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

Saito, Takahiro

doi:10.5802/aif.3629

The Hodge filtration of a monodromic mixed Hodge module and the irregular Hodge filtration
[La filtration de Hodge d’un module de Hodge mixte monodromique et la filtration de Hodge irrégulière]

Saito, Takahiro ¹

¹ Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502 (Japan)

Annales de l'Institut Fourier, Online first, 68 p.

Résumé
Abstract

Pour un fibré vectoriel algébrique $E$ sur une variété algébrique lisse $X$ , un $D$ -module monodromique sur $E$ est décomposé en une somme directe de certains $O$ -modules sur $X$ . Nous montrons que la filtration de Hodge d’un module de Hodge mixte monodromique est décomposée par rapport à la décomposition du $D$ -module sous-jacent. En utilisant ce résultat, nous munissons la transformée de Fourier–Laplace $M^{\land}$ du $D$ -module sous-jacent $M$ à un module de Hodge mixte monodromique d’une structure de module de Hodge mixte. De plus, nous décrivons explicitement la filtration de Hodge irrégulière sur $M^{\land}$ et montrons qu’elle coïncide avec la filtration de Hodge aux indices entiers.

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 $M^{\land}$ 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 $M^{\land}$ concretely and show that it coincides with the Hodge filtration at all integer indices.

Reçu le : 2022-05-09
Révisé le : 2022-10-18
Accepté le : 2022-12-16
Première publication : 2024-05-17

DOI : 10.5802/aif.3629

Classification : 14F10, 32S35, 32S40
Keywords: $D$-module, Perverse sheaf, Mixed Hodge module, Mixed twistor $D$-module, Irregular Hodge filtration.
Mot clés : $D$-module, faisceau pervers, module de Hodge mixte, $D$-module de twisteur mixte, filtration de Hodge irrégulière.

Affiliations des auteurs :

Saito, Takahiro ¹

¹ Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502 (Japan)

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     author = {Saito, Takahiro},
     title = {The {Hodge} filtration of a monodromic mixed {Hodge} module and the irregular {Hodge} filtration},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2024},
     doi = {10.5802/aif.3629},
     language = {en},
     note = {Online first},
}

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Saito, Takahiro. The Hodge filtration of a monodromic mixed Hodge module and the irregular Hodge filtration. Annales de l'Institut Fourier, Online first, 68 p.

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