# 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
How to Cite This Entry:
Tits bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tits_bundle&oldid=48980
This article was adapted from an original article by D.N. Akhiezer (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article