Finiteness of cominuscule quantum K-theory

TL;DR

This paper proves that in the quantum K-theory of cominuscule homogeneous spaces, the product of two Schubert classes results in a finite sum, by analyzing boundary Gromov-Witten varieties and their singularities.

Contribution

It establishes the finiteness of quantum K-theory products for cominuscule spaces through geometric analysis of Gromov-Witten varieties and singularity properties.

Findings

Boundary Gromov-Witten varieties have rational singularities.

Such varieties are either empty or unirational.

Boundary Gromov-Witten varieties with three points are rationally connected.

Abstract

The product of two Schubert classes in the quantum K-theory ring of a homogeneous space X = G/P is a formal power series with coefficients in the Grothendieck ring of algebraic vector bundles on X. We show that if X is cominuscule, then this power series has only finitely many non-zero terms. The proof is based on a geometric study of boundary Gromov-Witten varieties in the Kontsevich moduli space, consisting of stable maps to X that take the marked points to general Schubert varieties and whose domains are reducible curves of genus zero. We show that all such varieties have rational singularities, and that boundary Gromov-Witten varieties defined by two Schubert varieties are either empty or unirational. We also prove a relative Kleiman-Bertini theorem for rational singularities, which is of independent interest. A key result is that when X is cominuscule, all boundary Gromov-Witten varieties defined by three single points in X are rationally connected.