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

Graf, Patrick

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

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.

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

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