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arXiv · 2008.13008

K\"ahler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups

Benoît Claudon·Patrick Graf·Henri Guenancia·Philipp Naumann
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Abstract

Let $X$ be a compact K\"ahler 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-\'etale 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-\'etale 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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Benoît Claudon, Patrick Graf, Henri Guenancia, Philipp Naumann. 2020-08-29. K\"ahler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups. https://doi.org/10.1515/crelle-2022-0001

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