Singular spaces with trivial canonical class

The Beauville-Bogomolov decomposition theorem asserts that any compact Kahler manifold with trivial canonical bundle is finitely covered by the product of a compact complex torus, simply connected Calabi-Yau manifolds, and simply connected irreducible holomorphic symplectic manifolds. The decomposition of the etale cover corresponds to a decomposition of the tangent bundle into a direct sum whose summands are integrable and stable with respect to any polarization.
Building on recent extension theorems for differential forms on singular spaces, in the talk I will sketch the proof of an analogous decomposition theorem for the tangent sheaf of a projective variety with canonical singularities and numerically trivial canonical class. This leads to questions about holomorphic foliations on non-compact manifolds with trivial canonical bundle, which I will also discuss in my talk. This is joint work with Stefan Kebekus and Thomas Peternell.