On the LipmanZariski conjecture for logarithmic vector fields on log canonical pairs
Abstract
We consider a version of the LipmanZariski 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 arXiv:
 arXiv:1712.04052
 Bibcode:
 2017arXiv171204052B
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Complex Variables