Finiteness of cominuscule quantum K -theory

Anders S. Buch; Pierre-Emmanuel Chaput; Leonardo C. Mihalcea; Nicolas Perrin

Annales scientifiques de l'École Normale Supérieure (2013)

  • Volume: 46, Issue: 3, page 477-494
  • ISSN: 0012-9593

Abstract

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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.

How to cite

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Buch, Anders S., et al. "Finiteness of cominuscule quantum $K$-theory." Annales scientifiques de l'École Normale Supérieure 46.3 (2013): 477-494. <http://eudml.org/doc/272224>.

@article{Buch2013,
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.},
author = {Buch, Anders S., Chaput, Pierre-Emmanuel, Mihalcea, Leonardo C., Perrin, Nicolas},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {quantum $K$-theory; Gromov-Witten varieties; rational singularities; rational connectedness; quantum Schubert calculus; cominuscule grassmannians},
language = {eng},
number = {3},
pages = {477-494},
publisher = {Société mathématique de France},
title = {Finiteness of cominuscule quantum $K$-theory},
url = {http://eudml.org/doc/272224},
volume = {46},
year = {2013},
}

TY - JOUR
AU - Buch, Anders S.
AU - Chaput, Pierre-Emmanuel
AU - Mihalcea, Leonardo C.
AU - Perrin, Nicolas
TI - Finiteness of cominuscule quantum $K$-theory
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2013
PB - Société mathématique de France
VL - 46
IS - 3
SP - 477
EP - 494
AB - 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.
LA - eng
KW - quantum $K$-theory; Gromov-Witten varieties; rational singularities; rational connectedness; quantum Schubert calculus; cominuscule grassmannians
UR - http://eudml.org/doc/272224
ER -

References

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