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arXiv · 1803.05063

Geometry of horospherical varieties of Picard rank one

Richard Gonzales·Clélia Pech·Nicolas Perrin·Alexander Samokhin
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Abstract

We study the geometry of non-homogeneous horospherical varieties. These have been classified by Pasquier and include the well-known odd symplectic Grassmannians. We focus our study on quantum cohomology, with a view towards Dubrovin's conjecture. In particular, we describe the cohomology groups of these varieties as well as a Chevalley formula, and prove that many Gromov-Witten invariants are enumerative. This enables us to prove that in many cases the quantum cohomology is semisimple. We give a presentation of the quantum cohomology ring for odd symplectic Grassmannians. The final section is devoted to the derived categories of coherent sheaves on horospherical varieties. We first discuss a general construction of exceptional bundles on these varieties. We then study in detail the case of the horospherical variety associated to the exceptional group $G_2$, and construct a full rectangular Lefschetz exceptional collection in the derived category.

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Richard Gonzales, Clélia Pech, Nicolas Perrin, Alexander Samokhin. 2018-03-13. Geometry of horospherical varieties of Picard rank one. https://doi.org/10.1093/imrn%2Frnaa331

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