Restrictions of mixed Hodge modules using generalized V-filtrations

TL;DR

This paper investigates generalized V-filtrations on mixed Hodge modules, compares them to classical filtrations, and applies these insights to analyze singularities and cohomology in algebraic geometry.

Contribution

It introduces a method to compute restriction functors using generalized V-filtrations and applies this to classify certain singularities and their properties.

Findings

Generalized V-filtrations can be used to compute restriction functors.

Comparison between generalized and classical V-filtrations via cyclic covers.

Classification of when weighted homogeneous isolated complete intersection singularities are k-Du Bois and k-rational.

Abstract

We study generalized $V$-filtrations, defined by Sabbah, on $\mathcal D$-modules underlying mixed Hodge modules on $X\times \mathbf A^r$. Using cyclic covers, we compare these filtrations to the usual $V$-filtration, which is better understood. The main result shows that these filtrations can be used to compute the restriction functors $\sigma^!, \sigma^*$, where $\sigma \colon X \times \{0\} \to X \times \mathbf A^r$ is the inclusion of the zero section. As an application, we use the restriction result to study singularities of complete intersection subvarieties. These filtrations can be used to study the local cohomology mixed Hodge module. In particular, we classify when weighted homogeneous isolated complete intersection singularities in $\mathbf A^n$ are $k$-Du Bois and $k$-rational.