A smoothness criterion for complex spaces in terms of differential forms

  • Håkan Samuelsson Kalm
  • Martin Sera


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 $\omega _X^1$.


Barlet, D., Le faisceau $\omega ^\cdot _X$ sur un espace analytique $X$ de dimension pure, in “Fonctions de plusieurs variables complexes, III (Sém. François Norguet, 1975–1977)”, Lecture Notes in Math., vol. 670, Springer, Berlin, 1978, pp. 187–204.

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
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