Local cohomology and singular cohomology of toric varieties via mixed Hodge modules
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866918071970562048 |
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| author | Kim, Hyunsuk Venkatesh, Sridhar |
| author_facet | Kim, Hyunsuk Venkatesh, Sridhar |
| contents | Given an affine toric variety $X$ embedded in a smooth variety, we prove a general result about the mixed Hodge module structure on the local cohomology sheaves of $X$. As a consequence, we prove that the singular cohomology of a proper toric variety is mixed of Hodge-Tate type. Additionally, using these Hodge module techniques, we derive a purely combinatorial result on rational polyhedral cones that has consequences regarding the depth of reflexive differentials on a toric variety.
We then study in detail two important subclasses of toric varieties: those corresponding to cones over simplicial polytopes and those corresponding to cones over simple polytopes. Here, we give a comprehensive description of the local cohomology in terms of the combinatorics of the associated cones, and calculate the Betti numbers (or more precisely, the Hodge-Du Bois diamond) of a projective toric variety associated to a simple polytope. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_10417 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local cohomology and singular cohomology of toric varieties via mixed Hodge modules Kim, Hyunsuk Venkatesh, Sridhar Algebraic Geometry Combinatorics 14B05, 14B15, 14F10, 14M25, 32S35, 52B20 Given an affine toric variety $X$ embedded in a smooth variety, we prove a general result about the mixed Hodge module structure on the local cohomology sheaves of $X$. As a consequence, we prove that the singular cohomology of a proper toric variety is mixed of Hodge-Tate type. Additionally, using these Hodge module techniques, we derive a purely combinatorial result on rational polyhedral cones that has consequences regarding the depth of reflexive differentials on a toric variety. We then study in detail two important subclasses of toric varieties: those corresponding to cones over simplicial polytopes and those corresponding to cones over simple polytopes. Here, we give a comprehensive description of the local cohomology in terms of the combinatorics of the associated cones, and calculate the Betti numbers (or more precisely, the Hodge-Du Bois diamond) of a projective toric variety associated to a simple polytope. |
| title | Local cohomology and singular cohomology of toric varieties via mixed Hodge modules |
| topic | Algebraic Geometry Combinatorics 14B05, 14B15, 14F10, 14M25, 32S35, 52B20 |
| url | https://arxiv.org/abs/2505.10417 |