We consider a version of the Lipman–Zariski conjecture for logarithmic vector fields and logarithmic -forms on pairs. Let be a pair consisting of a normal complex variety and an effective Weil divisor such that the sheaf of logarithmic vector fields (or dually the sheaf of reflexive logarithmic -forms) is locally free. We prove that in this case the following holds: if is dlt, then is necessarily smooth and is snc. If is lc or the logarithmic -forms are locally generated by closed forms, then the pair is toroidal.
Nous considérons une version de la conjecture de Lipman–Zariski pour des champs de vecteurs logarithmiques et des 1-formes logarithmiques. Soit une paire, où est une variété complexe normale et est un diviseur de Weil effectif, tels que le faisceau des champs de vecteurs logarithmiques (ou de façon duale le faisceau des -formes logarithmiques réflexives) est localement libre. Nous démontrons le suivant dans ce cas : si est dlt, alors est nécessairement lisse et est snc. Si est lc ou si les -formes logarithmiques sont engendrées localement par des formes fermées, alors la paire est toroïdale.
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Keywords: Lipman–Zariski conjecture, logarithmic vector fields, logarithmic 1-forms, toroidal varieties
Mot 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/} }
TY - JOUR AU - Bergner, Hannah TI - On the Lipman–Zariski conjecture for logarithmic vector fields on log canonical pairs JO - Annales de l'Institut Fourier PY - 2021 SP - 407 EP - 446 VL - 71 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3366/ DO - 10.5802/aif.3366 LA - en ID - AIF_2021__71_1_407_0 ER -
%0 Journal Article %A Bergner, Hannah %T On the Lipman–Zariski conjecture for logarithmic vector fields on log canonical pairs %J Annales de l'Institut Fourier %D 2021 %P 407-446 %V 71 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3366/ %R 10.5802/aif.3366 %G en %F AIF_2021__71_1_407_0
Bergner, Hannah. On the Lipman–Zariski conjecture for logarithmic vector fields on log canonical pairs. Annales de l'Institut Fourier, Volume 71 (2021) no. 1, pp. 407-446. doi : 10.5802/aif.3366. https://aif.centre-mersenne.org/articles/10.5802/aif.3366/
[1] Geometrical methods in the theory of ordinary differential equations, Grundlehren der Mathematischen Wissenschaften, 250, Springer, 1988 (Translated from the Russian by J. Szücs) | DOI | MR | Zbl
[2] The Zariski–Lipman conjecture for log canonical spaces, Bull. Lond. Math. Soc., Volume 46 (2014) no. 4, pp. 827-835 | DOI | MR | Zbl
[3] Characterizing normal crossing hypersurfaces, Math. Ann., Volume 361 (2015) no. 3-4, pp. 995-1020 | DOI | MR | Zbl
[4] Complex analytic geometry, Lecture Notes in Mathematics, 538, Springer, 1976 | MR | Zbl
[5] Bogomolov–Sommese vanishing on log canonical pairs, J. Reine Angew. Math., Volume 702 (2015), pp. 109-142 | DOI | MR | Zbl
[6] An optimal extension theorem for 1-forms and the Lipman–Zariski conjecture, Doc. Math., Volume 19 (2014), pp. 815-830 | MR | Zbl
[7] Potentially Du Bois spaces, J. Singul., Volume 8 (2014), pp. 117-134 | MR | Zbl
[8] Normal crossing properties of complex hypersurfaces via logarithmic residues, Compos. Math., Volume 150 (2014) no. 9, pp. 1607-1622 | DOI | MR | Zbl
[9] Extension theorems for differential forms and Bogomolov–Sommese vanishing on log canonical varieties, Compos. Math., Volume 146 (2010) no. 1, pp. 193-219 | DOI | MR | Zbl
[10] Differential forms on log canonical spaces, Publ. Math., Inst. Hautes Étud. Sci. (2011) no. 114, pp. 87-169 | DOI | Numdam | MR | Zbl
[11] Semistability of the tangent sheaf of singular varieties, Algebr. Geom., Volume 3 (2016) no. 5, pp. 508-542 | DOI | MR | Zbl
[12] Geometric invariant theory on Stein spaces, Math. Ann., Volume 289 (1991) no. 4, pp. 631-662 | DOI | MR | Zbl
[13] A weak version of the Lipman–Zariski conjecture, Math. Z., Volume 278 (2014) no. 3-4, pp. 893-899 | DOI | MR | Zbl
[14] Infinitesimale Transformationsgruppen komplexer Räume, Math. Ann., Volume 160 (1965), pp. 72-92 | DOI | MR | Zbl
[15] Singularities of pairs, Algebraic geometry–Santa Cruz 1995 (Proceedings of Symposia in Pure Mathematics), Volume 62, American Mathematical Society, 1997, pp. 221-287 | DOI | MR | Zbl
[16] Lectures on resolution of singularities, Annals of Mathematics Studies, 166, Princeton University Press, 2007 | MR | Zbl
[17] Singularities of the minimal model program, Cambridge Tracts in Mathematics, 200, Cambridge University Press, 2013 (With the collaboration of S. Kovács) | DOI | MR | Zbl
[18] Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, 1998 (With the collaboration of C. H. Clemens and A. Corti, translated from the 1998 Japanese original) | DOI | MR | Zbl
[19] Free derivation modules on algebraic varieties, Am. J. Math., Volume 87 (1965), pp. 874-898 | DOI | MR | Zbl
[20] Nevanlinna theory in several complex variables and Diophantine approximation, Grundlehren der Mathematischen Wissenschaften, 350, Springer, 2014 | DOI | MR | Zbl
[21] Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 15, Springer, 1988 (An introduction to the theory of toric varieties, Translated from the Japanese) | MR | Zbl
[22] Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 27 (1980) no. 2, pp. 265-291 | MR | Zbl
[23] Reductive group actions on Stein spaces, Math. Ann., Volume 259 (1982) no. 1, pp. 79-97 | DOI | MR | Zbl
[24] On manifolds with trivial logarithmic tangent bundle, Osaka J. Math., Volume 41 (2004) no. 2, pp. 473-484 | MR | Zbl
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