2025/05/15 by Hyunsuk Kim, Kim, Hyunsuk, Sridhar Venkatesh +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics
paper · pdf · doi:10.48550/arxiv.2505.10417
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.