A smoothness criterion for complex spaces in terms of differential forms
DOI:
https://doi.org/10.7146/math.scand.a-119216Abstract
For a reduced pure dimensional complex space $X$, we show that if Barlet's recently introduced sheaf $\alpha _X^1$ of holomorphic $1$-forms or the sheaf of germs of weakly holomorphic $1$-forms is locally free, then $X$ is smooth. Moreover, we discuss the connection to Barlet's well-known sheaf $\omega _X^1$.
References
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Barlet, D., The sheaf $\alpha ^\bullet _X$, J. Singul. 18 (2018), 50–83. https://doi.org/10.5427/jsing.2018.18e
Flenner, H., Extendability of differential forms on nonisolated singularities, Invent. Math. 94 (1988), no. 2, 317–326. https://doi.org/10.1007/BF01394328
Greb, D., Kebekus, S., and Kovács, S. J., Extension theorems for differential forms and Bogomolov-Sommese vanishing on log canonical varieties, Compos. Math. 146 (2010), no. 1, 193–219. https://doi.org/10.1112/S0010437X09004321
Greb, D., Kebekus, S., Kovács, S. J., and Peternell, T., Differential forms on log canonical spaces, Publ. Math. Inst. Hautes Études Sci. (2011), no. 114, 87–169. https://doi.org/10.1007/s10240-011-0036-0
Griffiths, P. A., Variations on a theorem of Abel, Invent. Math. 35 (1976), 321–390. https://doi.org/10.1007/BF01390145
Kersken, M., Ein Regularitätskriterium für analytische Algebren, Arch. Math. (Basel) 51 (1988), no. 5, 434–439. https://doi.org/10.1007/BF01198627
Kollár, J., Lectures on resolution of singularities, Annals of Mathematics Studies, vol. 166, Princeton University Press, Princeton, NJ, 2007.
Lipman, J., Free derivation modules on algebraic varieties, Amer. J. Math. 87 (1965), 874–898. https://doi.org/10.2307/2373252
Pinkham, H. C., Singularités rationnelles de surfaces, in “Séminaire sur les Singularités des Surfaces: l'École Polytechnique, Palaiseau, 1976–1977” (Demazure, M., Pinkham, H. C., and Teissier, B., eds.), Lecture Notes in Mathematics, vol. 777, Springer, Berlin, 1980, pp. 147–178.
Riemenschneider, O., Characterizing Moišezon spaces by almost positive coherent analytic sheaves, Math. Z. 123 (1971), 263–284. https://doi.org/10.1007/BF01114795
Rossi, H., Picard variety of an isolated singular point, Rice Univ. Stud. 54 (1968), no. 4, 63–73.
Sera, M. L., A generalization of Takegoshi's relative vanishing theorem, J. Geom. Anal. 26 (2016), no. 3, 1891–1912. https://doi.org/10.1007/s12220-015-9612-8
van Straten, D. and Steenbrink, J., Extendability of holomorphic differential forms near isolated hypersurface singularities, Abh. Math. Sem. Univ. Hamburg 55 (1985), 97–110. https://doi.org/10.1007/BF02941491
Wahl, J. M., Vanishing theorems for resolutions of surface singularities, Invent. Math. 31 (1975), no. 1, 17–41. https://doi.org/10.1007/BF01389864
Yau, S. S. T., Various numerical invariants for isolated singularities, Amer. J. Math. 104 (1982), no. 5, 1063–1100. https://doi.org/10.2307/2374084
Barlet, D., The sheaf $\alpha ^\bullet _X$, J. Singul. 18 (2018), 50–83. https://doi.org/10.5427/jsing.2018.18e
Flenner, H., Extendability of differential forms on nonisolated singularities, Invent. Math. 94 (1988), no. 2, 317–326. https://doi.org/10.1007/BF01394328
Greb, D., Kebekus, S., and Kovács, S. J., Extension theorems for differential forms and Bogomolov-Sommese vanishing on log canonical varieties, Compos. Math. 146 (2010), no. 1, 193–219. https://doi.org/10.1112/S0010437X09004321
Greb, D., Kebekus, S., Kovács, S. J., and Peternell, T., Differential forms on log canonical spaces, Publ. Math. Inst. Hautes Études Sci. (2011), no. 114, 87–169. https://doi.org/10.1007/s10240-011-0036-0
Griffiths, P. A., Variations on a theorem of Abel, Invent. Math. 35 (1976), 321–390. https://doi.org/10.1007/BF01390145
Kersken, M., Ein Regularitätskriterium für analytische Algebren, Arch. Math. (Basel) 51 (1988), no. 5, 434–439. https://doi.org/10.1007/BF01198627
Kollár, J., Lectures on resolution of singularities, Annals of Mathematics Studies, vol. 166, Princeton University Press, Princeton, NJ, 2007.
Lipman, J., Free derivation modules on algebraic varieties, Amer. J. Math. 87 (1965), 874–898. https://doi.org/10.2307/2373252
Pinkham, H. C., Singularités rationnelles de surfaces, in “Séminaire sur les Singularités des Surfaces: l'École Polytechnique, Palaiseau, 1976–1977” (Demazure, M., Pinkham, H. C., and Teissier, B., eds.), Lecture Notes in Mathematics, vol. 777, Springer, Berlin, 1980, pp. 147–178.
Riemenschneider, O., Characterizing Moišezon spaces by almost positive coherent analytic sheaves, Math. Z. 123 (1971), 263–284. https://doi.org/10.1007/BF01114795
Rossi, H., Picard variety of an isolated singular point, Rice Univ. Stud. 54 (1968), no. 4, 63–73.
Sera, M. L., A generalization of Takegoshi's relative vanishing theorem, J. Geom. Anal. 26 (2016), no. 3, 1891–1912. https://doi.org/10.1007/s12220-015-9612-8
van Straten, D. and Steenbrink, J., Extendability of holomorphic differential forms near isolated hypersurface singularities, Abh. Math. Sem. Univ. Hamburg 55 (1985), 97–110. https://doi.org/10.1007/BF02941491
Wahl, J. M., Vanishing theorems for resolutions of surface singularities, Invent. Math. 31 (1975), no. 1, 17–41. https://doi.org/10.1007/BF01389864
Yau, S. S. T., Various numerical invariants for isolated singularities, Amer. J. Math. 104 (1982), no. 5, 1063–1100. https://doi.org/10.2307/2374084
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Published
2020-05-06
How to Cite
Samuelsson Kalm, H., & Sera, M. (2020). A smoothness criterion for complex spaces in terms of differential forms. MATHEMATICA SCANDINAVICA, 126(2), 221–228. https://doi.org/10.7146/math.scand.a-119216
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