Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-25T10:34:28.407Z Has data issue: false hasContentIssue false
  • English
  • Français

On completeness of holomorphic principal bundles

Published online by Cambridge University Press:  22 January 2016

Shigeru Takeuchi
Shigeru Takeuchi*
Affiliation:
Gifu 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.

In this paper we shall investigate the structure of complex Lie groups from function theoretical points of view. A. Morimoto proved in [10] that every connected complex Lie group G has the smallest closed normal connected complex Lie subgroup Ge, such that the factor group G/Ge is Stein. On the other hand there hold the following two basic structure theorems (A1) and (A2) for a connected algebraic group G (cf. [12]). (A1): G has the smallest normal algebraic subgroup N such that the factor group G/N is an affine algebraic group. Moreover N is a connected central subgroup. (A2): G has the unique maximal connected affine algebraic subgroup L, where L is normal and the factor group G/L is an abelian variety.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 56 , January 1975 , pp. 121 - 138
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1975

References

[1] Andreotti, A., and Grauert, H., “Théorèmes de finitude pour la cohomologie des espaces complexes,” Bull. Soc. Math. France 90 (1962), 139–259.Google Scholar
[2] Bourbaki, N., Groupes et algebres de Lie (Chap. 1), Hermann, Paris, 1961.Google Scholar
[3] Iwasawa, K., “On some types of topological groups,” Ann. Math. 50 (1949), 507–558.CrossRefGoogle Scholar
[4] Kazama, H., “On pseudo convexity of complex abelian Lie groups,” J. Math. Soc. Japan 25 (1973), 329–333.Google Scholar
[5] Kazama, H., “On pseudoconvexity of complex Lie groups,” Mem. Fac. Sci. Kyushu Univ. 27 (1973), 241–247.Google Scholar
[6] Kopfermann, K., “Maximale Untergruppen Abelscher komplexer Liescher Gruppen,” Schr. Math. Inst. Univ. Münster 39 (1964).Google Scholar
[7] Matsushima, Y., “Espaces homogènes de Stein des groupes de Lie complexes,” Nagoya Math. J. 16 (1960), 205–218.Google Scholar
[8] Matsushima, Y., and Morimoto, A., “Sur certains espaces fibres holomorphes sur une variété de Stein,” Bull. Soc. Math. France 88 (1960), 137–155.Google Scholar
[9] Morimoto, A., “On various complex Lie groups,” Sugaku 15 (1963-1964), 202-214. (in Japanese)Google Scholar
[10] Morimoto, A., “Non-compact complex Lie groups without non-constant holomorphic functions,” Proc. Conf. Complex Analysis at Univ. Minn., Springer, Berlin, 1965; pp. 257–272.Google Scholar
[11] Narasimhan, R., “The Levi problems for complex spaces II,” Math. Ann. 146 (1962), 195–216.CrossRefGoogle Scholar
[12] Rosenlicht, M., “Some basic theorems on algebraic groups,” Amer. J. Math. 78 (1956), 401–443.Google Scholar
[13] Vesentini, E., On Levi convexity of complex manifolds and cohomology vanishing theorem. Tata Institute Lecture Notes, 1967.Google Scholar
[14] Villani, V., “Su alcune proprietà coomologiche dei fasci coerenti su uno spazio complesso,” Rend. Sem. Mat. Universita di Padova 35 (1964), 47–55.Google Scholar
[15] Villani, V., “Fibrati vettoriali olomorfi su una varietà complessa q-completa,” Ann. Scoula Norm. Sup. Pisa S. 3 vol. 20 (1966), 15–23.Google Scholar