Subvarieties of homogeneous and almost homogeneous manifolds

Abstract

The general context of this paper is that of compact subvarieties of a complex manifold X which is equipped with the action of a Lie group G of holomorphic transformations. Here we restrict our attention to the case where X is homogeneous or at least almost homogeneous. For example, in Chapter 2 we consider the analytic hypersurfaces H(X), i. e., the lcodimensional complex analytic subsets, in a homogeneous space X = G / H. Although the results are primarily formulated in the language of complex geometry, we are mainly interested in determining the complex analytic objects which define the hypersurfaces, e.g., holomorphic functions coming from associated Stein manifolds, rational functions from related projective varieties, Θ-functions on abelian groups, Fourier-Jacobi series,... Difficulties arise, for example, because the ambient manifolds are in general non-compact, non-Kahlerian, and the relevant cohomology groups are infinite-dimensional.