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Homogeneous Complex Manifolds with more than One End

Published online by Cambridge University Press:  20 November 2018

B. Gilligan ,
K. Oeljeklaus  and
W. Richthofer
B. Gilligan
Affiliation:
University of Regina, Regina, Saskatchewan
K. Oeljeklaus
Affiliation:
University of Regina, Regina, Saskatchewan
W. Richthofer
Affiliation:
Ruhr-Universität Bochum, Bochum, F.R.G.
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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].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 41 , Issue 1 , 01 February 1989 , pp. 163 - 177
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Abe, Y., (H, C)-groups with positive line bundles, Nagoya Math. Journal 107 (1987), 1–11.Google Scholar
2. Abe, Y., Holomorphic sections of line bundles over (H, C)-groups, Manu. math. 60 (1988), 379–385.Google Scholar
3. Ahiezer, D., Invariant analytic hypersurfaces in complex nilpotent Lie groups, Ann. Glob. Analysis and Geometry 2 (1984), 129–140.Google Scholar
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. Blanchard, A., Sur les variétés analytiques complexes, Ann. Sci. Ecole Norm. Sup. 73 (1956), 157–202.Google Scholar
6. Borel, A., Les bouts des espaces homogènes de groupes de Lie, Ann. of Math. 58 (1953), 443–457.Google Scholar
7. Borel, A. and Remmert, R., Ueber kompakte homogène Kählersche Mannigfaltigkeiten, Math. Ann. 145 (1962), 429–439.Google Scholar
8. Freudenthal, H., Ueber die Enden topologischer Räume und Gruppen, Math. Z. 33 (1931), 692–713.Google Scholar
9. Gilligan, B., Ends of complex homogeneous manifolds having nonconstant holomorphic functions, Arch. Math. 37 (1981), 544–555.Google Scholar
10. Gilligan, B. and Huckleberry, A., Complex homogeneous manifolds with two ends, Mich. J. Math. 25(1981), 183–196.Google Scholar
11. Grauert, H. and Remmert, R., Ueber kompakte homogène komplexe Mannigfaltigkeiten, Arch. Math. 13 (1962), 498–507.Google Scholar
12. Huckleberry, A.T. and Margulis, G.A., Invariant analytic hypersurfaces, Inv. Math. 71 (1983), 235–240.Google Scholar
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. 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. Kodaira, K., On Kàhler varieties of restricted type (an intrinsic characterisation of algebraic varieties), Ann. Math. 60 (1954), 28–48.Google Scholar
16. Margulis, G.A., Discrete subgroups of real semisimple Lie groups, Mat. Sb. 80 (1969), 600–615.Google Scholar
17. Matsushima, Y., Sur les espaces homogènes kählériens d'un groupe de Lie reductive, Nagoya Math. J. 11 (1957), 53–60.Google Scholar
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. Oeljeklaus, K. and Richthofer, W., Homogeneous complex surfaces, Math. Ann. 268 (1984), 273–292.Google Scholar
20. Oeljeklaus, K. and Richthofer, W., On the structure of complex solvmanifolds, J. Diff. Geom. 27 (1988), 399–421.Google Scholar
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. Richthofer, W., Homogene CR-Mannigfaltigkeiten, Dissertation, Bochum (1985).Google Scholar
23. Richthofer, W., Currents in homogeneous manifolds, to appear.Google Scholar
24. Winkelmann, J., The classification of three-dimensional homogeneous complex manifolds, Dissertation, Bochum (1987).Google Scholar