The generalized Lipman-Zariski problem

Abstract

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 $1 \le p \le n - 1$, the sheaf $\Omega_X^{[p]}$ of reflexive differential $p$-forms is free. Does this imply that $(x \in X)$ 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 $\Omega_X^p$ 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.