An optimal extension theorem for 1-forms and the Lipman-Zariski conjecture

By Patrick Graf, S'andor J. Kov'acs. Documenta Mathematica, 2013

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Let $X$ be a normal variety. Assume that for some reduced divisor $D \subset X$, logarithmic 1-forms defined on the snc locus of $(X, D)$ extend to a log resolution $\tilde X \to X$ as logarithmic differential forms. We prove that then the Lipman-Zariski conjecture holds for $X$. This result applies in particular if $X$ has log canonical singularities. Furthermore, we give an example of a 2-form defined on the smooth locus of a three-dimensional log canonical pair $(X, \emptyset)$ which acquires a logarithmic pole along an exceptional divisor of discrepancy zero, thereby improving on a similar example of Greb, Kebekus, Kov\'acs and Peternell.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/46f7189db038232fa5d4dec94971560f9323816e