A description of monodromic mixed Hodge modules

TL;DR

This paper studies monodromic mixed Hodge modules on smooth algebraic varieties, showing their Hodge filtrations decompose and establishing an equivalence with gluing data, with applications to Fourier-Laplace transforms.

Contribution

It introduces a decomposition of Hodge filtrations for monodromic mixed Hodge modules and proves an equivalence with gluing data categories, advancing understanding of their structure.

Findings

Hodge filtration decomposes for monodromic mixed Hodge modules

An equivalence of categories with gluing data is established

Fourier-Laplace transform inherits a mixed Hodge module structure

Abstract

For a smooth algebraic variety $X$, a monodromic $D$-module on $X\times \mathbb{C}$ is decomposed into a direct sum of some $D$-modules on $X$. We show that the Hodge filtration of a mixed Hodge module on $X\times \mathbb{C}$ whose underlying $D$-module is monodromic is also decomposed. Moreover, we show that there is an equivalence of categories between the category of monodromic mixed Hodge modules on $X\times \mathbb{C}$ and the category of ``gluing data''. As an application, we endow the Fourier-Laplace transformation of the underlying $D$-module of a monodromic mixed Hodge module with a mixed Hodge module structure.