The minimal exponent and $k$-rationality for local complete intersections

Qianyu Chen; Bradley Dirks; Mircea Mustaţă

doi:10.5802/jep.267

The minimal exponent and

k

-rationality for local complete intersections

Qianyu Chen ¹; Bradley Dirks ¹; Mircea Mustaţă ¹

¹Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USA

Journal de l’École polytechnique — Mathématiques, Volume 11 (2024), pp. 849-873

Abstract (Original language)
Résumé

We show that if $Z$ is a local complete intersection subvariety of a smooth complex variety $X$ , of pure codimension $r$ , then $Z$ has $k$ -rational singularities if and only if $\tilde{α} (Z) > k + r$ , where $\tilde{α} (Z)$ is the minimal exponent of $Z$ . We also characterize this condition in terms of the Hodge filtration on the intersection complex Hodge module of $Z$ . Furthermore, we show that if $Z$ has $k$ -rational singularities, then the Hodge filtration on the local cohomology sheaf $ℋ_{Z}^{r} (𝒪_{X})$ is generated at level $dim (X) - ⌈ \tilde{α} (Z) ⌉ - 1$ and, assuming that $k \geq 1$ and $Z$ is singular, of dimension $d$ , that $ℋ^{k} ({\underset{̲}{Ω}}_{Z}^{d - k}) \neq 0$ . All these results have been known for hypersurfaces in smooth varieties.

Nous montrons que si $Z$ est une sous-variété localement intersection complète d’une variété complexe lisse $X$ , de codimension pure $r$ , alors $Z$ possède des singularités $k$ -rationnelles si et seulement si $\tilde{α} (Z) > k + r$ , où $\tilde{α} (Z)$ est l’exposant minimal de $Z$ . Nous caractérisons également cette condition en termes de filtration de Hodge sur le module de Hodge associé au complexe d’intersection de $Z$ . De plus, nous montrons que si $Z$ est à singularités $k$ -rationnelles, alors la filtration de Hodge sur le faisceau de cohomologie locale $ℋ_{Z}^{r} (𝒪_{X})$ est engendré au niveau $dim (X) - ⌈ \tilde{α} (Z) ⌉ - 1$ et, si de plus $k \geq 1$ et $Z$ est singulière, de dimension $d$ , que $ℋ^{k} ({\underset{̲}{Ω}}_{Z}^{d - k}) \neq 0$ . Tous ces résultats sont connus pour les hypersurfaces dans les variétés lisses.

Article information
Cite this article

Received: 2023-05-07
Accepted: 2024-06-05
Published online: 2024-08-27

MR Zbl

DOI: 10.5802/jep.267

Classification: 14F10, 14B05, 14J17, 32S35
Keywords: Minimal exponent, higher rational singularities, higher Du Bois singularities, Hodge modules, V-filtration
Mots-clés : Exposant minimal, singularités rationnelles supérieures, singularités de Du Bois supérieures, modules de Hodge, V-filtration

Author's affiliations:

Qianyu Chen ¹ ; Bradley Dirks ¹ ; Mircea Mustaţă ¹

¹ Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

Qianyu Chen; Bradley Dirks; Mircea Mustaţă. The minimal exponent and $k$-rationality for local complete intersections. Journal de l’École polytechnique — Mathématiques, Volume 11 (2024), pp. 849-873. doi: 10.5802/jep.267

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