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Kähler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups

Abstract : Let $X$ be a compact Kähler space with klt singularities and vanishing first Chern class. We prove the Bochner principle for holomorphic tensors on the smooth locus of $X$: any such tensor is parallel with respect to the singular Ricci-flat metrics. As a consequence, after a finite quasi-étale cover $X$ splits off a complex torus of the maximum possible dimension. We then proceed to decompose the tangent sheaf of $X$ according to its holonomy representation. In particular, we classify those $X$ which have strongly stable tangent sheaf: up to quasi-étale covers, these are either irreducible Calabi--Yau or irreducible holomorphic symplectic. As an application of these results, we show that if $X$ has dimension four, then it satisfies Campana's Abelianity Conjecture.
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https://hal.archives-ouvertes.fr/hal-02936352
Contributor : Henri Guenancia Connect in order to contact the contributor
Submitted on : Thursday, July 21, 2022 - 4:51:29 PM
Last modification on : Tuesday, July 26, 2022 - 3:56:25 AM

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Benoît Claudon, Patrick Graf, Henri Guenancia, Philipp Naumann. Kähler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups. Journal für die reine und angewandte Mathematik, Walter de Gruyter, 2022, pp.245-275. ⟨10.1515/crelle-2022-0001⟩. ⟨hal-02936352v2⟩

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