The generalized Lipman-Zariski problem

We propose and study a generalized version of the Lipman-Zariski conjecture: let $(x\in X)$ be an $n$-dimensional singularity such that for some integer $1 \le p \le n - 1$,thesheaf$\Omega_X^{[p]}$ofreflexivedifferential$p$-formsisfree\mathrm{.}Doesthisimplythat$(x \in X)$ 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 $(x\in X)$ is a hypersurface singularity with singular locus of codimension at least three, we give an affirmative answer to the above question for any $1\le p\le n-1$. The proof of this fact relies on a description of the torsion and cotorsion of the sheaves ${\mathrm{\Omega }}_X^p$ of K"ahler 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.