On compact complex coset spaces of reductive Lie groups

Abstract

1. The statement of theorems. Let G be a connected complex Lie group and let B be a closed complex Lie subgroup in G. The left coset space G/B is a complex manifold, which will be called a complex coset space. We denote by Bo the identity connected component of B and U the normalizer of Bo. The canonical projection p of G/B onto G/ U defines a holomorphic fibre bundle, and the complex Lie group U/Bo acts on G/B as the structure group. We denote by (G/B, p, G/ U) this holomorphic fibre bundle. Suppose that the complex coset space G/B is compact. Then, by a recent result of Borel-Remmert [1, Satz 7'], it turns out that the base space G/ U is a Kaehler C-space, that is, a simply connected compact complex coset space admitting a Kaehler metric, such that the group of isometries is transitive on it. Since G/ U is simply connected, U must be connected. The complex coset space of the connected complex Lie group U/Bo by the discrete subgroup B/Bo can be regarded as the standard fibre of (G/B, p, G/ U). Making use of this result of Borel-Remmert we derive the following.