Complexifying Lie group actions on homogeneous manifolds of non-compact dimension two

Abstract

If $X$ is a connected complex manifold with $d_X = 2$ that admits the holomorphic and transitive action of a (connected) Lie group $G$, then the action extends to an action of the complexification $\hat{G}$ of $G$ on $X$ except when either the unit disk or else a strictly pseudoconcave homogeneous complex manifold is involved as base or fiber in some homogeneous fibration of $X$.