A decomposition theorem for $\mathbb{Q}$-Fano Kähler–Einstein varieties

Stéphane Druel; Henri Guenancia; Mihai Păun

doi:10.5802/crmath.612

Article de recherche - Géométrie algébrique

A decomposition theorem for

ℚ

-Fano Kähler–Einstein varieties
[Un théorème de décomposition pour les variétés

ℚ

-Fano qui admettent une métrique de Kähler–Einstein]

Stéphane Druel ¹ ; Henri Guenancia ² ; Mihai Păun ³

¹ Univ Lyon, CNRS, Université Claude Bernard Lyon 1, UMR 5208, Institut Camille Jordan, F-69622 Villeurbanne, France
² Institut de Mathématiques de Toulouse, Université Paul Sabatier, 31062 Toulouse Cedex 9, France
³ Lehrstuhl für Mathematik VIII, Universität Bayreuth, 95440 Bayreuth, Germany

Comptes Rendus. Mathématique, Volume 362 (2024) no. S1, pp. 93-118.

Résumé
Abstract

Soit X une variété $ℚ$ -Fano admettant une métrique de Kähler–Einstein. Nous montrons, qu’à un revêtement fini quasi-étale près, X est un produit de variétés $ℚ$ -Fano admettant une métrique de Kähler–Einstein dont le fibré tangent est stable relativement au diviseur anticanonique. La démonstration repose notamment sur un théorème de décompostion pour les feuilletages algébriquement intégrables. Nous montrons également que l’extension canonique de $T_{X}$ par $𝒪_{X}$ est polystable à nouveau relativement au diviseur anticanonique.

Let $X$ be a $ℚ$ -Fano variety admitting a Kähler–Einstein metric. We prove that up to a finite quasi-étale cover, $X$ splits isometrically as a product of Kähler–Einstein $ℚ$ -Fano varieties whose tangent sheaf is stable with respect to the anticanonical polarization. This relies among other things on a very general splitting theorem for algebraically integrable foliations. We also prove that the canonical extension of $T_{X}$ by $𝒪_{X}$ is polystable with respect to the anticanonical polarization.

Reçu le : 2022-12-31
Accepté le : 2024-01-03
Publié le : 2024-06-06

DOI : 10.5802/crmath.612

Classification : 14B05, 14J45, 32Q20, 37F75
Keywords: $\mathbb{Q}$-Fano varieties, singular Kähler–Einstein metrics, stable reflexive sheaves, algebraically integrable foliations
Mot clés : Variétés $\mathbb{Q}$-Fano, métriques de Kähler–Einstein singulières, faisceaux réflexifs stables, feuilletages algébriquement intégrables

Affiliations des auteurs :

Stéphane Druel ¹ ; Henri Guenancia ² ; Mihai Păun ³

¹ Univ Lyon, CNRS, Université Claude Bernard Lyon 1, UMR 5208, Institut Camille Jordan, F-69622 Villeurbanne, France
² Institut de Mathématiques de Toulouse, Université Paul Sabatier, 31062 Toulouse Cedex 9, France
³ Lehrstuhl für Mathematik VIII, Universität Bayreuth, 95440 Bayreuth, Germany

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {St\'ephane Druel and Henri Guenancia and Mihai P\u{a}un},
     title = {A decomposition theorem for $\mathbb{Q}${-Fano} {K\"ahler{\textendash}Einstein} varieties},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {93--118},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     number = {S1},
     year = {2024},
     doi = {10.5802/crmath.612},
     language = {en},
}

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Stéphane Druel; Henri Guenancia; Mihai Păun. A decomposition theorem for $\mathbb{Q}$-Fano Kähler–Einstein varieties. Comptes Rendus. Mathématique, Volume 362 (2024) no. S1, pp. 93-118. doi : 10.5802/crmath.612. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.612/

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