Homogeneous Spaces from a Complex Analytic Viewpoint

Abstract

Manifolds having many automorphisms play a fundamental role in geometry. If X is a compact complex manifold, then the group Aut(X) of holomorphic automorphisms of X is, when equipped with the compact-open topology, a complex Lie group. If G = Aut(X), then an orbit G(p) , p∈X, may be holomorphical ly identified with the quotient manifold G/H, where H := {g ∈ G|g(p) = p} is the isotropy group of the G-action at p. Thus, studying quotients G/H of a complex Lie group G by a closed subgroup H becomes relevant. A natural first step is to analyze the structure of compact homogeneous spaces X = G/H. In the early 1950’s, with development of Lie theory along with the theory of algebraic groups, a number of very sharp results were obtained by algebraic methods. The works of Borel (e.g., [10]), Goto [22], Tits [64] and Wang [67] are typical of this direction. Later, but still in an algebraic geometry spirit, many general methods were developed (e.g., see the papers of Hochschild, Mostow, Rosenlicht et al.).