Globalization of Holomorphic Actions on Principal Bundles

Suppose G is a Lie group acting as a group of holomorphic automorphisms on a holomorphic principal bundle P → X. We show that if there is a holomorphic action of the complexification G C of G on. X , this lifts to a holomorphic action of G C on the bundle P → X. Two applications are presented. We prove that given any connected homogeneous complex manifold G/H with more than one end, the complexification G C 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.