Restrictions of mixed Hodge modules using generalized V-filtrations

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 $\ddot{I}{ƒ}^{!},\ddot{I}{ƒ}^{\ast }$, where $\ddot{I}ƒ:X\times \{0\}\to X\times {\mathbf{A}}^r$ 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.

36 pages; v2: Improved the result of Theorem A to both restriction functors and its exposition. Fixed typos. 42 pages; v3: Final version. Improved exposition and corrected typos following referee suggestions, to appear in Advances in Mathematics