The generalized Lipman-Zariski problem

We propose and study a generalized version of the Lipman-Zariski conjecture: let be an -dimensional singularity such that for some integer , the sheaf of reflexive differential -forms is free. Does this imply that is smooth? We give an example showing that the answer is no even for and a terminal threefold. However, we prove that if , then there are only finitely many log canonical counterexamples in each dimension, and all of these are isolated and terminal. As an application, we show that if is a projective klt variety of dimension such that the sheaf of -forms on its smooth locus is flat, then is a quotient of an Abelian variety. On the other hand, if is a hypersurface singularity with singular locus of codimension at least three, we give an affirmative answer to the above question for any . The proof of this fact relies on a description of the torsion and cotorsion of the sheaves of Kahler differentials on a hypersurface in terms of a Koszul complex. As a corollary, we obtain that for a normal hypersurface singularity, the torsion in degree is isomorphic to the cotorsion in degree via the residue map.