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arXiv · 2505.10417

Local cohomology and singular cohomology of toric varieties via mixed Hodge modules

Hyunsuk Kim·Sridhar Venkatesh
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Abstract

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.

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Hyunsuk Kim, Sridhar Venkatesh. 2025-05-15. Local cohomology and singular cohomology of toric varieties via mixed Hodge modules. https://arxiv.org/abs/2505.10417

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