Curve neighborhoods of Schubert varieties

TL;DR

This paper explicitly characterizes the union of rational curves passing through Schubert varieties using Hecke products, providing formulas for Gromov-Witten invariants and insights into quantum cohomology.

Contribution

It identifies the Schubert variety formed by rational curves via Hecke products and applies this to derive formulas for Gromov-Witten invariants and quantum cohomology.

Findings

Explicit formula for two-point Gromov-Witten invariants

New proof of the quantum Chevalley formula

Formula for minimal degree of rational curves in cominuscule varieties

Abstract

A previous result of the authors with Chaput and Perrin states that the union of all rational curves of fixed degree passing through a Schubert variety in a homogeneous space G/P is again a Schubert variety. In this paper we identify this Schubert variety explicitly in terms of the Hecke product of Weyl group elements. We apply our result to give an explicit formula for any two-point Gromov-Witten invariant as well as a new proof of the quantum Chevalley formula and its equivariant generalization. We also recover a formula for the minimal degree of a rational curve between two given points in a cominuscule variety.