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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Quantum cohomology
of projective bundles over $\mathbb{P}^{n}$


Authors: Zhenbo Qin and Yongbin Ruan
Journal: Trans. Amer. Math. Soc. 350 (1998), 3615-3638
MSC (1991): Primary 58D99, 14J60; Secondary 14F05, 14J45
MathSciNet review: 1422617
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Abstract: In this paper we study the quantum cohomology ring of certain projective bundles over the complex projective space $\mathbb{P}^{n}$. Using excessive intersection theory, we compute the leading coefficients in the relations among the generators of the quantum cohomology ring structure. In particular, Batyrev's conjectural formula for quantum cohomology of projective bundles associated to direct sum of line bundles over $\mathbb{P}^{n}$ is partially verified. Moreover, relations between the quantum cohomology ring structure and Mori's theory of extremal rays are observed. The results could shed some light on the quantum cohomology for general projective bundles.


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Additional Information

Zhenbo Qin
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
Email: zq@math.okstate.edu

Yongbin Ruan
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
Email: ruan@math.wisc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-98-01968-0
PII: S 0002-9947(98)01968-0
Received by editor(s): September 1, 1996
Additional Notes: Both authors were partially supported by NSF grants. The second author also had a Sloan fellowship.
Article copyright: © Copyright 1998 American Mathematical Society