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.

arXiv comment28 pages; v2: exposition improved following the referee's suggestions, new title, section 7 shortened and section 8 removed; to appear in Crelle

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• Numerical characterization of complex torus quotients
• Equality in the Miyaoka-Yau inequality and uniformization of non-positively curved klt pairs
• Semispecial tensors and quotients of the polydisc
• Fundamental groups of log Calabi-Yau surfaces
• A decomposition theorem for singular Kähler spaces with trivial first Chern class of dimension at most four