Finiteness of cominuscule quantum K-theory

By Anders Skovsted Buch, P. Chaput, Leonardo Constantin Mihalcea, N. Perrin. 2010

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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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3d1e4bb5e7ab3a6b4516056b3f9a3a09c590a595

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

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Citations in the corpus

Curve neighborhoods of Schubert varieties (2013, background): “curves of degree d in X that meet Ω. Recent developments suggest that this variety is a key object in the study of the quantum K-theory ring of X. The study of Γd(Ω) was initiated in the recent paper [5] by Chaput, Perrin, and the authors. Using that the Kontsevich moduli space of stable maps to X i”