On the differential geometry of homogeneous vector bundles

Abstract

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.