On the Lipman–Zariski conjecture for logarithmic vector fields on log canonical pairs
[Sur la conjecture de Lipman-Zarisiki pour les champs de vecteurs logarithmiques sur les paires log-canoniques]
Annales de l'Institut Fourier, Tome 71 (2021) no. 1, pp. 407-446.

Nous considérons une version de la conjecture de Lipman–Zariski pour des champs de vecteurs logarithmiques et des 1-formes logarithmiques. Soit (X,D) une paire, où X est une variété complexe normale et D est un diviseur de Weil effectif, tels que le faisceau des champs de vecteurs logarithmiques (ou de façon duale le faisceau des 1-formes logarithmiques réflexives) est localement libre. Nous démontrons le suivant dans ce cas : si (X,D) est dlt, alors X est nécessairement lisse et D est snc. Si (X,D) est lc ou si les 1-formes logarithmiques sont engendrées localement par des formes fermées, alors la paire (X,D) est toroïdale.

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 the pair (X,D) is toroidal.

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DOI : https://doi.org/10.5802/aif.3366
Classification : 14B05,  32M05,  32M25,  32S05
Mots clés : conjecture de Lipman–Zariski, champs de vecteurs logarithmiques, 1-formes logarithmiques, variétés toroïdales
@article{AIF_2021__71_1_407_0,
     author = {Bergner, Hannah},
     title = {On the {Lipman{\textendash}Zariski} conjecture for logarithmic vector fields on log canonical pairs},
     journal = {Annales de l'Institut Fourier},
     pages = {407--446},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {71},
     number = {1},
     year = {2021},
     doi = {10.5802/aif.3366},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3366/}
}
Bergner, Hannah. On the Lipman–Zariski conjecture for logarithmic vector fields on log canonical pairs. Annales de l'Institut Fourier, Tome 71 (2021) no. 1, pp. 407-446. doi : 10.5802/aif.3366. https://aif.centre-mersenne.org/articles/10.5802/aif.3366/

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