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Quantum cohomology of flag manifolds. (English) Zbl 1061.14061
This paper contributes to the study of the small quantum cohomology ring of partial flag manifolds over \({\mathbb C}\).
The classical Schubert calculus provides a description of the classical cohomology ring of a partial flag manifold in terms of generators and relations. The generators are determined by Chern classes of tautological vector bundles. A central role is played by the Giambelli formula, which expresses the Schubert classes as polynomials in the generators. These results have been extended to the quantum cohomology ring by I. Ciocan-Fontanine [Duke Math. J. 98, No. 3, 485–524 (1999; Zbl 0969.14039)] and S. Fomin, S. Gelfand and A. Postnikov [J. Am. Math. Soc. 10, No. 3, 565–596 (1997; Zbl 0912.14018)].
In the paper under review, simplified and more natural proofs for these results are provided. The key idea is to study the geometry of the relationship between quantum Schubert polynomials and W. Fulton’s universal Schubert polynomials, which solve a certain degeneracy locus problem [Duke Math. J. 96, No. 3, 575–594 (1999; Zbl 0981.14022)]. The proofs are based on calculations in the cohomology of so-called hyperquot schemes. These schemes compactify moduli spaces of maps from \({\mathbb P}^ 1\) to the flag manifold in a different way than Kontsevich’s stable map spaces do.

MSC:
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14N15 Classical problems, Schubert calculus
14M15 Grassmannians, Schubert varieties, flag manifolds
05E15 Combinatorial aspects of groups and algebras (MSC2010)
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References:
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