Cohomological Dimension of Homogeneous Spaces of Complex Lie Groups

In [4] we obtained a formula which represents the completeness of complex Lie groups, and in [3] we generalized it to some types of homogeneous spaces of complex Lie groups. The purpose of the present short note is to give a proof of the generalized formula without the assumption in [3]. Throughout we denote (complex) Lie groups by Roman capital letters and their Lie algebras by the corresponding German small letters respectively. Let (G, H) be a pair of a connected complex Lie group G and a connected closed complex subgroup H of G. Let (K, L) be a pair of maximal compact subgroups K and L of G and H respectively. We consider only such K that contains L. We denote the canonical surjection from g onto the quotient complex vector space g/Ij by 71. Denoting the complex dimension of any complex object X by d(X), we can give the following indices of a G-homogeneous complex manifold G/H: