Abstract:
In this paper we study compact complex homogeneous spaces having a complex torus for the fiber of the canonical fibration (Tits fibration). We prove that the cohomology of such a space $X$ with coefficients in the sheaf of germs of holomorphic sections of the homogeneous linear fibration $\mathbf E$ is nonzero only if $\mathbf E$ is the inverse image of some fibration $\widetilde{\mathbf E}$ over a base $D$ of the canonical fibration. In this case the representation in $H^*(X,\mathbf E)$ can be computed using a spectral sequence if we know the representation in $H^*(D,\widetilde{\mathbf E})$. The resulting theorem generalizes Griffiths' result for $C$-spaces.
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