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

11 Dec 2017  ·  Bergner Hannah ·

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. read more

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Algebraic Geometry Complex Variables