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On quantum cohomology rings of partial flag varieties. (English) Zbl 0969.14039
In this article a unified description of the structure of the small cohomology rings for all projective homogeneous spaces $$SL_n(\mathbb C)/P$$ (with $$P$$ a parabolic subgroup) is given. First the results on the classical cohomology rings are recalled. Then the algebraic structure of the quantum cohomology ring is studied. Important results are the general quantum versions of the Giambelli and Pieri formulas of the classical cohomology (classical Schubert calculus). They are obtained via geometric computations of certain Gromov-Witten invariants, which are realized as intersection numbers on hyperquot schemes.

##### MSC:
 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) 14M15 Grassmannians, Schubert varieties, flag manifolds 14N15 Classical problems, Schubert calculus 14M17 Homogeneous spaces and generalizations 53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
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