Quantum cohomology of Grassmannians.

*(English)*Zbl 1050.14053The aim of this article is to supply simpler proofs of the main theorems about the (small) quantum cohomology ring of a Grassmann variety. This includes A. Bertram’s quantum version of the Pieri and Giambelli formulas [Adv. Math. 128, 289–305 (1997; Zbl 0945.14031)]. In contrast to Bertram’s proofs, which require the use of quot schemes, the presented proofs in this article stay with more elementary algebraic geometric methods and do not use any moduli space techniques. Essentially everything is based only on the definition of the Gromov-Witten invariants.

The author shows that the quantum Pieri formula is a consequence of the classical Pieri formula. Furthermore, he shows that the quantum Giambelli formula follows immediately from the quantum Pieri formula together with the classical Giambelli formula and associativity of the quantum cohomology. Also a proof is given of the Grassmannian case of a formula of Fulton and Woodward for the minimal \(q\)-power which appears in a quantum product of two Schubert classes. A proof of a simple version of the rim-hook algorithm is supplied. Finally, the presentation in terms of generators and relations of the quantum cohomology of Grassmannians is obtained.

The idea of the author is to start from the simple fact that if a rational curve of degree \(d\) passes through a Schubert variety in the Grassmannian \(\text{Gr}(l,\mathbb C^n)\), then the linear span of the points of this curve gives rise to a point in \(\text{Gr}(l+d,\mathbb C^n)\) which must lie in a related Schubert variety. This idea can be used to conclude in many cases that no curves pass trough three Schubert varieties in general position because the intersection of the related Schubert varieties is empty. In particular, the quantum Giambelli formula can be deduced by knowing that certain Gromov-Witten invariants are zero, which follows because the codimensions of the related Schubert varieties add up to more than the dimension of \(\text{Gr}(l+d,\mathbb C^n)\).

The author shows that the quantum Pieri formula is a consequence of the classical Pieri formula. Furthermore, he shows that the quantum Giambelli formula follows immediately from the quantum Pieri formula together with the classical Giambelli formula and associativity of the quantum cohomology. Also a proof is given of the Grassmannian case of a formula of Fulton and Woodward for the minimal \(q\)-power which appears in a quantum product of two Schubert classes. A proof of a simple version of the rim-hook algorithm is supplied. Finally, the presentation in terms of generators and relations of the quantum cohomology of Grassmannians is obtained.

The idea of the author is to start from the simple fact that if a rational curve of degree \(d\) passes through a Schubert variety in the Grassmannian \(\text{Gr}(l,\mathbb C^n)\), then the linear span of the points of this curve gives rise to a point in \(\text{Gr}(l+d,\mathbb C^n)\) which must lie in a related Schubert variety. This idea can be used to conclude in many cases that no curves pass trough three Schubert varieties in general position because the intersection of the related Schubert varieties is empty. In particular, the quantum Giambelli formula can be deduced by knowing that certain Gromov-Witten invariants are zero, which follows because the codimensions of the related Schubert varieties add up to more than the dimension of \(\text{Gr}(l+d,\mathbb C^n)\).

Reviewer: Martin Schlichenmaier (Luxembourg)

##### MSC:

14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |

14M15 | Grassmannians, Schubert varieties, flag manifolds |

05E05 | Symmetric functions and generalizations |