Compact Homogeneous Manifolds

Abstract

In this chapter we study geometric properties of compact homogeneous complex manifolds. It is natural to begin with flag manifolds, which are defined as the coset spaces S/P, where S is a connected complex semisimple Lie group, P ⊂ S a parabolic subgroup. Their description requires some work with roots systems, after which we prove that a flag manifold admits an equivariant projective embedding. Furthermore, flag manifolds can be characterized as projective homogeneous manifolds, which are rational and/or simply connected. We also discuss their automorphism groups, though the proof of one important theorem stated here will be given later in Chapter 4.