On quantum cohomology of Grassmannians of isotropic lines, unfoldings of $A_n$-singularities, and Lefschetz exceptional collections


The subject of this paper is the big quantum cohomology rings of symplectic isotropic Grassmannians $IG(2,2n)$. We show that these rings are regular. In particular, by “generic smoothness”, we obtain a conceptual proof of generic semisimplicity of the big quantum cohomology for $IG(2,2n)$. Further, by a general result of Hertling, the regularity of these rings implies that they have a description in terms of isolated hypersurface singularities, which we show in this case to be of type $A_{n-1}$. By the homological mirror symmetry conjecture, these results suggest the existence of a very special full exceptional collection in the derived category of coherent sheaves on $IG(2,2n)$. Such a collection is constructed in the appendix by Alexander Kuznetsov.

Ann. Inst. Fourier (Grenoble) 69 (2019), no. 3, 955–991