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

TL;DR

This paper investigates the mixed Hodge module structure on local cohomology sheaves of affine toric varieties and explores their singular cohomology, deriving combinatorial results and detailed descriptions for specific subclasses of toric varieties.

Contribution

It establishes a general framework for the mixed Hodge module structure on local cohomology of affine toric varieties and applies it to compute cohomological invariants for special classes.

Findings

Singular cohomology of proper toric varieties is of Hodge-Tate type.

Provides combinatorial criteria related to rational polyhedral cones.

Calculates Betti numbers for toric varieties over simple polytopes.

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.