Ideals of the cohomology rings of Hilbert schemes and their applications
Authors:
WeiPing Li, Zhenbo Qin and Weiqiang Wang
Journal:
Trans. Amer. Math. Soc. 356 (2004), 245265
MSC (2000):
Primary 14C05; Secondary 14F25, 17B69
Published electronically:
August 26, 2003
MathSciNet review:
2020031
Fulltext 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 quasiprojective 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 ChenRuan orbifold cohomology ring of the symmetric product. In particular, we prove a ring isomorphism between and for a large class of smooth quasiprojective surfaces with numerically trivial canonical class.
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 W. Chen, Y. Ruan, A new cohomology theory for orbifold, arXiv:math.AG/0004129.
 [ES]
 G. Ellingsrud, S. Strømme, Towards the Chow ring of the Hilbert scheme of , J. Reine Angew. Math. 441 (1993), 3344. MR 94i:14004
 [FG]
 B. Fantechi, L. Göttsche, Orbifold cohomology for global quotients, Duke Math. J. 117 (2003), 197227.
 [FH]
 H. Farahat, G. Higman, The centres of symmetric group rings, Proc. Roy. Soc. (A) 250 (1959), 212221. MR 21:2697
 [Got]
 L. Göttsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann. 286 (1990), 193207. MR 91h:14007
 [Gro]
 I. Grojnowski, Instantons and affine algebras I: the Hilbert scheme and vertex operators, Math. Res. Lett. 3 (1996), 275291. MR 97f:14041
 [Lehn]
 M. Lehn, Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Invent. Math. 136 (1999), 157207. MR 2000h:14003
 [LS1]
 M. Lehn, C. Sorger, Symmetric groups and the cup product on the cohomology of Hilbert schemes, Duke Math. J. 110 (2001), 345357. MR 2002i:14004
 [LS2]
 M. Lehn, C. Sorger, The cup product of the Hilbert scheme for surfaces, Invent. Math. 152 (2003) 305329.
 [LQW1]
 W.P. Li, Z. Qin, W. Wang, Vertex algebras and the cohomology ring structure of Hilbert schemes of points on surfaces, Math. Ann. 324 (2002), no. 1, 105133. MR 2003h:14009
 [LQW2]
 W.P. Li, Z. Qin, W. Wang, Generators for the cohomology ring of Hilbert schemes of points on surfaces, Intern. Math. Res. Notices 20 (2001), 10571074. MR 2002j:14003
 [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), 217234.
 [LQW4]
 W.P. Li, Z. Qin, W. Wang, Hilbert schemes and algebras, Intern. Math. Res. Notices 27 (2002), 14271456. MR 2003d:17033
 [Na1]
 H. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. Math. 145 (1997), 379388. MR 98h:14006
 [Na2]
 H. Nakajima, Lectures on Hilbert schemes of points on surfaces, Univ. Lect. Ser. 18, Amer. Math. Soc. (1999). MR 2001b:14007
 [Ru1]
 Y. Ruan, Stringy geometry and topology of orbifolds, Symposium in Honor of C. H. Clemens (Salt Lake City, UT, 2000), 187233, 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), 233257, Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002.
 [Uri]
 B. Uribe, Orbifold cohomology of the symmetric product, Preprint, math.AT/0109125.
 [VW]
 C. Vafa, E. Witten, A strong coupling test of duality, Nucl. Phys. B 431 (1994), 377. MR 95k:81138
 [Vas]
 E. Vasserot, Sur l'anneau de cohomologie du schéma de Hilbert de , C. R. Acad. Sci. Paris, Sér. I Math. 332 (2001), 712. MR 2001k:14012
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 W. Wang, The FarahatHigman ring of wreath products and Hilbert schemes, Preprint 2002, arXiv:math.QA/0205071.
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Additional Information
WeiPing 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:
http://dx.doi.org/10.1090/S0002994703034226
PII:
S 00029947(03)034226
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
