Homogeneous Complex Manifolds Research Articles

On homogeneous complex manifolds with negative definite canonical Hermitian form

On homogeneous complex manifolds with negative definite canonical Hermitian form

Topology of Some Kähler Manifolds II

Topology of positively curved compact Kähler manifolds had been studied by several authors (cf. [6; 2]); these manifolds are simply connected and their second Betti number is one [1]. We will restrict ourselves to the study of some compact homogeneous Kähler manifolds. The aim of this paper is to supplement some results in [9]. We prove, among other results, that a compact, simply connected homogeneous complex manifold whose Euler number is a prime p ≥ 2 is isomorphic to the complex projective space Pp-1(C); in the p-1 case of surfaces, we prove that a compact, simply connected, homogeneous almost complex surface with Euler-Poincaré characteristic positive, is hermitian symmetric.

Locally homogeneous complex manifolds

Locally homogeneous complex manifolds

On Complex Homogeneous Manifolds

Compact complex homogeneous manifolds have been studied in great detail by Borel, Goto, Remmert and Wang (cf., (13)): it was shown that every compact, connected complex homogeneous manifold M is a holomorphic fiber bundle over a projective algebraic homogeneous manifold B with a connected, complex parallelizable fiber F.

Topology of Complex Manifolds

The aim of this paper is to collect some interesting unsolved problems in the topology of compact complex manifolds which are scattered in literature; needless to remark, some of them are known to specialists. Besides, the readermay consult the well-known list of problems by F. Hirzebruch [ 5 ] which has stimulated remarkable progress in this direction. But the topology of complex manifolds has always baffled workers in this field and is still far from satisfactory. Of course, there is remarkable progress in the case of homogeneous compact complex manifolds. We start with the following result.

Some differential geometric remarks on complex homogeneous manifolds

Some differential geometric remarks on complex homogeneous manifolds

On the differential geometry of homogeneous vector bundles

This paper falls roughly into three parts. The first section (??I-III) may be considered as an extension of the works of Nomizu and Wang [15; 19] to bundles over homogeneous complex manifolds. These results lay the groundwork for the succeeding paragraphs. The second section (??IV-VI) gives differential-geometric derivations of various properties of homogeneous complex manifolds. The last section (??VII-IX) gives some applications of differential geometry to homogeneous vector bundles in the sense of [4] and to the study of sheaf cohomology. To be more explicit, we let X be a homogeneous complex manifold which may be written as the coset space of complex Lie groups A, B (X = A/ B) and also as the coset space of compact Lie groups M, V (X = M/ V) where M is semi-simple. The first part treats the following question: If P -> X is any analytic principal bundle to which the action of A on X lifts, we ask for a algebraic description of those connexions x in P which are M-invariant and compatible with the complex structures involved. (It will be seen that the adjective computable is for us crucial.) Moreover, given such a x, we seek its form, and, in the case where P is the principal tangent bundle, we ask for the complex torsion of x. In the second part, we single out two types of connexions in such a P for further study. The first are those connexions arising from an invariant Hermitian metric in P. By examining these connexions, we are able to give a unified differentialgeometric treatment of several aspects in the theory of homogeneous complex manifolds. The complex torsion plays an important role here.The other connexion, which we call the canonical complex connexion, is the complex analogue of the Nomizu canonical affine connexion; it is important for applications to sheaf theory. In the third part, we first give a geometric realization of the curvature class, a sheaf cohomology class defined by Atiyah in [1]. Next, using a metric geometry, we discuss a vanishing theorem; the chief application here is the non-Kahler case. In the last paragraph, we give a new type of application of differential geometry to sheaf cohomology; this application arises from a connextion between the holonomy algebra of a linear connexion and the coboundary maps in an exact cohomology sequence.

Some geometric and analytic properties of homogeneous complex manifolds: Part I: Sheaves and cohomology

Some geometric and analytic properties of homogeneous complex manifolds: Part I: Sheaves and cohomology

Some geometric and analytic properties of homogeneous complex manifolds: Part II: Deformation and bundle theory

Some geometric and analytic properties of homogeneous complex manifolds: Part II: Deformation and bundle theory

On Automorphism Groups of Homogeneous Complex Manifolds

On Automorphism Groups of Homogeneous Complex Manifolds

On automorphism groups of homogeneous complex manifolds

The reproducing property of the Bergman kernel function [1 ] plays an essential role in our proof. As we consider not only bounded domains in Cn, but also general complex manifolds, we replace the kernel function by the kernel form (see [2 ] for the definition and properties of the kernel form). If AT is a domain in Cn, then F can be identified with the space F* of square integrable holomorphic functions (with respect to the Euclidean measure of Cn). For our purpose, it is, however, desirable to use F even for a domain in Cn, because every holomorphic transformation of M induces, in a natural way, a unitary transformation of F. We recall that, by a square integrable holomorphic n-form f, we mean

A Fibering of a Class of Homogeneous Complex Manifolds

A Fibering of a Class of Homogeneous Complex Manifolds

A fibering of a class of homogeneous complex manifolds

A fibering of a class of homogeneous complex manifolds