Osaka Journal of Mathematics

Quotients of bounded homogeneous domains by cyclic groups

Christian Miebach

Full-text: Open access


Let $D$ be a bounded homogeneous domain in $\mathbb{C}^{n}$ and let $\varphi$ be an automorphism of $D$ which generates a discrete subgroup $\Gamma$ of $\Aut_{\mathcal{O}}(D)$. It is shown that the complex space $D/\Gamma$ is Stein.

Article information

Osaka J. Math. Volume 47, Number 2 (2010), 331-352.

First available in Project Euclid: 23 June 2010

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32M10: Homogeneous complex manifolds [See also 14M17, 57T15] 32E10: Stein spaces, Stein manifolds


Miebach, Christian. Quotients of bounded homogeneous domains by cyclic groups. Osaka J. Math. 47 (2010), no. 2, 331--352.

Export citation


  • D.N. Akhiezer: Lie Group Actions in Complex Analysis, Aspects of Mathematics, E27, Vieweg, Braunschweig, 1995.
  • A. Borel: Kählerian coset spaces of semisimple Lie groups, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 1147–1151.
  • H. Cartan: Sur les Groupes de Transformations Analytiques, (Exposés mathématiques IX.) Actual. scient. et industr. 198, Hermann, Paris, 1935.
  • C. De Fabritiis: A family of complex manifolds covered by $\Delta_{n}$, Complex Variables Theory Appl. 36 (1998), 233–252.
  • C. De Fabritiis and A. Iannuzzi: Quotients of the unit ball of $\mathbb{C}^{n}$ for a free action of $\mathbb{Z}$, J. Anal. Math. 85 (2001), 213–224.
  • F. Docquier and H. Grauert: Leisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten, Math. Ann. 140 (1960), 94–123.
  • H. Grauert and R. Remmert: Theory of Stein Spaces, Classics in Mathematics, Translated from the German by Alan Huckleberry, Reprint of the 1979 translation, Springer, Berlin, 2004.
  • P. Heinzner and A. Iannuzzi: Integration of local actions on holomorphic fiber spaces, Nagoya Math. J. 146 (1997), 31–53.
  • A.T. Huckleberry and E. Oeljeklaus: Homogeneous spaces from a complex analytic viewpoint; in Manifolds and Lie Groups (Notre Dame, Ind., 1980), Progr. Math. 14, Birkhäuser, Boston, Mass, 1981, 159–186.
  • G. Hochschild: The Structure of Lie Groups, Holden-Day, San Francisco, 1965.
  • A. Iannuzzi: Characterizations of $G$-tube domains, Manuscripta Math. 98 (1999), 425–445.
  • H. Ishi: On symplectic representations of normal $j$-algebras and their application to Xu's realizations of Siegel domains, Differential Geom. Appl. 24 (2006), 588–612.
  • A. Iannuzzi, A. Spiro and S. Trapani: Complexifications of holomorphic actions and the Bergman metric, Internat. J. Math. 15 (2004), 735–747.
  • S. Kaneyuki: On the automorphism groups of homogeneuous bounded domains, J. Fac. Sci. Univ. Tokyo Sect. I 14 (1967), 89–130.
  • S. Kaneyuki: Homogeneous Bounded Domains and Siegel Domains, Lecture Notes in Math. 241, Springer, Berlin, 1971.
  • W. Kaup, Y. Matsushima and T. Ochiai: On the automorphisms and equivalences of generalized Siegel domains, Amer. J. Math. 92 (1970), 475–498.
  • B. Kostant: On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. École Norm. Sup. (4) 6 (1973), 413–455.
  • Y. Matsushima and A. Morimoto: Sur certains espaces fibrés holomorphes sur une variété de Stein, Bull. Soc. Math. France 88 (1960), 137–155.
  • R.S. Palais: On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961), 295–323.
  • I.I. Pyateskii-Shapiro: Automorphic Functions and the Geometry of Classical Domains, Translated from the Russian. Mathematics and Its Applications 8, Gordon and Breach, New York, 1969.
  • H.L. Royden: Holomorphic fiber bundles with hyperbolic fiber, Proc. Amer. Math. Soc. 43 (1974), 311–312.
  • N. Steenrod: The Topology of Fibre Bundles, Princeton Mathematical Series 14 Princeton Univ. Press, Princeton, N.J., 1951.
  • K. Stein: Überlagerungen holomorph-vollständiger komplexer Räume, Arch. Math. (Basel) 7 (1956), 354–361.
  • È.B. Vinberg, S.G. Gindikin and I.I. Pjateckiĭ -Šapiro: Classification and canonical realization of complex homogeneous bounded domains, Trudy Moskov. Mat. Obšč. 12 (1963), 359–388.
  • È.B. Vinberg: The Morozov-Borel theorem for real Lie groups, Dokl. Akad. Nauk SSSR 141 (1961), 270–273.
  • È.B. Vinberg (ed.): Lie Groups and Lie Algebras, III, Encyclopaedia Math. Sci. 41, Springer, Berlin, 1994.
  • P.C. Yang: Geometry of tube domains; in Complex Analysis of Several Variables (Madison, Wis., 1982), Proc. Sympos. Pure Math. 41, Amer. Math. Soc. Providence, RI, 1984, 277–283.