# ANNALES DE L'INSTITUT FOURIER

On quantum cohomology of Grassmannians of isotropic lines, unfoldings of ${A}_{n}$-singularities, and Lefschetz exceptional collections
Annales de l'Institut Fourier, to appear, 37 p.

The subject of this paper is the big quantum cohomology rings of symplectic isotropic Grassmannians $\text{IG}\left(2,2n\right)$. 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 $\text{IG}\left(2,2n\right)$. 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 $\text{IG}\left(2,2n\right)$. Such a collection is constructed in the appendix by Alexander Kuznetsov.

Dans cet article, nous nous intéressons au gros anneau de cohomologie quantique de $\text{IG}\left(2,2n\right)$, la grassmanienne symplectique des droites isotropes. Nous montrons que cet anneau est régulier et en déduisons par « lissité générique » une preuve conceptuelle de la semi-simplicité générique du gros anneau de cohomologie quantique de $\text{IG}\left(2,2n\right)$. Par ailleurs, par un résultat général de Hertling, cette régularité donne une description de cet anneau en termes de singularités isolées d’hypersurfaces et nous montrons que les singularités qui apparaissent sont de type ${A}_{n-1}$. La conjecture de symétrie miroir homologique prédit l’existence de suites exceptionnelles très spéciales dans la catégorie dérivée des faisceaux cohérents de $\text{IG}\left(2,2n\right)$. L’existence de telles collections est démontrée en appendice par Alexander Kuznetsov.

Accepted : 2017-09-14
Classification:  14N35,  53D45
Keywords: semisimplicity of quantum cohomology, unfoldings of singularities, Lefschetz exceptional collections
@unpublished{AIF_0__0_0_A27_0,
author = {Cruz Morales, John Alexander and Mellit, Anton and Perrin, Nicolas and Smirnov, Maxim},
title = {On quantum cohomology of Grassmannians of isotropic lines, unfoldings of $A\_n$-singularities, and Lefschetz exceptional collections},
note = {to appear in \emph{Annales de l'Institut Fourier}},
}

Cruz Morales, John Alexander; Mellit, Anton; Perrin, Nicolas; Smirnov, Maxim. On quantum cohomology of Grassmannians of isotropic lines, unfoldings of $A_n$-singularities, and Lefschetz exceptional collections. Annales de l'Institut Fourier, to appear, 37 p.

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