×

On the quantum cohomology of adjoint varieties. (English) Zbl 1267.14065

Let \(G\) be a semi-simple algebraic group and \(R\) its root system. Denote by \(\Theta\) (resp. \(\theta\)) the highest root (resp. the highest short root) of \(R\). A dominant weight is said to be adjoint if it is equal to \(\Theta\) and coadjoint if it is equal to \(\theta\). Similarly, if \(P\) is the parabolic subgroup of \(G\) associated to the adjoint (resp. coadjoint) weight, then \(G/P\) is called an adjoint (resp. coadjoint) homogeneous space. Examples are the orthogonal and symplectic Grassmannians of lines, odd-dimensional quadrics and projective spaces, the two-step flag variety \(F(1,n;n+1)\) as well as six exceptional homogeneous spaces.
The paper under review studies various aspects of the (small) quantum cohomology of (co)adjoint varieties, in line with previous work on minuscule homogeneous spaces by the same authors and L. Manivel [Transform. Groups 13, No. 1, 47–89 (2008; Zbl 1147.14023)]. Let \(q\) denote the quantum parameter of a (co)adjoint variety \(X=G/P\). The main results are a simplified formula for the quantum multiplication by the hyperplane class of \(X\), a bound on the possible degrees in \(q\) of the quantum product of two Schubert classes of \(X\), and a presentation of the quantum cohomology ring for the six exceptional (co)adjoint varieties – such a presentation being already known for the classical (co)adjoint varieties from [A.S. Buch, A. Kresch and H. Tamvakis, Invent. Math. 178, No. 2, 345–405 (2009; Zbl 1193.14071)].
As a consequence of the presentation of the quantum cohomology ring, the authors are able to tell whether the (small) quantum cohomology ring of (co)adjoint varieties is semi-simple. Semi-simplicity of the quantum cohomology is an important problem. For instance, it appears in a conjecture of B. Dubrovin [Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, vol. II, 315–326 (1998 ; Zbl 0916.32018)] on Fano varieties, in relation with properties of their derived categories. Another consequence of semi-simplicity for homogeneous spaces \(G/P\) is a phenomenon known as strange duality, i.e. the existence of an involution of the localised quantum cohomology ring \(\text{QH}^*(G/P)_{q\neq 0}\). Strange duality for (co)minuscule homogeneous spaces has been studied in [P. E. Chaput, L. Manivel and N. Perrin, Canadian J. Math. 62, No. 6, 1246–1263 (2010; Zbl 1219.14060)]. Here the authors prove that the quantum cohomology ring of coadjoint varieties is almost never semi-simple, while it always is for adjoint non-coadjoint varieties. In the latter case they give a partial description of strange duality.

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
PDFBibTeX XMLCite
Full Text: DOI arXiv Link