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arXiv · 1705.01819

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

John Alexander Cruz Morales·Alexander Kuznetsov·Anton Mellit·Nicolas Perrin·Maxim Smirnov
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Abstract

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

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John Alexander Cruz Morales, Alexander Kuznetsov, Anton Mellit, Nicolas Perrin, Maxim Smirnov. 2017-05-04. On quantum cohomology of Grassmannians of isotropic lines, unfoldings of $A_n$-singularities, and Lefschetz exceptional collections. https://arxiv.org/abs/1705.01819

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