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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-03-03422-6
Published electronically: August 26, 2003
MathSciNet review: 2020031
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Abstract: We study the ideals of the rational cohomology ring of the Hilbert scheme $X^{[n]}$ of $n$ points on a smooth projective surface $X$. As an application, for a large class of smooth quasi-projective surfaces $X$, we show that every cup product structure constant of $H^*(X^{[n]})$ is independent of $n$; moreover, we obtain two sets of ring generators for the cohomology ring $H^*(X^{[n]})$.

Similar results are established for the Chen-Ruan orbifold cohomology ring of the symmetric product. In particular, we prove a ring isomorphism between $H^*(X^{[n]}; \mathbb{C} )$ and $H^*_{\rm orb}(X^n/S_n; \mathbb{C} )$ for a large class of smooth quasi-projective surfaces with numerically trivial canonical class.


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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

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