Feuilleter
no. 5
A decomposition theorem for singular Kähler spaces with trivial first Chern class of dimension at most four
Patrick Graf1
1 Fachbereich 08, Johannes-Gutenberg-Universität Mainz, Staudingerweg 9, 55099 Mainz, Germany
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 32 (2023) no. 5, pp. 893-909.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/afst.1757
Classification : 32J27, 14E30, 14J32
Mots-clés : Kähler spaces, klt singularities, vanishing first Chern class, unobstructed deformations, decomposition theorem

Patrick Graf 1

1 Fachbereich 08, Johannes-Gutenberg-Universität Mainz, Staudingerweg 9, 55099 Mainz, Germany
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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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, Série 6, Tome 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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