Nagoya Mathematical Journal

Some remarks on complex Lie groups

H. Kazama, D. K. Kim, and C. Y. Oh

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Abstract

First we show that any complex Lie group is complete Kähler. Moreover we obtain a plurisubharmonic exhaustion function on a complex Lie group as follows. Let ${\frak k}$ the real Lie algebra of a maximal compact real Lie subgroup $K$ of a complex Lie group $G$. Put $q:=\dim_ {\Bbb C} {\frak k} \cap \sqrt{-1} {\frak k}$. Then we obtain that there exists a plurisubharmonic, strongly $(q + 1)$-pseudoconvex in the sense of Andreotti-Grauert and $K$-invariant exhaustion function on $G$.

Article information

Source
Nagoya Math. J., Volume 157 (2000), 47-57.

Dates
First available in Project Euclid: 27 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1114631342

Mathematical Reviews number (MathSciNet)
MR1752474

Zentralblatt MATH identifier
0957.32010

Subjects
Primary: 32M05: Complex Lie groups, automorphism groups acting on complex spaces [See also 22E10]
Secondary: 32U10: Plurisubharmonic exhaustion functions

Citation

Kazama, H.; Kim, D. K.; Oh, C. Y. Some remarks on complex Lie groups. Nagoya Math. J. 157 (2000), 47--57. https://projecteuclid.org/euclid.nmj/1114631342


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References

  • A. Andreotti and H. Grauert, Théorèmes de finitude pour la cohomologie des espaces complexes , Bull. Soc. Math. France, 99 , 193–259 (1962).
  • K. Iwasawa, On some types of topological groups , Annals of Math., 50 , 507–558 (1949).
  • H. Kazama, On pseudoconvexity of complex abelian Lie groups , J. Math. Soc. Japan, 25 , 329–333 (1973).
  • H. Kazama, On pseudoconvexity of complex Lie groups , Mem. Fac. Sci. Kyushu Univ., 27 , 241–247 (1973).
  • Y. Matsushima, Espaces homogènes de Stein des groupes de Lie complexes , Nagoya Math. J., 16 , 205–218 (1960).
  • Y. Matsushima and A. Morimoto, Sur certains espaces fibrés holomorphes sur une variété de Stein , Bull. Soc. Math. France, 88 , 137–155 (1960).
  • A. Morimoto, Non-compact complex Lie groups without non-constant holomorphic functions , Proc. Conf. on Complex Analysis, Minneapolis, 256–272 (1965).
  • A. Morimoto, On the classification of non-compact complex abelian Lie groups , Trans. Amer. Math. Soc., 123 , 200–228 (1966).
  • S. Nakano, On the inverse of monoidal transformation , Publ.RIMS, 6 (1970), 483–502.
  • S. Takeuchi, On completeness of holomorphic principal bundles , Nagoya Math. J., 57 (1974), 121–138.