Homogeneous Complex Manifolds with more than One End

Gilligan, B. ; Oeljeklaus, K. ; Richthofer, W.

Canadian journal of mathematics, Tome 41 (1989) no. 1, pp. 163-177

Résumé

For homogeneous spaces of a (real) Lie group one of the fundamental results concerning ends (in the sense of Freudenthal [8] ) is due to A. Borel [6]. He showed that if X = G/H is the homogeneous space of a connected Lie group G by a closed connected subgroup H, then X has at most two ends. And if X does have two ends, then it is diffeomorphic to the product of R with the orbit of a maximal compact subgroup of G.In the setting of homogeneous complex manifolds the basic idea should be to find conditions which imply that the space has at most two ends and then, when the space has exactly two ends, to display the ends via bundles involving C* and compact homogeneous complex manifolds. An analytic condition which ensures that a homogeneous complex manifold X has at most two ends is that X have non-constant holomorphic functions and the structure of such a space with exactly two ends is determined, namely, it fibers over an affine homogeneous cone with its vertex removed with the fiber being compact [9], [13].

Détail

DOI : 10.4153/CJM-1989-008-4

Bibliographie

[1] 1. Abe, Y., (H, C)-groups with positive line bundles, Nagoya Math. Journal 107 (1987), 1–11. Google Scholar

[2] 2. Abe, Y., Holomorphic sections of line bundles over (H, C)-groups, Manu. math. 60 (1988), 379–385. Google Scholar

[3] 3. Ahiezer, D., Invariant analytic hypersurfaces in complex nilpotent Lie groups, Ann. Glob. Analysis and Geometry 2 (1984), 129–140. Google Scholar

[4] 4. Berteloot, F. and Oeljeklaus, K., Invariant plurisubharmonic functions and hypersurfaces on semi-simple complex Lie groups, Math. Ann. 281 (1988), 513–530. Google Scholar

[5] 5. Blanchard, A., Sur les variétés analytiques complexes, Ann. Sci. Ecole Norm. Sup. 73 (1956), 157–202. Google Scholar

[6] 6. Borel, A., Les bouts des espaces homogènes de groupes de Lie, Ann. of Math. 58 (1953), 443–457. Google Scholar

[7] 7. Borel, A. and Remmert, R., Ueber kompakte homogène Kählersche Mannigfaltigkeiten, Math. Ann. 145 (1962), 429–439. Google Scholar

[8] 8. Freudenthal, H., Ueber die Enden topologischer Räume und Gruppen, Math. Z. 33 (1931), 692–713. Google Scholar

[9] 9. Gilligan, B., Ends of complex homogeneous manifolds having nonconstant holomorphic functions, Arch. Math. 37 (1981), 544–555. Google Scholar

[10] 10. Gilligan, B. and Huckleberry, A., Complex homogeneous manifolds with two ends, Mich. J. Math. 25(1981), 183–196. Google Scholar

[11] 11. Grauert, H. and Remmert, R., Ueber kompakte homogène komplexe Mannigfaltigkeiten, Arch. Math. 13 (1962), 498–507. Google Scholar

[12] 12. Huckleberry, A.T. and Margulis, G.A., Invariant analytic hypersurfaces, Inv. Math. 71 (1983), 235–240. Google Scholar

[13] 13. Huckleberry, A.T. and Oeljeklaus, E., Classification theorems for almost homogeneous spaces, Institut Elie Cartan 9. Université de Nancy, Nancy (1984). Google Scholar

[14] 14. Huckleberry, A.T. and Richthofer, W., Recent developments in homogeneous CR-hypersurfaces, Contributions to several complex variables, 149–177. Aspects of Math. E9. Vieweg, Braunschweig (1986). Google Scholar

[15] 15. Kodaira, K., On Kàhler varieties of restricted type (an intrinsic characterisation of algebraic varieties), Ann. Math. 60 (1954), 28–48. Google Scholar

[16] 16. Margulis, G.A., Discrete subgroups of real semisimple Lie groups, Mat. Sb. 80 (1969), 600–615. Google Scholar

[17] 17. Matsushima, Y., Sur les espaces homogènes kählériens d'un groupe de Lie reductive, Nagoya Math. J. 11 (1957), 53–60. Google Scholar

[18] 18. Oeljeklaus, K., Hyperflàchen und Geradenbündel auf homogenen komplexen Mannigfaltigkeiten, Schriftenreihe des Mathematischen Instituts der Universität Münster, Ser. 2, Heft 36, Munster (1985). Google Scholar

[19] 19. Oeljeklaus, K. and Richthofer, W., Homogeneous complex surfaces, Math. Ann. 268 (1984), 273–292. Google Scholar

[20] 20. Oeljeklaus, K. and Richthofer, W., On the structure of complex solvmanifolds, J. Diff. Geom. 27 (1988), 399–421. Google Scholar

[21] 21. Oeljeklaus, K. and Richthofer, W., Recent results on homogeneous complex manifolds, Complex Analysis III, Proc. Spec. Year, College Park/Md. (1985–86., (Springer-Verlag, New York), 78–119. Google Scholar

[22] 22. Richthofer, W., Homogene CR-Mannigfaltigkeiten, Dissertation, Bochum (1985). Google Scholar

[23] 23. Richthofer, W., Currents in homogeneous manifolds, to appear. Google Scholar

[24] 24. Winkelmann, J., The classification of three-dimensional homogeneous complex manifolds, Dissertation, Bochum (1987). Google Scholar