Geometry of horospherical varieties of Picard rank one
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2018
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| _version_ | 1866916515584933888 |
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| author | Gonzales, Richard Pech, Clélia Perrin, Nicolas Samokhin, Alexander |
| author_facet | Gonzales, Richard Pech, Clélia Perrin, Nicolas Samokhin, Alexander |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1803_05063 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Geometry of horospherical varieties of Picard rank one Gonzales, Richard Pech, Clélia Perrin, Nicolas Samokhin, Alexander Algebraic Geometry Representation Theory 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. |
| title | Geometry of horospherical varieties of Picard rank one |
| topic | Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/1803.05063 |