Abstract
Let X be a connected compact complex manifold. We call X a homogeneous manifold if there exists a complex Liegroup G of biholomorphic automorphisms of X such that for any two points p, q∈X there is a g∈G with: g(p)=q. We give some non-trivial examples of homogeneous manifolds. All examples are non-kaehlerian manifolds. Such examples have been given first by Iwasawa (as far as we know). The last section of this note contains a proposition, which shows that only very special non-semisimple Liegroups are able to act transitivily in the sense stated above, on a compact complex manifold. This is remarkable also in contrast to the semisimple case (cf. [3]).
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Literatur
AUSLANDER, L.: On radicals of discrete subgroups of Liegroups. Am. J. Math. 85, 145–150 (1963)
BARTH, W. u. M. OTTE: Über fastuniforme Untergruppen komplexer Liegruppen und auflösbare komplexe Mannigfaltigkeiten. Comm. Math. Helv. 44, 269–281 (1969)
BOREL, A.: Compact Clifford-Klein forms of symmetric spaces. Topology 2, 111–122 (1963)
BOREL, A.: Introduction aux groups arithmétiques. Paris 1969
BOREL, A. u. HARISH-CHANDRA: Arithmetic subgroups of algebraic groups. Ann. of Math. 75, 485–535 (1962)
BOREL, A. u. J. TITS: Groupes réductifs. J.H.E.S. Publ. Math. 27, 55–152 (1965)
BOREL, A. u. R. REMMERT: Über kompakte homogene Kählersche Mannigfaltigkeiten. Math. Ann. 145, 429–439 (1962)
BOREWICS, S. u. J. SAFAREVIC: Zahlentheorie. Basel 1966
FREUDENTHAL, H. u. H. de VRIES: Linear Lie groups. New-York 1969
HARDER, G.: Bericht über neuere Resultate der Galoiskohomologie halbeinfacher Gruppen. Jahresbericht d. DMV 70, 182–216 (1968)
MOSTOW, G.: Factor spaces of solvable groups. Ann. of Math. 60, 410–441 (1954)
SELBERG, A.: On discontinuous groups in higher dimensional spaces. Contr. to Function theory, Tata Inst. Bombay 1960, 147–164
TITS, J.: Espaces homogenes complexes compacts. Comm. Math. Helv. 37, 111–120 (1962)
TITS, J.: Classification of algebraic semi-simple groups. Proc. Symp. Pure Math. 9, 33–62 (1966)
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Otte, M., Potters, J. Beispiele homogener Mannigfaltigkeiten. Manuscripta Math 10, 117–127 (1973). https://doi.org/10.1007/BF01475037
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DOI: https://doi.org/10.1007/BF01475037