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

Let 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 : any such tensor is parallel with respect to the singular Ricci-flat metrics. As a consequence, after a finite quasi-étale cover splits off a complex torus of the maximum possible dimension. We then proceed to decompose the tangent sheaf of according to its holonomy representation. In particular, we classify those 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 has dimension four, then it satisfies Campana’s Abelianity Conjecture.