Tits bundle

The holomorphic fibration of a compact connected homogeneous complex space $ X $ over a homogeneous projective rational variety $ D $, which is universal in the class of all such fibrations. Universality in this case means that the projection $ \pi ^ \prime : X \rightarrow D ^ \prime $ of any fibration in this class is representable as $ \pi ^ \prime = \phi \circ \pi $, where $ \pi : X \rightarrow D $ is the projection of the Tits bundle and $ \phi : D \rightarrow D ^ \prime $ is some holomorphic fibering.

An explicit construction of the Tits bundle is carried out as follows. Let $ G $ be a connected complex Lie group acting holomorphically and transitively on $ X $, and let $ U $ be the isotropy subgroup of some point in $ X $. The normalizer $ P $ of the connected component of the identity of $ U $ is a parabolic subgroup of $ G $, i.e. $ P $ contains a maximal connected solvable subgroup (cf. [1], [2]). The base space $ D $ of the Tits bundle is defined to be the quotient space $ D = G/P $, and the projection $ \pi : X \rightarrow D $ is induced by the inclusion of the subgroup $ U \subset P $. This construction is due to J. Tits [1], who also proved universality for this bundle.

The fibre of the Tits bundle is complex-parallelizable. If the space $ X $ is simply-connected, then this fibre is a complex torus. If $ X $ admits a transitive group $ G $ equal to its own commutator subgroup, then the Tits bundle coincides with the meromorphic reduction bundle (cf. [3]). This means that all meromorphic functions on $ X $ are constant on the fibres of the Tits bundle. In the case where the complex compact homogeneous space $ X $ is Kähler, the fibre of the Tits bundle is a complex torus (moreover, it is the Albanese variety of $ X $), and the bundle itself is analytically trivial [2]. Thus, a compact Kähler homogeneous space is the product of a projective rational homogeneous variety and a complex torus.

References

[1]	J. Tits, "Espaces homogènes complexes compacts" Comment. Math. Helv. , 37 (1962) pp. 111–120
[2]	A. Borel, R. Remmert, "Ueber kompakte homogene Kählerische Mannigfaltigkeiten" Math. Ann. , 145 (1962) pp. 429–439
[3]	H. Grauert, R. Remmert, "Über kompakte homogene komplexe Mannigfaltigkeiten" Arch. Math. , 13 (1962) pp. 498–507