Quantum K-theory of Incidence Varieties

Abstract

We prove a conjecture of Buch and Mihalcea in the case of the incidence variety X=Fl(1,n-1;n) and determine the structure of its (T-equivariant) quantum K-theory ring. Our results are an interplay between geometry and combinatorics. The geometric side concerns Gromov-Witten varieties of 3-pointed genus 0 stable maps to X with markings sent to Schubert varieties, while on the combinatorial side are formulas for the (equivariant) quantum K-theory ring of X. We prove that the Gromov-Witten variety is rationally connected when one of the defining Schubert varieties is a divisor and another is a point. This implies that the (equivariant) K-theoretic Gromov-Witten invariants defined by two Schubert classes and a Schubert divisor class can be computed in the ordinary (equivariant) K-theory ring of X. We derive a positive Chevalley formula for the equivariant quantum K-theory ring of X and a positive Littlewood-Richardson rule for the non-equivariant quantum K-theory ring of X. The Littlewood-Richardson rule in turn implies that non-empty Gromov-Witten varieties given by Schubert varieties in general position have arithmetic genus 0.

Figures (2)

• Figure 1: Hasse diagram of $(W^P, \leq)$ when $n=5$.
• Figure 2: Hasse diagrams of $(I(v), \leq)$ for various values of $v$.

Theorems & Definitions (78)

• Conjecture 1.1
• Theorem 1.2: $\doteq$ \ref{['thm:ratcon']}
• Theorem 1.3: $\doteq$ \ref{['thm:qkTchev']}
• Theorem 1.4: $=$ \ref{['thm:LR']}
• Proposition 2.1
• proof
• Corollary 2.2
• proof
• Lemma 2.3
• proof