Restrictions of mixed Hodge modules using generalized V-filtrations
Qianyu Chen, Bradley Dirks, Sebastian Olano
TL;DR
This work extends Sabbah's generalized ${}^L V$-filtrations to the setting of mixed Hodge modules on $X\times\mathbf{A}^r$, connecting them to the classical $V$-filtration via cyclic coverings. It proves that these filtrations compute the restriction functors $\sigma^!$ and $\sigma^*$ along the zero section, and develops a robust toolkit of ${}^L V$-Koszul complexes and specialization constructions that behave compatibly with Hodge and weight filtrations. Leveraging deformation to the normal bundle and cyclic covers, the authors reduce to the $L$-monodromic case and obtain explicit descriptions of local cohomology along complete intersections, yielding new criteria for $k$-Du Bois and $k$-rational singularities and giving sharp bounds for the minimal exponent in weighted homogeneous isolated complete intersections. These results connect D-module restrictions, V-filtration theory, and singularity invariants in a unified framework with concrete consequences for the study of local cohomology Hodge modules and singularity classification.
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 $σ^!, σ^*$, where $σ\colon X \times \{0\} \to X \times \mathbf A^r$ 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.