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A weak version of the Lipman-Zariski conjecture. (English) Zbl 1327.32016
Summary: 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.
##### MSC:
 32C15 Complex spaces 32C20 Normal analytic spaces 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 32S05 Local complex singularities
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