Hodge–Lyubeznik numbers

Ricardo García López; Claude Sabbah

doi:10.5802/crmath.724

Article de recherche - Géométrie algébrique

Hodge–Lyubeznik numbers
[Nombres de Hodge–Lyubeznik]

Ricardo García López ¹ ; Claude Sabbah ²

¹ Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Vía, 585, E-08007 Barcelona, Spain
² CMLS, CNRS, École polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau cedex, France

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 213-221.

Résumé
Abstract

Nous définissons un raffinement en théorie de Hodge des nombres de Lyubeznik pour les anneaux locaux de variétés algébriques complexes. Nous prouvons que ces nombres sont indépendants des choix faits dans leur définition et que, pour l’anneau local d’une singularité isolée, ils peuvent être exprimés en termes de nombres de Hodge de la cohomologie de l’entrelac de la singularité. Nous donnons des exemples de singularités isolées ayant les mêmes nombres de Lyubeznik mais des nombres de Hodge–Lyubeznik différents.

We define a Hodge-theoretical refinement of the Lyubeznik numbers for local rings of complex algebraic varieties. We prove that these numbers are independent of the choices made in their definition and that, for the local ring of an isolated singularity, they can be expressed in terms of the Hodge numbers of the cohomology of the link of the singularity. We give examples of isolated singularities with the same Lyubeznik numbers but different Hodge–Lyubeznik numbers.

Reçu le : 2024-10-08
Révisé le : 2025-01-22
Accepté le : 2025-01-26
Publié le : 2025-03-25

DOI : 10.5802/crmath.724

Classification : 13D45, 14B15, 32S35
Keywords: Hodge numbers, Lyubeznik numbers, mixed Hodge modules
Mots-clés : Nombres de Hodge, nombres de Lyubeznik, modules de Hodge mixtes

Affiliations des auteurs :

Ricardo García López ¹ ; Claude Sabbah ²

¹ Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Vía, 585, E-08007 Barcelona, Spain
² CMLS, CNRS, École polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau cedex, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2025__363_G2_213_0,
     author = {Ricardo Garc{\'\i}a L\'opez and Claude Sabbah},
     title = {Hodge{\textendash}Lyubeznik numbers},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {213--221},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.724},
     language = {en},
}

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Ricardo García López; Claude Sabbah. Hodge–Lyubeznik numbers. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 213-221. doi : 10.5802/crmath.724. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.724/

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