Theory of Functions on Complex Manifolds

Beginning with Valiron [262], the value distribution theory of algebroid functions of one complex variable has been studied extensively. A few contributions are listed under references. The Nevalinna theory of meromorphic functions on a complex manifold M or meromorphic mappings from M into a projective variety has been studied (see Griffith-King [75], Stoll [239], Vitter [265]). Here we will concentrate on the value distribution theory and its application of algebroid functions in several complex variables.

Let Ĝ be a connected complex Lie group and X = Ĝ/Ĥ a complex manifold homogeneous under a holomorphic Ĝ-action. In order to understand X, e.g. how it fits into a fine classification, details of its function theory, etc., one should use as much Lie theoretic information about Ĝ as is possible. In particular it is often useful to study the orbit structure of real subgroups of Ĝ. Such orbits are usually not complex sub-manifolds of X.