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Books Year : 2018

## Irregular Hodge theory. Avec la collaboration de Jeng-Daw Yu

Claude Sabbah

#### 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.

#### Domains

Mathematics [math] Algebraic Geometry [math.AG]

### Dates and versions

hal-01727630 , version 1 (09-03-2018)

### Identifiers

• HAL Id : hal-01727630 , version 1
• ARXIV :

### Cite

Claude Sabbah. Irregular Hodge theory. Avec la collaboration de Jeng-Daw Yu. Société mathématique de France, 156, 2018, Mémoires de la Société mathématique de France. ⟨hal-01727630⟩

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