A decomposition theorem for singular Kähler spaces with trivial first Chern class of dimension at most four

Let X be a compact Kähler fourfold with klt singularities and vanishing first Chern class, smooth in codimension two. We show that X admits a Beauville–Bogomolov decomposition: a finite quasi-étale cover of X splits as a product of a complex torus and singular Calabi–Yau and irreducible holomorphic symplectic varieties. We also prove that X has small projective deformations and the fundamental group of X is projective. To obtain these results, we propose and study a new version of the Lipman–Zariski conjecture.