The Hodge filtration of a monodromic mixed Hodge module and the irregular Hodge filtration

TL;DR

This paper studies the structure of monodromic mixed Hodge modules, showing how their Hodge filtrations decompose and how the irregular Hodge filtration on their Fourier-Laplace transforms can be explicitly described, revealing deep structural properties.

Contribution

It establishes the decomposition of the Hodge filtration for monodromic mixed Hodge modules and provides a concrete description of the irregular Hodge filtration on their Fourier-Laplace transforms.

Findings

Hodge filtration decomposes with the underlying D-module

Fourier-Laplace transform inherits a mixed Hodge module structure

Irregular Hodge filtration coincides with the Hodge filtration at integer indices

Abstract

For an algebraic vector bundle $E$ over a smooth algebraic variety $X$, a monodromic $D$-module on $E$ is decomposed into a direct sum of some $O$-modules on $X$. We show that the Hodge filtration of a monodromic mixed Hodge module is decomposed with respect to the decomposition of the underlying $D$-module. By using this result, we endow the Fourier-Laplace transform $M^{\wedge}$ of the underlying $D$-module $M$ of a monodromic mixed Hodge module with a mixed Hodge module structure. Moreover, we describe the irregular Hodge filtration on $M^{\wedge}$ concretely and show that it coincides with the Hodge filtration at all integer indices.