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The Zariski-Lipman conjecture for log canonical spaces. (English) Zbl 1357.14009
In the paper under the review, the author proves a special case of the Zariski-Lipman conjecture (which asserts that a complex variety \(X\) with a locally free tangent sheaf \(T_X\) is necessarily smooth). His main result states as follows: let \(X\) be a log canonical space such that the tangent sheaf \(T_X\) is locally free, then \(X\) is smooth. The log canonical spaces are related to the minimal model programm. Let us mention that the author obtain his main result by proving the following more general theorem: let \((X,B)\) be a log canonical pair such that the tangent sheaf \(T_X\) is locally free, then \(X\) is smooth.

MSC:
14B05 Singularities in algebraic geometry
32B05 Analytic algebras and generalizations, preparation theorems
32S65 Singularities of holomorphic vector fields and foliations
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References:
[1] Araujo, On Fano foliations, Adv. Math. 238 pp 70– (2013) · Zbl 1282.14085 · doi:10.1016/j.aim.2013.02.003
[2] Beltrametti, The adjunction theory of complex projective varieties (1995) · Zbl 0845.14003 · doi:10.1515/9783110871746
[3] Camacho, Invariant varieties through singularities of holomorphic vector fields, Ann. of Math. 115 ((2)) pp 579– (1982) · Zbl 0503.32007 · doi:10.2307/2007013
[4] Flenner, Extendability of differential forms on nonisolated singularities, Invent. Math. 94 pp 317– (1988) · Zbl 0658.14009 · doi:10.1007/BF01394328
[5] P. Graf An optimal extension theorem for 1-forms and the Lipman-Zariski conjecture arXiv:1301.7315
[6] Greb, Extension theorems for differential forms and Bogomolov-Sommese vanishing on log canonical varieties, Compos. Math. 146 pp 193– (2010) · Zbl 1194.14056 · doi:10.1112/S0010437X09004321
[7] Greb, Differential forms on log canonical spaces, Publ. Math. Inst. Hautes Études Sci. 114 pp 87– (2011) · Zbl 1258.14021 · doi:10.1007/s10240-011-0036-0
[8] Hochster, The Zariski-Lipman conjecture for homogeneous complete intersections, Proc. Amer. Math. Soc. 49 pp 261– (1975) · Zbl 0311.13007
[9] Källström, The Zariski-Lipman conjecture for complete intersections, J. Algebra 337 pp 169– (2011) · Zbl 1250.14003 · doi:10.1016/j.jalgebra.2011.05.003
[10] Kebekus, Families of canonically polarized varieties over surfaces, Invent. Math. 172 pp 657– (2008) · Zbl 1140.14031 · doi:10.1007/s00222-008-0128-8
[11] Kollár, Lectures on resolution of singularities (2007)
[12] Kollár, Birational geometry of algebraic varieties (1998) · doi:10.1017/CBO9780511662560
[13] Lipman, Free derivation modules on algebraic varieties, Amer. J. Math. 87 pp 874– (1965) · Zbl 0146.17301 · doi:10.2307/2373252
[14] Matsuki, Introduction to the Mori program (2002) · Zbl 0988.14007 · doi:10.1007/978-1-4757-5602-9
[15] Scheja, Differentielle Eigenschaften der Lokalisierungen analytischer Algebren, Math. Ann. 197 pp 137– (1972) · Zbl 0223.14002 · doi:10.1007/BF01419591
[16] Suwa, Indices of holomorphic vector fields relative to invariant curves on surfaces, Proc. Amer. Math. Soc. 123 pp 2989– (1995) · Zbl 0866.32016 · doi:10.1090/S0002-9939-1995-1291793-0
[17] van Straten, Extendability of holomorphic differential forms near isolated hypersurface singularities, Abh. Math. Sem. Univ. Hamburg 55 pp 97– (1985) · Zbl 0584.32018 · doi:10.1007/BF02941491
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