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

Let $X$ be a normal variety. Assume that for some reduced divisor $D \subset X$,logarithmic1-formsdefinedonthesnclocusof$(X, D)$ 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 $(X,\mathrm{\varnothing })$ 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.