Geodesic
Comology of compact complex homogeneous spaces
Sbornik. Mathematics, Tome 13 (1971) no. 2, pp. 285-296 
D. N. Akhiezer. Comology of compact complex homogeneous spaces. Sbornik. Mathematics, Tome 13 (1971) no. 2, pp. 285-296. http://geodesic.mathdoc.fr/item/SM_1971_13_2_a5/
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     author = {D. N. Akhiezer},
     title = {Comology of compact complex homogeneous spaces},
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     volume = {13},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1971_13_2_a5/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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. Bibliography: 8 titles.

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[2] R. Bott, “Homogeneous vector bundles”, Ann. Math., 66:2 (1957), 203–248 | DOI | MR | Zbl

[3] P. A. Griffiths, “Some geometric and analytic properties of homogeneous complex manifolds. I”, Acta Math., 110:1–2 (1963), 115–155 | DOI | MR | Zbl

[4] F. Hirzebruch, Topological methods in algebraic geometry, Springer-Verlag, 1966 | MR

[5] Zh. P. Serr, Algebry Li i gruppy Li, Mir, Moskva, 1969 | MR | Zbl

[6] M. Ise, “On the geometry of Hopf manifolds”, Osaka Math. J., 12:2 (1960), 387–402 | MR | Zbl

[7] M. Ise, “Some properties of complex analytic vector bundles over compact complex, homogeneous spaces”, Osaka Math. J., 12:2 (1960), 217–252 | MR | Zbl

[8] S. Murakami, “Sur certains espaces fibres principaux differentiables et holomorphes”, Nagoya Math. J., 15 (1959), 171–199 | MR | Zbl