Toward a classification of compact complex homogeneous spaces

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.