Irregular Hodge theory

TL;DR

This paper develops a theory of irregular Hodge filtrations for holonomic D-modules, extending classical mixed Hodge modules to irregular cases and proving key properties like stability under functors and degeneration.

Contribution

It introduces a canonical irregular Hodge filtration for a broad class of holonomic D-modules, generalizing previous results and including explicit cases like irregular mixed Hodge structures.

Findings

Irregular Hodge filtration exists for a wide class of D-modules.

The filtration satisfies the E_1-degeneration property.

Explicit formulas for irregular mixed Hodge structures are provided.

Abstract

We introduce a category of possibly irregular holonomic D-modules which can be endowed in a canonical way with an irregular Hodge filtration. Mixed Hodge modules with their Hodge filtration naturally belong to this category, as well as their twist by $\exp\varphi$ for any meromorphic function $\varphi$. This category is stable by various standard functors, which produce many more filtered objects. The irregular Hodge filtration satisfies the $E_1$-degeneration property by a projective morphism. This generalizes some results proved by Esnault-Sabbah-Yu arxiv:1302.4537 and Sabbah-Yu arxiv:1406.1339. We also show that those rigid irreducible holonomic D-modules on the complex projective line whose local formal monodromies have eigenvalues of absolute value one, are equipped with such an irregular Hodge filtration in a canonical way, up to a shift of the filtration. In a chapter written jointly with Jeng-Daw~Yu, we make explicit the case of irregular mixed Hodge structures, for which we prove in particular a Thom-Sebastiani formula.