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.


32E10 Stein spaces
32M10 Homogeneous complex manifolds
32M05 Complex Lie groups, group actions on complex spaces
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[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
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