On holomorphically separable complex solv-manifolds

Alan T. Huckleberry; E. Oeljeklaus

Annales de l'institut Fourier (1986)

  • Volume: 36, Issue: 3, page 57-65
  • ISSN: 0373-0956

Abstract

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Let G be a solvable complex Lie group and H a closed complex subgroup of G . If the global holomorphic functions of the complex manifold X : G / H locally separate points on X , then X is a Stein manifold. Moreover there is a subgroup H ^ of finite index in H with π 1 ( G / H ^ ) nilpotent. In special situations (e.g. if H is discrete) H normalizes H ^ and H / H ^ is abelian.

How to cite

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Huckleberry, Alan T., and Oeljeklaus, E.. "On holomorphically separable complex solv-manifolds." Annales de l'institut Fourier 36.3 (1986): 57-65. <http://eudml.org/doc/74728>.

@article{Huckleberry1986,
abstract = {Let $G$ be a solvable complex Lie group and $H$ a closed complex subgroup of $G$. If the global holomorphic functions of the complex manifold $X:G/H$ locally separate points on $X$, then $X$ is a Stein manifold. Moreover there is a subgroup $\widehat\{H\}$ of finite index in $H$ with $\pi _1(G/\widehat\{H\})$ nilpotent. In special situations (e.g. if $H$ is discrete) $H$ normalizes $\widehat\{H\}$ and $H/\widehat\{H\}$ is abelian.},
author = {Huckleberry, Alan T., Oeljeklaus, E.},
journal = {Annales de l'institut Fourier},
keywords = {solv-manifolds; Steinness of homogeneous space; solvable complex Lie group},
language = {eng},
number = {3},
pages = {57-65},
publisher = {Association des Annales de l'Institut Fourier},
title = {On holomorphically separable complex solv-manifolds},
url = {http://eudml.org/doc/74728},
volume = {36},
year = {1986},
}

TY - JOUR
AU - Huckleberry, Alan T.
AU - Oeljeklaus, E.
TI - On holomorphically separable complex solv-manifolds
JO - Annales de l'institut Fourier
PY - 1986
PB - Association des Annales de l'Institut Fourier
VL - 36
IS - 3
SP - 57
EP - 65
AB - Let $G$ be a solvable complex Lie group and $H$ a closed complex subgroup of $G$. If the global holomorphic functions of the complex manifold $X:G/H$ locally separate points on $X$, then $X$ is a Stein manifold. Moreover there is a subgroup $\widehat{H}$ of finite index in $H$ with $\pi _1(G/\widehat{H})$ nilpotent. In special situations (e.g. if $H$ is discrete) $H$ normalizes $\widehat{H}$ and $H/\widehat{H}$ is abelian.
LA - eng
KW - solv-manifolds; Steinness of homogeneous space; solvable complex Lie group
UR - http://eudml.org/doc/74728
ER -

References

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  1. [1] A. BOREL, Linear algebraic groups, Benjamin, New York, 1969. Zbl0186.33201MR40 #4273
  2. [2] G. COEURÉ, J. LOEB, A counterexample to the Serre problem with a bounded domain of C2 as fiber, Ann. of Math., 122 (1985), 329-334. Zbl0585.32030MR87c:32033
  3. [3] B. GILLIGAN, A.T. HUCKLEBERRY, On non-compact complex nilmanifolds, Math. Ann., 238 (1978), 39-49. Zbl0405.32009MR80a:32021
  4. [4] H. GRAUERT, Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann., 135 (1958), 263-273. Zbl0081.07401MR20 #4661
  5. [5] G. HOCHSCHILD, G.D. MOSTOW, On the algebra of representative functions of an analytic group, II, Am. J. Math., 86 (1964), 869-887. Zbl0152.01301MR34 #287
  6. [6] A.T. HUCKLEBERRY, E. OELJEKLAUS, Homogeneous spaces from a complex analytic viewpoint, Progress in Mathematics, Birkhäuser Vol. 14 (1981), 159-186. Zbl0527.32020MR84i:32045
  7. [7] J. LOEB, Actions d'une forme de Lie réelle d'un groupe de Lie complexe sur les fonctions plurisousharmoniques, Annales de l'Institut Fourier, 35-4 (1985), 59-97. Zbl0563.32013MR87c:32035
  8. [8] Y. MATSUSHIMA, Espaces homogènes de Stein des groupes de Lie complexes I, Nagoya Math. J., 16 (1960), 205-218. Zbl0094.28201MR22 #739
  9. [9] Y. MATSUSHIMA, A. MORIMOTO, Sur certains espaces fibrés holomorphes sur une variété de Stein, Bull. Soc. Math. France, 88 (1960), 137-155. Zbl0094.28104MR23 #A1061
  10. [10] G.D. MOSTOW, Factor spaces of solvable groups, Ann. of Math., 60 (1954), 1-27. Zbl0057.26103MR15,853g
  11. [11] D. SNOW, Stein quotients of connected complex Lie groups, Manuskripta Math., 50 (1985), 185-214. Zbl0582.32020MR86m:32050
  12. [12] K. STEIN, Überlagerungen holomorph-vollständiger komplexer Räume, Arch. Math., 7 (1956), 354-361. Zbl0072.08002MR18,933a
  13. [13] V. VARADARAJAN, Lie groups, Lie algebras, and their representations, Prentice Hall, Englewood Cliffs, 1974. Zbl0371.22001

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