A weak version of the Lipman–Zariski conjecture

The complex-analytic version of the Lipman–Zariski conjecture says that a complex space is smooth if its tangent sheaf is locally free. We prove the following weak version of the conjecture: A normal complex space is smooth if its tangent sheaf is locally free and locally admits a basis consisting of pairwise commuting vector fields. The main tool used in the proof of our result is a new extension theorem for reflexive differential forms on a normal complex space. It says that a closed holomorphic differential form of degree one defined on the smooth locus of a normal complex space can be extended to a holomorphic differential form on any resolution of singularities of the complex space.