The generalized Lipman–Zariski problem

We propose and study a generalized version of the Lipman–Zariski conjecture: let ( x ∈ X ) be an n -dimensional singularity such that for some integer 1 ≤ p ≤ n - 1 , the sheaf Ω X [ p ] of reflexive differential p -forms is free. Does this imply that ( x ∈ X ) is smooth? We give an example showing that the answer is no even for p = 2 and X a terminal threefold. However, we prove that if p = n - 1 , 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 X is a projective klt variety of dimension n such that the sheaf of ( n - 1 ) -forms on its smooth locus is flat, then X is a quotient of an Abelian variety. On the other hand, if ( x ∈ X ) is a hypersurface singularity with singular locus of codimension at least three, we give an affirmative answer to the above question for any 1 ≤ p ≤ n - 1 . The proof of this fact relies on a description of the torsion and cotorsion of the sheaves Ω X p of Kähler 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 p is isomorphic to the cotorsion in degree p - 1 via the residue map.