Quantum Cohomology of Flag Manifolds

In this paper, we study the (small) quantum cohomology ring of the partial flag manifold. We give proofs of the presentation of the ring and of the quantum Giambelli formula for Schubert varieties. These are known results, but our proofs are more natural and direct than the previous ones. One of our goals is to give evidence of a relationship between universal Schubert polynomials, which give the answer to a degeneracy locus problem, and quantum Schubert polynomials, which appear in quantum cohomology. It has been known that the universal Schubert polynomials specialize to both the ordinary and the quantum Schubert polynomials, but previous reasons for this have been purely algebraic [Fu]. The quantum cohomology ring of a projective manifold X is a deformation of the ordinary cohomology ring of X. The classical Schubert calculus, consisting of Giambelli and Pieri-type formulas which give the multiplicative structure of the cohomology ring of the flag manifold, has been used as a tool to solve enumerative problems. Similarly, the entries in the quantum multiplication table count rational curves on a flag manifold of a given multidegree which meet three general Schubert varieties. These numbers can be interpreted as intersection numbers on appropriate moduli spaces of holomorphic maps from the projective line P1 to the flag manifold. In order to understand these intersections, various compactifications of the moduli space of maps have been studied, for example the stable maps of Kontsevich. However, in the case of partial flag manifolds, including Grassmannians and complete flag manifolds, there are smooth compactifications called hyperquot schemes, which generalize Grothendieck’s Quot scheme [G]. They have been studied by Ciocan-Fontanine [C-F1] [C-F2], Laumon [Lau], Kim [K], and in [C]. Most of what is known about the quantum cohomology of flag manifolds rely heavily on computations in the cohomology of hyperquot schemes. We obtain our results through a further study of the intersection theory of hyperquot schemes. For the sake of notation, we first state and prove our results for the case of the complete flag manifold. Most of the statements hold verbatim for the general case of partial flag manifolds, and many of the proofs need only slight modifications. We give ingredients to extend the arguments in the final section of this paper. The exceptions to this are found in sections 9 and 10, whose constructions and results apply only to complete flag manifolds, and whose methods do not generalize. In section 11, we give an alternate approach completely bypasses this special argument. We include sections 9 and 10 because they may be of outside interest as we introduce and study a new set of degeneracy loci on the hyperquot scheme.