Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-11T10:01:10.128Z Has data issue: false hasContentIssue false
  • English
  • Français

On The Holomorphic Automorphism Groups of Complex Spaces

Published online by Cambridge University Press:  22 January 2016

Hirotaka Fujimoto
Hirotaka Fujimoto*
Affiliation:
Nagoya University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For a complex space X we consider the group Aut (X) of all automorphisms of X, where an automorphism means a holomorphic automorphism, i.e. an injective holomorphic mapping of X onto X itself with the holomorphic inverse. In 1935, H. Cartan showed that Aut (X) has a structure of a real Lie group if X is a bounded domain in CN([7]) and, in 1946, S. Bochner and D. Montgomery got the analogous result for a compact complex manifold X ([2] and [3]). Afterwards, the latter was generalized by R.C. Gunning ([11]) and H. Kerner ([16]), and the former by W. Kaup ([14]), to complex spaces. The purpose of this paper is to generalize these results to the case of complex spaces with weaker conditions. For brevity, we restrict ourselves to the study of σ-compact irreducible complex spaces only.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 33 , November 1968 , pp. 85 - 106
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1968

References

[1] Andreotti, A. and Grauert, H., Théorèmes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France, 90 (1962), 193–259.Google Scholar
[2] Bochner, S. and Montgomery, D., Locally compact groups of difierentiable transformations, Ann. Math., 47 (1946), 639–653.Google Scholar
[3] Bochner, S. and Montgomery, D., Groups on analytic manifolds, Ann. Math., 48 (1947), 659–669.Google Scholar
[4] Bourbaki, N., Fasc. X, Topologie générale, Chap. 10, Hermann, Paris, 1961.Google Scholar
[5] Carathéodory, C., Über die Abbildungen, die durch Systeme von analytischen Funktionen von mehreren Veränderlichen erzeugt werden, Math. Z., 34 (1931), 758–792.CrossRefGoogle Scholar
[6] Cartan, H., Sur les fonctions de plusieurs variables complexes. L’itération des transformations intérieures d’un domaine borné, Math. Z., 35 (1932), 760–773.CrossRefGoogle Scholar
[7] Cartan, H., Sur les groupes de transformations analytiques, Act. Sci. Ind., 198, Paris, 1935.Google Scholar
[8] Fujimoto, H., On the continuation of analytic sets, J. Math. Soc. Japan, 18 (1966), 51–85.CrossRefGoogle Scholar
[9] Grauert, H., Charakterisierung der holomorph-vollständigen komplexer Räume, Math. Ann., 129 (1955), 233–259.CrossRefGoogle Scholar
[10] Grauert, H. and Remmert, R., Komplexe Räume, Math. Ann., 136 (1958) 245–318.Google Scholar
[11] Gunning, R.C., On Vitali’s theorem for complex spaces with singularities, J. Math. Mech., 8 (1959), 133–141.Google Scholar
[12] Husain, T., Introduction to topological groups, W.B. Saunders Company, Philadelphia and London, 1966.Google Scholar
[13] Kasahara, K., On Hartogs-Osgood’s theorem for Stein spaces, J. Math. Soc. Japan, 17 (1965), 297–312.CrossRefGoogle Scholar
[14] Kaup, W., Reelle Transformationsgruppen und invariante Metriken auf komplexen Räumen, Inv. math., 3 (1967), 43–70.CrossRefGoogle Scholar
[15] Kerner, H., Über die Fortsetzung holomorpher Abbildungen, Arch. Math., 11 (1960), 44–49.Google Scholar
[16] Kerner, H., Über die Automorphismengruppen kompakter komplexer Räume, Arch. Math., 11 (1960), 282–288.CrossRefGoogle Scholar
[17] Kuranishi, M., On conditions of differentiability of locally compact groups, Nagoya Math. J., 1 (1950) 71–81.CrossRefGoogle Scholar
[18] Montgomery, D. and Zippin, L., Topological transformation groups, Interscience Publ., Inc., New York, 1955.Google Scholar
[19] Remmert, R., Holomorphe und meromorphe Abbildungen komplexer Raurne, Math. Ann., 133 (1957), 328–370.Google Scholar