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In this paper we discuss some geometric and analytic properties of a class of locally homogeneous complex manifolds. Our original motivation came from algebraic geometry where certain non-compact, homogeneous complex manifolds arose naturally from the period matrices of general algebraic varieties in a similar fashion to the appearance of the Siegel upper-half-space from the periods of algebraic curves. However, these manifolds arc generally not Hermitian symmetric domains and, because of this, several interesting new phenomena turn up.The following is a description of the manifolds we have in mind. Let Gc be a connected, complex semi-simple Lie group and Bc Gc a parabolic subgroup. Then, as is well known, the coset space X = Gc/B is a compact, homogeneous algebraic manifold. If G~ Gc is a con- One case is when G=M is a maximal compact subgroup of Gc. Then necessarily F ={e), and D=X is the whole compact algebraic manifold. These varieties have been the subject of considerable study, and their basic properties are well known. The opposite extreme occurs when G has no compact factors. These non-compact homogeneous domains D then include the Hermitian symmetric spaces, about which quite a bit is known, and also include important and interesting non-classical domains which have been discussed relatively little. It is these manifolds which are our main interest; however, since the (1) During the preparation of this

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