On the Lipman-Zariski conjecture for logarithmic vector fields on log canonical pairs

TL;DR

This paper investigates the Lipman-Zariski conjecture in the context of logarithmic vector fields on pairs, establishing smoothness and toroidal structures under certain conditions for log canonical pairs.

Contribution

It proves that for dlt pairs with locally free logarithmic vector fields, the underlying variety is smooth and the divisor has simple normal crossings, extending the conjecture's validity.

Findings

If (X,D) is dlt with locally free logarithmic vector fields, then X is smooth and D has snc.

For lc pairs or when logarithmic 1-forms are generated by closed forms, (X,⌊D⌋) is toroidal.

Abstract

We consider a version of the Lipman-Zariski conjecture for logarithmic vector fields and logarithmic $1$-forms on pairs. Let $(X,D)$ be a pair consisting of a normal complex variety $X$ and an effective Weil divisor $D$ such that the sheaf of logarithmic vector fields (or dually the sheaf of reflexive logarithmic $1$-forms) is locally free. We prove that in this case the following holds: If $(X,D)$ is dlt, then $X$ is necessarily smooth and $\lfloor D\rfloor $ is snc. If $(X,D)$ is lc or the logarithmic $1$-forms are locally generated by closed forms, then $(X,\lfloor D\rfloor)$ is toroidal.