A decomposition theorem for singular Kähler  spaces with trivial first Chern class  of dimension at most four

Patrick Graf

doi:10.5802/afst.1757

A decomposition theorem for singular Kähler spaces with trivial first Chern class of dimension at most four

Patrick Graf ¹

¹ Fachbereich 08, Johannes-Gutenberg-Universität Mainz, Staudingerweg 9, 55099 Mainz, Germany

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 5, pp. 893-909.

Abstract
Résumé

Let $X$ be a compact Kähler fourfold with klt singularities and vanishing first Chern class, smooth in codimension two. We show that $X$ admits a Beauville–Bogomolov decomposition: a finite quasi-étale cover of $X$ splits as a product of a complex torus and singular Calabi–Yau and irreducible holomorphic symplectic varieties. We also prove that $X$ has small projective deformations and the fundamental group of $X$ is projective. To obtain these results, we propose and study a new version of the Lipman–Zariski conjecture.

Soit $X$ une variété kählérienne compacte de dimension quatre, avec des singularités klt et première classe de Chern nulle, lisse en codimension deux. Nous montrons que $X$ admet une decomposition de Beauville–Bogomolov : à un revêtement quasi-étale fini près, $X$ est un produit d’un tore complexe et des variétés singulières de Calabi–Yau et holomorphes symplectiques irréductibles. Nous prouvons aussi que $X$ admet des deformations projectives petites et que le groupe fondamental de $X$ est projective. Pour obtenir ces resultats, nous proposons et étudions une nouvelle version de la conjecture de Lipman–Zariski.

Received: 2021-04-15
Accepted: 2021-10-27
Published online: 2024-05-16

DOI: 10.5802/afst.1757

Classification: 32J27, 14E30, 14J32
Keywords: Kähler spaces, klt singularities, vanishing first Chern class, unobstructed deformations, decomposition theorem

Author's affiliations:

Patrick Graf ¹

¹ Fachbereich 08, Johannes-Gutenberg-Universität Mainz, Staudingerweg 9, 55099 Mainz, Germany

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Patrick Graf},
     title = {A decomposition theorem for singular {K\"ahler}  spaces with trivial first {Chern} class  of dimension at most four},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {893--909},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
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Patrick Graf. A decomposition theorem for singular Kähler  spaces with trivial first Chern class  of dimension at most four. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 5, pp. 893-909. doi : 10.5802/afst.1757. https://afst.centre-mersenne.org/articles/10.5802/afst.1757/

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