On holomorphically separable complex solv-manifolds
Annales de l'Institut Fourier, Volume 36 (1986) no. 3, p. 57-65

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.

Soit G un groupe de Lie complexe résoluble et H un sous-groupe complexe fermé de G. Si les fonctions holomorphes sur la variété complexe X:=G/H séparent localement les points de X, alors X est une variété de Stein. De plus, il existe un sous-groupe H ^ d’indice fini dans H avec π 1 (G/H) nilpotent. Dans des cas particuliers (par exemple si H est discret), H normalise H ^ et H/H ^ est abélien.

@article{AIF_1986__36_3_57_0,
     author = {Huckleberry, Alan T. and Oeljeklaus, E.},
     title = {On holomorphically separable complex solv-manifolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {36},
     number = {3},
     year = {1986},
     pages = {57-65},
     doi = {10.5802/aif.1059},
     zbl = {0571.32012},
     mrnumber = {88b:32069},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1986__36_3_57_0}
}
Huckleberry, Alan T.; Oeljeklaus, E. On holomorphically separable complex solv-manifolds. Annales de l'Institut Fourier, Volume 36 (1986) no. 3, pp. 57-65. doi : 10.5802/aif.1059. http://www.numdam.org/item/AIF_1986__36_3_57_0/

[1] A. Borel, Linear algebraic groups, Benjamin, New York, 1969. | MR 40 #4273 | 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. | MR 87c:32033 | Zbl 0585.32030

[3] B. Gilligan, A.T. Huckleberry, On non-compact complex nilmanifolds, Math. Ann., 238 (1978), 39-49. | MR 80a:32021 | Zbl 0405.32009

[4] H. Grauert, Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann., 135 (1958), 263-273. | MR 20 #4661 | 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. | MR 34 #287 | 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. | MR 84i:32045 | 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. | Numdam | MR 87c:32035 | Zbl 0563.32013

[8] Y. Matsushima, Espaces homogènes de Stein des groupes de Lie complexes I, Nagoya Math. J., 16 (1960), 205-218. | MR 22 #739 | 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. | Numdam | MR 23 #A1061 | Zbl 0094.28104

[10] G.D. Mostow, Factor spaces of solvable groups, Ann. of Math., 60 (1954), 1-27. | MR 15,853g | Zbl 0057.26103

[11] D. Snow, Stein quotients of connected complex Lie groups, Manuskripta Math., 50 (1985), 185-214. | MR 86m:32050 | Zbl 0582.32020

[12] K. Stein, Überlagerungen holomorph-vollständiger komplexer Räume, Arch. Math., 7 (1956), 354-361. | MR 18,933a | Zbl 0072.08002

[13] V. Varadarajan, Lie groups, Lie algebras, and their representations, Prentice Hall, Englewood Cliffs, 1974. | Zbl 0371.22001