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

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 ⌊D⌋ is snc. If (X, D) is lc or the logarithmic 1-forms are locally generated by closed forms, then (X, ⌊D⌋) is toroidal.