On Compact Complex Coset Spaces of Reductive Lie Groups
نویسندگان
چکیده
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 BoThe 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 [l, 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/B0 by the discrete subgroup B/B0 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.
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