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

TL;DR

This paper proves that under certain extension conditions for logarithmic 1-forms on a normal variety, the Lipman-Zariski conjecture is valid, especially for varieties with log canonical singularities, and provides a new example of 2-forms with logarithmic poles.

Contribution

It establishes a new criterion for the Lipman-Zariski conjecture based on extension of logarithmic 1-forms and constructs an improved example of 2-forms with poles.

Findings

The Lipman-Zariski conjecture holds if logarithmic 1-forms extend to a log resolution.

The result applies to varieties with log canonical singularities.

An example of a 2-form with a logarithmic pole along an exceptional divisor is provided.

Abstract

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.