[Na1]**Hiraku Nakajima,*Heisenberg algebra and Hilbert schemes of points on projective surfaces*, Ann. of Math. (2)**145**(1997), no. 2, 379–388. MR**1441880**, 10.2307/2951818**[Na2]**Hiraku Nakajima,*Lectures on Hilbert schemes of points on surfaces*, University Lecture Series, vol. 18, American Mathematical Society, Providence, RI, 1999. MR**1711344****[Ru1]**Y. Ruan,*Stringy geometry and topology of orbifolds*, Symposium in Honor of C. H. Clemens (Salt Lake City, UT, 2000), 187-233, Contemp. Math.,**312**, Amer. Math. Soc., Providence, RI, 2002.**[Ru2]**Y. Ruan,*Cohomology ring of crepant resolutions of orbifolds*, Preprint, arXiv:ath.AG/0108195.**[QW]**Z. Qin, W. Wang,*Hilbert schemes and symmetric products: a dictionary*, Orbifolds in Mathematics and Physics (Madison, WI, 2001), 233-257, Contemp. Math.,**310**, Amer. Math. Soc., Providence, RI, 2002.**[Uri]**B. Uribe,*Orbifold cohomology of the symmetric product*, Preprint, math.AT/0109125.**[VW]**Cumrun Vafa and Edward Witten,*A strong coupling test of 𝑆-duality*, Nuclear Phys. B**431**(1994), no. 1-2, 3–77. MR**1305096**, 10.1016/0550-3213(94)90097-3**[Vas]**Eric Vasserot,*Sur l’anneau de cohomologie du schéma de Hilbert de 𝐂²*, C. R. Acad. Sci. Paris Sér. I Math.**332**(2001), no. 1, 7–12 (French, with English and French summaries). MR**1805619**, 10.1016/S0764-4442(00)01766-3**[Wa]**W. Wang,*The Farahat-Higman ring of wreath products and Hilbert schemes*, Preprint 2002, arXiv:math.QA/0205071.

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Additional Information

**Wei-Ping Li**

Affiliation:
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

Email:
mawpli@ust.hk

**Zhenbo Qin**

Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Email:
zq@math.missouri.edu

**Weiqiang Wang**

Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904

Email:
ww9c@virginia.edu

DOI:
https://doi.org/10.1090/S0002-9947-03-03422-6

Keywords:
Heisenberg algebra,
Hilbert scheme,
cohomology ring.

Received by editor(s):
July 5, 2002

Published electronically:
August 26, 2003

Additional Notes:
The first author was partially supported by the grant HKUST6170/99P

The second author was partially supported by an NSF grant

The third author was partially supported by an NSF grant

Article copyright:
© Copyright 2003
American Mathematical Society