×

On holomorphically separable complex solv-manifolds. (English) Zbl 0571.32012

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 \(\tilde H\) of finite index in H with \(\pi_ 1(G/\tilde H)\) nilpotent. In special situations (e.g. if H is discrete) H normalizes \(\tilde H\) and \(H/\tilde H\) is abelian.

MSC:

32E10 Stein spaces
32M10 Homogeneous complex manifolds
32M05 Complex Lie groups, group actions on complex spaces
PDF BibTeX XML Cite
Full Text: DOI Numdam Numdam EuDML

References:

[1] A. BOREL, Linear algebraic groups, Benjamin, New York, 1969. · Zbl 0186.33201
[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. · Zbl 0585.32030
[3] B. GILLIGAN, A.T. HUCKLEBERRY, On non-compact complex nilmanifolds, Math. Ann., 238 (1978), 39-49. · Zbl 0405.32009
[4] H. GRAUERT, Analytische faserungen über holomorph-vollständigen Räumen, Math. Ann., 135 (1958), 263-273. · Zbl 0081.07401
[5] G. HOCHSCHILD, G.D. MOSTOW, On the algebra of representative functions of an analytic group, II, Am. J. Math., 86 (1964), 869-887. · Zbl 0152.01301
[6] A.T. HUCKLEBERRY, E. OELJEKLAUS, Homogeneous spaces from a complex analytic viewpoint, Progress in Mathematics, Birkhäuser Vol. 14 (1981), 159-186. · Zbl 0527.32020
[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. · Zbl 0563.32013
[8] Y. MATSUSHIMA, Espaces homogènes de Stein des groupes de Lie complexes I, Nagoya Math. J., 16 (1960), 205-218. · Zbl 0094.28201
[9] Y. MATSUSHIMA, A. MORIMOTO, Sur certains espaces fibrés holomorphes sur une variété de Stein, Bull. Soc. Math. France, 88 (1960), 137-155. · Zbl 0094.28104
[10] G.D. MOSTOW, Factor spaces of solvable groups, Ann. of Math., 60 (1954), 1-27. · Zbl 0057.26103
[11] D. SNOW, Stein quotients of connected complex Lie groups, Manuskripta Math., 50 (1985), 185-214. · Zbl 0582.32020
[12] K. STEIN, Überlagerungen holomorph-vollständiger komplexer Räume, Arch. Math., 7 (1956), 354-361. · Zbl 0072.08002
[13] V. VARADARAJAN, Lie groups, Lie algebras, and their representations, Prentice Hall, Englewood Cliffs, 1974. · Zbl 0371.22001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.