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The holomorphic fibration of a compact connected homogeneous complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t0929101.png" /> over a homogeneous projective rational variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t0929102.png" />, which is universal in the class of all such fibrations. Universality in this case means that the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t0929103.png" /> of any fibration in this class is representable as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t0929104.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t0929105.png" /> is the projection of the Tits bundle and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t0929106.png" /> is some holomorphic fibering.
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An explicit construction of the Tits bundle is carried out as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t0929107.png" /> be a connected complex Lie group acting holomorphically and transitively on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t0929108.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t0929109.png" /> be the isotropy subgroup of some point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t09291010.png" />. The normalizer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t09291011.png" /> of the connected component of the identity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t09291012.png" /> is a parabolic subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t09291013.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t09291014.png" /> contains a maximal connected solvable subgroup (cf. [[#References|[1]]], [[#References|[2]]]). The base space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t09291015.png" /> of the Tits bundle is defined to be the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t09291016.png" />, and the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t09291017.png" /> is induced by the inclusion of the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t09291018.png" />. This construction is due to J. Tits [[#References|[1]]], who also proved universality for this bundle.
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The fibre of the Tits bundle is complex-parallelizable. If the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t09291019.png" /> is simply-connected, then this fibre is a complex torus. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t09291020.png" /> admits a transitive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t09291021.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t09291022.png" /> are constant on the fibres of the Tits bundle. In the case where the complex compact homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t09291023.png" /> is Kähler, the fibre of the Tits bundle is a complex torus (moreover, it is the [[Albanese variety|Albanese variety]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092910/t09291024.png" />), 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.
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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====
 
====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>
 
<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>

Latest revision as of 08:25, 6 June 2020


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