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**Finiteness of cominuscule quantum \(K\)-theory.
(Finitude de la \(K\)-théorie quantique cominuscule.)**
*(English.
French summary)*
Zbl 1282.14016

This paper is concerned with the quantum \(K\)-theory of cominuscule homogeneous spaces. To help the reader, who is not specialist in the subject, to catch the flavor of the paper under review, some basic vocabulary is needed. Recall that if \(G\) is an algebraic group, an algebraic subgroup \(P\) of \(G\) is said to be parabolic if the quotient \(G/P\) is a complete algebraic variety. Any parabolic subgroup contains a Borel subgroup \(B\) which contains a maximal torus \(T\) of \(G\). The pair \((G,T)\) defines weights (which are characters of \(T\) satisfying certain non triviality conditions) and a root system. A fundamental weight \(\omega\) is said to be minuscule if and only if \(|<\omega, \alpha>|\leq 0\), for each positive root \(\alpha\), where \(<,>\) is the pairing induced by the natural duality between the characters and the co-characters of \(T\). A fundamental weigth \(\omega\) is said to be co-minuscule if and only if \(<\omega, \alpha^\vee>=1\), where \(\alpha\) denotes the highest root. To each such weight a parabolic subgroup \(P_\omega\) of \(G\) can be associated, and the corresponding quotient \(G/P_\omega\) is said to be a (co)minuscule homogeneous space.

Example of cominuscules homogeneous varieties are type A Grassmannians, Lagrangian grassmannians \(LG(n,2n)\) and, more exotically, the two exceptional homogeneous spaces: the Cayley Plane and the Freudenthal variety.

The paper under review deals with the finiteness of the cominuscule quantum \(K\)-theory. The product of two Schubert classes in the quantum \(K\)-theory of a homogeneous space \(X=G/P\) is a formal power series with coefficients in the Grothendieck ring of algebraic vector bundles on \(X\). The remarkable result proven by the authors is that if \(X\) is cominuscule then the power series expansion of the product has only finitely many non zero terms (Theorem 1 stated in the Introduction and proven in Section 5). The proof consists in a fine analysis of the geometry of the Schubert varieties of cominuscule homogeneous spaces. They all have at most rational singularities. Furthermore boundary Gromov-Witten varietes defined by two Schubert varieties are either empty or unirational. General details on the nature of the theorems and their proofs are provided in the comprehensive introduction to the paper. In order to prove their main results the authors are led to use an adaptation of a result by M. Brion [J. Algebra 258, No. 1, 137–159 (2002; Zbl 1052.14054)] about a Kleiman-Bertini’s like theorem regarding rational singularities (Theorem 2.5), which is interesting in its own. Section 3 is devoted to the geometry of the Gromov-Witten varieties. Section 4 is very important as it supplies the list of the Gromov-Witten varieties of cominuscule spaces. Indeed, this is what the authors need to reach the climax of the paper in the last section, where the main theorem about the finiteness of the expansion of the product in quantum \(K\)-theory, stated in the introduction, is finally proven.

Example of cominuscules homogeneous varieties are type A Grassmannians, Lagrangian grassmannians \(LG(n,2n)\) and, more exotically, the two exceptional homogeneous spaces: the Cayley Plane and the Freudenthal variety.

The paper under review deals with the finiteness of the cominuscule quantum \(K\)-theory. The product of two Schubert classes in the quantum \(K\)-theory of a homogeneous space \(X=G/P\) is a formal power series with coefficients in the Grothendieck ring of algebraic vector bundles on \(X\). The remarkable result proven by the authors is that if \(X\) is cominuscule then the power series expansion of the product has only finitely many non zero terms (Theorem 1 stated in the Introduction and proven in Section 5). The proof consists in a fine analysis of the geometry of the Schubert varieties of cominuscule homogeneous spaces. They all have at most rational singularities. Furthermore boundary Gromov-Witten varietes defined by two Schubert varieties are either empty or unirational. General details on the nature of the theorems and their proofs are provided in the comprehensive introduction to the paper. In order to prove their main results the authors are led to use an adaptation of a result by M. Brion [J. Algebra 258, No. 1, 137–159 (2002; Zbl 1052.14054)] about a Kleiman-Bertini’s like theorem regarding rational singularities (Theorem 2.5), which is interesting in its own. Section 3 is devoted to the geometry of the Gromov-Witten varieties. Section 4 is very important as it supplies the list of the Gromov-Witten varieties of cominuscule spaces. Indeed, this is what the authors need to reach the climax of the paper in the last section, where the main theorem about the finiteness of the expansion of the product in quantum \(K\)-theory, stated in the introduction, is finally proven.

Reviewer: Letterio Gatto (Torino)

### MSC:

14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |

14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |

14N15 | Classical problems, Schubert calculus |

14M15 | Grassmannians, Schubert varieties, flag manifolds |