Two remarks on Kähler homogeneous manifolds

Bruce Gilligan[1]; Karl Oeljeklaus[2]

  • [1] Dept. of Mathematics and Statistics, University of Regina, Regina, Canada S4S 0A2
  • [2] Centre de Mathématiques et d’Informatique, CNRS-UMR 6632 (LATP), 39, rue Joliot-Curie, Université de Provence, 13453 Marseille Cedex 13 France

Annales de la faculté des sciences de Toulouse Mathématiques (2008)

  • Volume: 17, Issue: 1, page 73-80
  • ISSN: 0240-2963

Abstract

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We prove that every Kähler solvmanifold has a finite covering whose holomorphic reduction is a principal bundle. An example is given that illustrates the necessity, in general, of passing to a proper covering. We also answer a stronger version of a question posed by Akhiezer for homogeneous spaces of nonsolvable algebraic groups in the case where the isotropy has the property that its intersection with the radical is Zariski dense in the radical.

How to cite

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Gilligan, Bruce, and Oeljeklaus, Karl. "Two remarks on Kähler homogeneous manifolds." Annales de la faculté des sciences de Toulouse Mathématiques 17.1 (2008): 73-80. <http://eudml.org/doc/10082>.

@article{Gilligan2008,
abstract = {We prove that every Kähler solvmanifold has a finite covering whose holomorphic reduction is a principal bundle. An example is given that illustrates the necessity, in general, of passing to a proper covering. We also answer a stronger version of a question posed by Akhiezer for homogeneous spaces of nonsolvable algebraic groups in the case where the isotropy has the property that its intersection with the radical is Zariski dense in the radical.},
affiliation = {Dept. of Mathematics and Statistics, University of Regina, Regina, Canada S4S 0A2; Centre de Mathématiques et d’Informatique, CNRS-UMR 6632 (LATP), 39, rue Joliot-Curie, Université de Provence, 13453 Marseille Cedex 13 France},
author = {Gilligan, Bruce, Oeljeklaus, Karl},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {complex Lie group; Kähler manifold; homogeneous space; solvmanifold},
language = {eng},
month = {6},
number = {1},
pages = {73-80},
publisher = {Université Paul Sabatier, Toulouse},
title = {Two remarks on Kähler homogeneous manifolds},
url = {http://eudml.org/doc/10082},
volume = {17},
year = {2008},
}

TY - JOUR
AU - Gilligan, Bruce
AU - Oeljeklaus, Karl
TI - Two remarks on Kähler homogeneous manifolds
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2008/6//
PB - Université Paul Sabatier, Toulouse
VL - 17
IS - 1
SP - 73
EP - 80
AB - We prove that every Kähler solvmanifold has a finite covering whose holomorphic reduction is a principal bundle. An example is given that illustrates the necessity, in general, of passing to a proper covering. We also answer a stronger version of a question posed by Akhiezer for homogeneous spaces of nonsolvable algebraic groups in the case where the isotropy has the property that its intersection with the radical is Zariski dense in the radical.
LA - eng
KW - complex Lie group; Kähler manifold; homogeneous space; solvmanifold
UR - http://eudml.org/doc/10082
ER -

References

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  1. Akhiezer (D.).— Invariant analytic hypersurfaces in complex nilpotent Lie groups, Ann. Global Anal. Geom., 2, p. 129-140 (1984). Zbl0576.32039MR777904
  2. Akhiezer (D.).— Invariant meromorphic functions on complex semisimple Lie groups, Invent. Math.65, p. 325-329 (1982). Zbl0479.32010MR643557
  3. Berteloot (F.).— Existence d’une structure kählérienne sur les variétés homogènes semi-simples, C.R. Acad. Sci. Paris, Sér. I, 305, p. 809-812 (1987). Zbl0635.32019
  4. Borel (A.) and Remmert (R.).— Über kompakte homogene Kählersche Mannigfaltigkeiten, Math. Ann.145, p. 429-439 (1962). Zbl0111.18001MR145557
  5. Gilligan (B.) and Huckleberry (A. T.).— On Non-Compact Complex Nil-Manifolds, Math. Ann.238, p. 39-49 (1978). Zbl0405.32009MR510305
  6. Hochschild (G.) and Mostow (G.D.).— On the algebra of representative functions of an analytic group. II, Amer. J. Math., 86, p. 869-887 (1964). Zbl0152.01301MR200392
  7. Huckleberry (A.) and Oeljeklaus (E.).— On holomorphically separable complex solvmanifolds, Ann. Inst. Fourier (Grenoble)36, p. 57-65 (1986). Zbl0571.32012MR865660
  8. Loeb (J-J).— Fonctions plurisousharmoniques sur un groupe de Lie complexe invariantes par une forme réelle, C.R. Acad. Sci. Paris, Sér. I, 299, p. 663-666 (1984). Zbl0616.31006MR770458
  9. Matsushima (Y.).— Espaces homogènes de Stein des groupes de Lie complexes, Nagoya Math. J.16, p. 205-218 (1960). Zbl0094.28201MR109854
  10. Onishchik (A. L.).— Complex envelopes of compact homogeneous spaces, Dokl. Acad. Nauk SSSR130, p. 726-729 (1960). Zbl0090.09401
  11. Oeljeklaus (K.) and Richthofer (W.).— On the Structure of Complex Solvmanifolds, J. Diff. Geom.27, p. 399-421 (1988). Zbl0619.32021MR940112
  12. Oeljeklaus (K.) and Richthofer (W.).— Recent results on homogeneous complex manifolds. Complex Analysis III, (College Park, Md., 1985-86), p. 78-119, Lecture Notes in Math. 1277, Springer, Berlin, 1987. Zbl0627.32026MR922335

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