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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. [[#References|[1]]], [[#References|[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 [[#References|[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. [[#References|[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|Albanese variety]] of  $  X $), 
and the bundle itself is analytically trivial [[#References|[2]]]. Thus, a compact Kähler homogeneous space is the product of a projective rational homogeneous variety and a complex torus.

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