Homogeneous Complex Manifolds with more than One End

Abstract

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].