DUALITY, HAAR PROGRAMS, AND FINITE SEQUENCE SPACES

THEOREM 0. Let En -En X be a homogeneous line bundle over a C-space of complex dimension n. Suppose that the first Chern class ci(Eo) is given by a negative semi-definite quadratic form of index k < n. Then HI(X, E*) = 0 for q < k. (If k = n, we have again Kodaira's theorem.) To apply Theorem 0 to C-spaces, a fairly extensive study of the differential geometry of homogeneous vector bundles is useful; these results may be of independent interest. The reason is that the Atiyah construction of the Chern classes in terms of forms does not work in the non-Kahler case and so one must use a curvature tensor in order to construct the forms. Using Theorem 0 and the fact that H*(X, E*) may be written in terms of Lie algebra cohomology, Theorem 1 is completed using several spectral sequences in Lie algebra cohomology. The Leray spectral sequence used by Bott does not seem to give the complete information here. As mentioned above, the details of the proofs together with other results and applications will appear later.