Mixed Hodge structure on local cohomology with support in determinantal varieties

TL;DR

This paper investigates the mixed Hodge module structure of local cohomology modules supported on determinantal varieties, revealing how weights are determined and providing formulas for the Hodge filtration's generation level.

Contribution

It introduces a method to compute the mixed Hodge module structure on local cohomology with determinantal support, highlighting the role of support and cohomological degree.

Findings

Weight of simple composition factors is uniquely determined by support and degree

Provides the equivariant structure of the Hodge filtration on local cohomology modules

Derives a formula for the generation level of the Hodge filtration

Abstract

We employ the inductive structure of determinantal varieties to calculate the mixed Hodge module structure of local cohomology modules with determinantal support. We show that the weight of a simple composition factor is uniquely determined by its support and cohomological degree. As a consequence, we obtain the equivariant structure of the Hodge filtration on each local cohomology module. Finally, as an application, we provide a formula for the generation level of the Hodge filtration on these modules.