Abstract
We consider a convexity notion for complex spaces X with respect to a holomorphic line bundle L over X. This definition has been introduced by Grauert and, when L is analytically trivial, we recover the standard holomorphic convexity. In this circle of ideas, we prove the counterpart of the classical Remmert’s reduction result for holomorphically convex spaces. In the same vein, we show that if H 0(X,L) separates each point of X, then X can be realized as a Riemann domain over the complex projective space Pn, where n is the complex dimension of X and L is the pull-back of O(1).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Andreotti: Théorèmes de dépendance algébrique sur les espaces complexes pseudoconcaves. Bull. Soc. Math. Fr. 91 (1963), 1–38. (In French.)
C. Bănică; O. Stănăşilă: Méthodes Algébriques dans la Théorie Globale des Espaces Complexes. Vol. 2. Traduit du Roumain. Collection “Varia Mathematica”, Gauthier-Villars, Paris, 1977. (In French.)
D. Barlet; A. Silva: Convexité holomorphe intermédiaire. Math. Ann. 296 (1993), 649–665. (In French. English summary.)
H. Cartan: Quotients of complex analytic spaces. Contrib. Function Theory. Int. Colloqu. Bombay, 1960, Tata Institute of Fundamental Research, Bombay, 1960, pp. 1–15.
H. Grauert: Bemerkenswerte pseudokonvexe Mannigfaltigkeiten. Math. Z. 81 (1963), 377–391. (In German.)
H. Grauert: Charakterisierung der holomorph vollständigen komplexen Räume. Math. Ann. 129 (1955), 233–259. (In German.)
B. Kaup: Über offene analytische Äquivalenzrelationen auf komplexen Räumen. Math. Ann. 183 (1969), 6–16. (In German.)
R. Remmert: Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes. C. R. Acad. Sci., Paris 243 (1956), 118–121. (In French.)
B. Shiffman: On the removal of singularities for analytic sets. Mich. Math. J. 15 (1968), 111–120.
Y.-T. Siu: Techniques of Extension of Analytic Objects. Lecture Notes in Pure and Applied Mathematics, Vol. 8, Marcel Dekker, New York, 1974.
T. Ueda: On the neighborhood of a compact complex curve with topologically trivial normal bundle. J. Math. Kyoto Univ. 22 (1983), 583–607.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Vâjâitu, V. Grauert’s line bundle convexity, reduction and Riemann domains. Czech Math J 66, 493–509 (2016). https://doi.org/10.1007/s10587-016-0271-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10587-016-0271-0