Summary
We prove that if X is a normal [reap. reduced, maximal] complex space and f: X→C is a holomorphic function, then f−1(c) is normal [resp. reduced, maximal] for all but countably many cεC. This Sard type theorem, together with a Bănică's result on the fibers of a flat map, allows us to prove Bertini type theorems for reduced and normal complex spaces.
Article PDF
Similar content being viewed by others
References
W.Adkins - A.Andbeotti - J. V.Leahy,Weakly normal complex spaces, Atti Acc. Naz. Lincei (1980).
A. Andreotti -F. Norguet,La convexité holomorphe dans l'espace analytique des cycles d'une variété algébrique, Ann. Sc. Norm. Sup. Pisa,21 (1967), pp. 31–82.
C.Bănică,Le lieu réduit et le lieu normal d'un morphisme, Proceedings of the Romanian-Firmish Seminar on Complex Analysis, Bucharest, 1976, Lect. Notes Math. 743, Springer-Verlag, 1979, pp. 389–398.
C. Bănică -M. Stoia,Gorenstein points of a flat morphism of complex spaces, Revue Roumaine Math. Pures Appl.,26, no. 5 (1981), pp. 687–690.
N. Bourbaki,Topologie générale. (Fascicule de résultats). Hermann, Paris, 1964.
C.Cumino - S.Greco - M.Manaresi,Bertini theorems for weak normality, to appear on Comp. Math.
G.Fischer,Complex Analytic Geometry, Lect. Notes Math. 538, Springer-Verlag, 1976.
H. Flenner,Die Sätze von Bertini für lokale Ringe, Math. Ann.,229 (1977), pp. 97–111.
J. Frisch,Points de platitude d'un morphisme d'espaces analytiques complexes, Inv. Math.,4 (1967), pp. 118–138.
W. E. Kuan,A note on a generic hyperplane section of an algebraic variety, Can. J. Math.,22, no. 5 (1970), pp. 1047–1053.
J. W.Milnor,Topology from the differentiale view-point, University Press of Virginia, 1965.
R. Remmert,Holomorphe und meromorphe Abbildungen komplexer Räume, Math. Ann.,133 (1957), pp. 328–370.
A. Seidenberg,The hyperplane sections of normal varieties, Trans. Amer. Math. Soc.,50 (1941), pp. 357–386.
Y. T. Siu,Noether-Lasker decomposition of coherent analytic subsheaves, Trans. Amer. Math. Soc.,135 (1969), pp. 375–385.
Y. T.Siu - G.Trautmann Gap sheaves and extensions of coherent analytic subsheaves Lect. Notes Math. 172, Springer-Verlag, 1971.
Author information
Authors and Affiliations
Additional information
This research was done when the author was a member of the G.N.S.A.G.A. of the C.N.R.
Rights and permissions
About this article
Cite this article
Manaresi, M. Sard and Bertini type theorems for complex spaces. Annali di Matematica pura ed applicata 131, 265–279 (1982). https://doi.org/10.1007/BF01765156
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01765156