On the holomorphic separability of discrete quotients of complex Lie groups

Abstract

Let G be a complex Lie group and H ~ G be a closed complex subgroup of G. We refer to the quotient X = G/H as a complex-homogeneous manifold. It is natural to ask under which conditions a complex homogeneous manifold X is Stein. First we give some known results. Matsushima [M] proved that if G is reductive, then X = G/H is a Stein manifold, if and only if the group H is reductive. In particular if dime H = 0, then G/H is Stein if and only if H is a finite group. Barth and Otte [BO] showed that for G reductive a complex homogeneous manifold G/H being holomorphically separable implies that H is algebraic in G. As the following example shows, the condition of holomorphical separability is not sufficient for Steinness: Let G = SL2(C),