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A weak version of the Lipman–Zariski conjecture

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Abstract

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.

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Acknowledgments

The author was motivated to think about the Lipman–Zariski conjecture following interesting discussions with Patrick Graf, Daniel Greb and Sebastian Goette. The author would like to thank especially Stefan Kebekus, Daniel Greb, Patrick Graf and the anonymous referee for carefully reading preliminary versions of this work. Karl Oeljeklaus kindly pointed to his and Richthofers results.

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  1. Mathematisches Institut, Eckerstraße 1, 79104 , Freiburg, Germany

    Clemens Jörder

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  1. Clemens Jörder
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Correspondence to Clemens Jörder.

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The author gratefully acknowledges support by the DFG-Forschergruppe 790 “Classification of Algebraic Surfaces and Compact Complex Manifolds” and the Graduiertenkolleg 1821 “Cohomological Methods in Geometry”.

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Jörder, C. A weak version of the Lipman–Zariski conjecture. Math. Z. 278, 893–899 (2014). https://doi.org/10.1007/s00209-014-1337-z

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  • DOI: https://doi.org/10.1007/s00209-014-1337-z

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