Toric vector bundles over a discrete valuation ring and Bruhat-Tits buildings
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866910923371839488 |
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| author | Kaveh, Kiumars Manon, Christopher Tsvelikhovskiy, Boris |
| author_facet | Kaveh, Kiumars Manon, Christopher Tsvelikhovskiy, Boris |
| contents | We give a classification of rank $r$ torus equivariant vector bundles $\mathcal{E}$ on a toric scheme $\mathfrak{X}$ over a discrete valuation ring $\mathcal{O}$, in terms of graded piecewise linear maps $Φ$ from the fan of $\mathfrak{X}$ to the (extended) building of $GL(r)$. This is an extension of Klyachko's classification of torus equivariant vector bundles on toric varieties over a field on one hand, and Mumford's classification of equivariant line bundles on toric schemes over $\mathcal{O}$ on the other hand. We also give a simple criterion for equivariant splitting of $\mathcal{E}$ into a sum of toric line bundles in terms of its piecewise linear map. Among other things, this work lays the foundations for study of arithmetic geometry of toric vector bundles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_04299 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Toric vector bundles over a discrete valuation ring and Bruhat-Tits buildings Kaveh, Kiumars Manon, Christopher Tsvelikhovskiy, Boris Algebraic Geometry 14M25, 51E24 We give a classification of rank $r$ torus equivariant vector bundles $\mathcal{E}$ on a toric scheme $\mathfrak{X}$ over a discrete valuation ring $\mathcal{O}$, in terms of graded piecewise linear maps $Φ$ from the fan of $\mathfrak{X}$ to the (extended) building of $GL(r)$. This is an extension of Klyachko's classification of torus equivariant vector bundles on toric varieties over a field on one hand, and Mumford's classification of equivariant line bundles on toric schemes over $\mathcal{O}$ on the other hand. We also give a simple criterion for equivariant splitting of $\mathcal{E}$ into a sum of toric line bundles in terms of its piecewise linear map. Among other things, this work lays the foundations for study of arithmetic geometry of toric vector bundles. |
| title | Toric vector bundles over a discrete valuation ring and Bruhat-Tits buildings |
| topic | Algebraic Geometry 14M25, 51E24 |
| url | https://arxiv.org/abs/2208.04299 |