Quantum cohomology of Grassmannians

Anders Buch, Department of Mathematics, Rutgers University

Abstract: The three-point, genus zero Gromov-Witten invariants on a flag variety X=G/P give the number of rational curves of fixed degree that meet three Schubert varieties in general position. The (small) quantum cohomology ring of X encodes these Gromov-Witten invariants as its structure constants, and provides an efficient tool for computing them, at least if one can obtain sufficient information about the structure of this ring. Although there are general algorithms for computing Gromov-Witten invariants, structure theorems are still lacking in many cases. One of the challenges is that quantum cohomology lacks functoriality, so it must be computed case by case, which has traditionally been done (by Bertram and others) by applying degeneracy loci formulas on compactified moduli spaces of parametrized rational curves. I will explain a simpler method for proving structure theorems based on my definition of the kernel and span of a curve. In an ongoing project with Kresch and Tamvakis, this has made it possible to obtain structure theorems for the quantum cohomology of all isotropic Grassmannians.