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Globalization of Holomorphic Actions on Principal Bundles
Authors:Bruce Gilligan  Peter Heinzner
Abstract:Suppose G is a Lie group acting as a group of holomorphic automorphisms on a holomorphic principal bundle PX. We show that if there is a holomorphic action of the complexification GC of G on. X, this lifts to a holomorphic action of GC on the bundle PX. Two applications are presented. We prove that given any connected homogeneous complex manifold G/H with more than one end, the complexification GC of G acts holomorphically and transitively on G/H. We also show that the ends of a homogeneous complex manifold G/H with more than two ends essentially come from a space of the form S/Γ, where Γ is a Zariski dense discrete subgroup of a semisimple complex Lie group S with S and Γ being explicitly constructed in terms of G and H.
Keywords:Complexifications of Lie groups  principal bundles  ends of homogeneous complex manifolds
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