Finiteness of cominuscule quantum K-theory
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 46 (2013) no. 3, p. 477-494

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.

Le produit de deux classes de Schubert dans l’anneau de K-théorie quantique d’un espace homogène X=G/P est une série formelle à coefficients dans l’anneau de Grothendieck des fibrés vectoriels algébriques au-dessus de X. Nous montrons que pour X cominuscule, cette série formelle n’a qu’un nombre fini de termes non nuls. La preuve repose sur une étude géométrique de certaines variétés de Gromov-Witten contenues dans le bord de l’espace de modules de Kontsevitch. Ces variétés paramètrent des applications stables à valeurs dans X, dont la courbe source est une union réductible de courbes rationnelles, et qui envoient les points marqués dans des sous-variétés de Schubert générales. Nous montrons que ces variétés de Gromov-Witten sont à singularités rationnelles et que celles définies par seulement deux sous-variétés de Schubert sont soit vides soit unirationnelles. Nous présentons également un énoncé relatif, de type Kleiman-Bertini pour les singularités rationnelles, d’intérêt indépendant. Un résultat-clé pour notre preuve est le fait que toutes les variétés de Gromov-Witten du bord de l’espace de modules de Kontsevitch, définies par trois variétés de Schubert ponctuelles dans X, sont rationnellement connexes.

DOI : https://doi.org/10.24033/asens.2194
Classification:  14N35,  19E08,  14N15,  14M15,  14M20,  14M22
Keywords: quantum K-theory, Gromov-Witten varieties, rational singularities, rational connectedness, quantum Schubert calculus, cominuscule grassmannians
@article{ASENS_2013_4_46_3_477_0,
     author = {Buch, Anders S. and Chaput, Pierre-Emmanuel and Mihalcea, Leonardo C. and Perrin, Nicolas},
     title = {Finiteness of cominuscule quantum $K$-theory},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 46},
     number = {3},
     year = {2013},
     pages = {477-494},
     doi = {10.24033/asens.2194},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2013_4_46_3_477_0}
}
Buch, Anders S.; Chaput, Pierre-Emmanuel; Mihalcea, Leonardo C.; Perrin, Nicolas. Finiteness of cominuscule quantum $K$-theory. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 46 (2013) no. 3, pp. 477-494. doi : 10.24033/asens.2194. http://www.numdam.org/item/ASENS_2013_4_46_3_477_0/

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