Complex subspaces of homogeneous complex manifolds II—Homotopy Results

Theorems of Barth-Lefschetz type for complex subspaces of homogeneous complex manifolds.

Barth, Larsen, and others showed that complex submanifolds of complex projective space, CP(N), of small codimension strongly resemble CP(N) both homotopically and cohomologically. These results are generalized to yield analogous results for complex subspaces of arbitrary homogeneous complex manifolds. One very special corollary that gives the flavor of the results is:Corollary. Let A and B be c...

A Note on Homogeneous Complex Contact Manifolds

Homogeneous complex manifolds and representations of semisimple lie groups.

Let G be a connected semisimple Lie group, K a maximal compact subgroup of G. Assume that rank K = rank G. We keep fixed a Cartan subgroup H of K; H is then also a Cartan subgroup of G. Denote the Lie algebras of G, K, H by go, fo, to, and their complexifications by g, E, b. The adjoint action of b determines a rootspace decomposition g = t) ®( ga, where A is the set of nonzero roots of the pai...

Some Geometric and Analytic Properties of Homogeneous Complex Manifolds

This is the first of two papers dealing with homogeneous complex manifolds; since the second work is a continuation of this one, we shall let the following introduction serve for both. The general problem is to study the geometric, analytic, and function-theoretic properties of homogeneous complex manifolds. The present paper, referred to as Part I, is concerned mainly with sheaves and cohomolo...

Notes on homogeneous vector bundles over complex flag manifolds

Let P be a parabolic subgroup of a semisimple complex Lie group G defined by a subset Σ of simple roots of G, and let Eφ be a homogeneous vector bundle over the flag manifold G/P corresponding to a linear representation φ of P . Using Bott’s theorem, we obtain sufficient conditions on φ in terms of the combinatorial structure of Σ for some cohomology groups of the sheaf of holomorphic sections ...