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

Let X be a compact Kahler 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.