An Optimal Extension Theorem for 1-Forms and the Lipman-Zariski Conjecture

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 $\wt X \to X$ 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, Kovács and Peternell.

2010 Mathematics Subject Classification: 14B05, 32S05

Keywords and Phrases: Singularities of the minimal model program, differential forms, Lipman-Zariski conjecture

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