On $V$-filtration, Hodge filtration and Fourier transform

TL;DR

This paper explores the interaction between $V$-filtration and Hodge filtration on mixed Hodge modules, providing formulas for functors, applications to monodromic modules, a proof of Skoda's theorem, and insights into Fourier transforms.

Contribution

It introduces a new formula relating $V$-filtration and Hodge filtration, and applies it to various problems in Hodge theory and Fourier analysis.

Findings

Derived formulas for $i^*$ and $i^!$ functors in terms of $V$-filtration.

Established a Hodge-theoretic proof of Skoda's theorem on multiplier ideals.

Analyzed the Fourier-Laplace transform of monodromic mixed Hodge modules.

Abstract

For $i : Z \to X$ a closed immersion of smooth varieties, we study how the $V$-filtration along $Z$ and the Hodge filtration on a mixed Hodge module $M$ on $X$ interact with each other. We also give a formula for the functors $i^*$, $i^!$ in terms of this $V$-filtration. As applications, we obtain results on the Hodge filtration of monodromic mixed Hodge modules and we give a Hodge theoretic proof of Skoda's theorem on multiplier ideals. Finally, we use the results to study the Fourier-Laplace transform of a monodromic mixed Hodge module.