Complex Homogeneous Spaces Research Articles

Remarks on Kähler-Einstein Manifolds

The main purpose of this note is to characterize a compact Káhler-Einstein manifold in terms of curvature form. The curvature form Q is an EndT valued differential form of type (1,1) which represents the curvature class of the manifold. We shall prove that the curvature form of a Káhler metric is the harmonic representative of the curvature class if and only if the Káhler metric is an Einstein metric in the generalized sense (g.s.), that is, if the Ricci form of the metric is parallel. It is well known that a Káhler metric is an Einstein metric in the g. s. if and only if it is locally product (globally, if the manifold is simply connected and complete) of Kàhler-Einstein metrics. We obtain an integral formula, involving the integral of the trace of some operators defined by the curvature tensor, which measures the deviation of a Káhler-Einstein metric from a Hermitian symmetric metric. In the final section we shall prove the uniqueness up to equivalence of Kãhler-Einstein metrics in a simply connected compact complex homogeneous space. This result was proved by Berger in the case of a complex projective space and our proof is completely different from Berger’s.

COHOMOLOGY OF COMPACT COMPLEX HOMOGENEOUS SPACES

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 with coefficients in the sheaf of germs of holomorphic sections of the homogeneous linear fibration is nonzero only if is the inverse image of some fibration over a base of the canonical fibration. In this case the representation in can be computed using a spectral sequence if we know the representation in . The resulting theorem generalizes Griffiths' result for -spaces.Bibliography: 8 items.

DECOMPOSITIONS OF REDUCTIVE LIE GROUPS

In this work we study decompositions G = G'G of reductive Lie groups G into the product of Lie subgroups G' and G. Such decompositions are fully described in case G' and G are reductive in G, or G is simple and G' and G are maximal. The results are applied to the classification of complex homogeneous spaces. Bibliography: 21 titles.

Some properties of complex analytic vector bundles over compact complex homogeneous spaces

Some properties of complex analytic vector bundles over compact complex homogeneous spaces

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In this note, we prove some results on the classification of compact complex homogeneous spaces. We first consider the case of a parallelizable space M=G/Γ, where G is a complex connected Lie group and Γ is a discrete cocompact subgroup of G. Using a generalization of results in [M. Otte, J. Potters, Manuscripta Math. 10 (1973) 117–127; D. Guan, Trans. Amer. Math. Soc. 354 (2002) 4493–4504, see also Electron. Res. Announc. Amer. Math. Soc. 3 (1997) 90], it will be shown that, up to a finite covering, G/Γ is a torus bundle over the product of two such quotients, one where G is semisimple, the other where the simple factors of the Levi subgroups of G are all of type Al. In the general case of compact complex homogeneous spaces, there is a similar decomposition into three types of building blocks.