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Smoothable varieties with torsion free canonical sheaf. (English) Zbl 0683.14013

Let X be a d-dimensional smoothable Cohen-Macaulay projective variety. Consider the canonical map \(c:\quad \Omega^ d_ X\to \omega_ X.\) Let L be an ample line bundle on X. By a clever calculation involving \(\chi (\Omega^ d_ X\otimes L^ N)\) for \(N\gg 0\), the author shows that if c is injective, then c is an isomorphism and X is smooth by a theorem of Kunz and Waldi.
Reviewer: L.Ein

MSC:

14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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References:

[1] R. Bassein, On smoothable curve singularities: local methods, Math. Ann. 230 (1977) · Zbl 0368.32006
[2] R. Berger, Differentialmodulin eindimensionaler localer Ringe, Math. Zeit. 81 (1963)
[3] R. Buchweitz, G. Greuel, The Milnor number and deformations of complex curve singularities, Invent. Math. 58 (1980) · Zbl 0458.32014
[4] R. Hartshorne, Algebraic geometry, Springer-Verlag (1977) · Zbl 0367.14001
[5] R. Hübl, A note the torsion of differential forms, preprint
[6] E. Kunz, R. Waldi, Regular differential forms, preprint · Zbl 0658.13019
[7] J. Lipman, Dualizing sheaves, differentials and residues on algebraic varieties, Astérisque 117 (1984) · Zbl 0562.14003
[8] D. Mumford, Abelian varieties, Tata Institute (1970) · Zbl 0223.14022
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