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\controldates{26-AUG-1997,26-AUG-1997,26-AUG-1997,26-AUG-1997}
 
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\issueinfo{3}{13}{January}{1997}
\dateposted{August 29, 1997}
\pagespan{90}{92}
\PII{S 1079-6762(97)00028-0}
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\newtheorem*{mthm1}{Main Theorem 1}
\newtheorem*{mthm2}{Main Theorem 2}
\newtheorem*{mthm3}{Main Theorem 3}
\newtheorem*{cor}{Corollary}

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\begin{document}
\title[CLASSIFICATION OF COMPACT COMPLEX HOMOGENEOUS SPACES]
{Classification of compact complex homogeneous spaces with invariant 
volumes}

%\author{Daniel Guan$^{1}$\\
%        e-mail: zguan@@math.princeton.edu\\
%        Fax: (609)258-1367\\
%        Phone: (609)683-8725(H) (609)258-6466(O)}
%\markboth{DANIEL GUAN}{CLASSIFICATION OF COMPACT COMPLEX
%HOMOGENEOUS SPACES}

\author{Daniel Guan}
\address{Department of Mathematics,
Princeton University,
Princeton, NJ 08544}
\email{zguan@@math.princeton.edu}
\thanks{Supported by NSF Grant DMS-9401755 and DMS-9627434.}

%\begin{document}

%\footnotetext[1]{Supported by NSF Grant DMS-9401755 and DMS-9627434.


%1991 Mathematics Subject Classification. 53C15, 57T15,
%53C30, 53C56, 53C50.
\subjclass{Primary 53C15, 57T15, 53C30, 53C56, 53C50}

%Key words and phrases.
%Invariant volume, homogeneous, product,
%fiber bundles, complex manifolds,
%parallelizable manifolds, discrete subgroups, classifications.}
\keywords{Invariant volume, homogeneous, product,
fiber bundles, complex manifolds,
parallelizable manifolds, discrete subgroups, classifications}

%\baselineskip20pt

%\maketitle

%\begin{quote}

%\issueinfo{3}{1}{July}{1997}

\copyrightinfo{1997}{American Mathematical Society}

\commby{Svetlana Katok}

\date{May 30, 1997}

%\begin{document}

\begin{abstract}
In this note we give a classification of
 compact complex homogeneous spaces
with invariant volume.
\end{abstract}
%\end{quote}

\maketitle


\section{Introduction}


  We call a $2n$-dimensional manifold $M$ a
{\em complex homogeneous space with invariant volume\/} if there is a
complex structure and a nonzero $2n$-form on $M$ such
that a transitive Lie transformation group
keeps both the complex structure and
the $2n$-form invariant. There are many papers published
in the direction
of classification of such manifolds, e.g., \cite{Bo}, 
\cite{DG1},
\cite{DG2},
\cite{DG3},
\cite{DN},
\cite{Gu1}, \cite{Gu2}, \cite{Ha1}, \cite{Hk}, 
\cite{HK}, \cite{Kz}, \cite{Mt1},
\cite{Mt2}, \cite{Wa2} and the
references there
(see also \cite{BR}, 
\cite{Gu1}, \cite{Gu2}, \cite{Gu4}, \cite{Ti}, \cite{Wa1} for related topics
involving compact
complex homogeneous spaces). In this paper we will finish the
classification in the compact
case.

A major break-through in this direction became possible
after the following two
results were established.
Firstly, the Hano-Kobayashi
fibration of a compact
complex homogeneous space with invariant volume
(we might also call it the Ricci form reduction) is holomorphic
and coincides with the anticanonical fibration (see
\cite{DG1}). Secondly,
one can classify compact complex homogeneous spaces
with invariant pseudo-K\"ahler structure (see \cite{DG1}, 
\cite{Hk} and 
\cite{Gu1},
\cite{Gu2},
also \cite{Gu4}).

The proof is much harder
than the corresponding proof in
the K\"ahler case in \cite{Mt1}. Namely, in the K\"ahler case one can
choose the transitive group to be compact.
Then the isotropy group is a subgroup of an orthogonal
group. In particular, both groups are reductive.

  In \cite{Hk} Huckleberry observed that one can handle
the pseudo-K\"ahler case using methods from symplectic geometry.
In particular, he applied
here the construction of the moment map.
In \cite{Gu1},
\cite{Gu2} we
observed that his method actually works for a compact complex homogeneous
space with an invariant symplectic structure.

  Huckleberry's method was used in \cite{Gu1},
\cite{Gu2} to get a structure
theorem for compact homogeneous complex manifolds with
$2$-cohomology classes $\omega$ such that $\omega ^{n} \ne 0$
in the top cohomology.
  This generalized the result of \cite{BR} for the K\"ahler case (one
does not
assume that the K\"ahler form is invariant).

  For a general compact complex homogeneous space with
invariant volume, the symplectic method does not apply.
However, our original method (see \cite{HK}, 
\cite{Mt2} and 
\cite{DG1})
gives a classification.

%\smallskip

%  {\bf MAIN THEOREM 1.} {\em
\begin{mthm1} 
\label{mthm1}
Every compact complex homogeneous
space with an
invariant volume form
is a homogeneous complex torus bundle over the product of a
projective
rational homogeneous space and a parallelizable manifold. Conversely,
every complex homogeneous manifold $M$ of
this kind admits
a transitive real transformation Lie group $G$, acting on $M$ by
holomorphic transforms and preserving a volume form on $M$. %\/}
\end{mthm1}
%\smallskip

  For more details concerning the structure theorem,
one might look at Sections 3,
4 and 5 of \cite{Gu3}.
We also note that every compact complex homogeneous space $M$ with
a 2-cohomology class such that its top power is nonzero in the top
cohomology
group, admits a transitive real
Lie transformation group $G$, which acts on $M$
by holomorphic transforms
and preserves a volume form.

  Our proof is better than the proof of both results in \cite{DG1} and
\cite{Gu1}.

  In \cite{Mt2} Matsushima considered the special case of a
semisimple group action. He proved that
if $G/H$ is a compact complex homogeneous
space with a $G$-invariant volume and if $G$ is semisimple,
then $G/H$ is a holomorphic fiber bundle
over a projective rational homogeneous space, the
typical fiber being a complex
parallelizable homogeneous space
of a reductive  complex Lie group.

Applying our Main Theorem %1 
\ref{mthm1} to this situation, we immediately see
that the result of Matsushima can be generalized to the case
when $G$ is reductive. Moreover, we have the following stronger result.

%\smallskip

%{\bf MAIN THEOREM 2.} {\em 
\begin{mthm2}\label{mthm2}
Assume that $G/H$ is a compact complex homogeneous
space with a $G$-invariant volume and $G$ is reductive. Then $G/H$ is
a holomorphic torus bundle over the product of a projective
rational homogeneous
space and a complex parallelizable homogeneous
space of a
semisimple complex
 Lie group. %\/}
\end{mthm2}
%\smallskip

Even if we drop here the assumption
that $G/H$ has a $G$-invariant volume form,
we still have a holomorphic
fibration of $G/H$ over a projective
rational base with parallelizable fiber (see \cite{BR}, 
\cite{Ti}).
J. Hano \cite{Ha2} %[Ha2]
 proved that the fiber is
of the form $L/\Gamma $, where $\Gamma $
is a discrete cocompact subgroup of a reductive complex Lie group $L$.
Our methods show that the converse is also true.
Namely, we have the following theorem.

%\smallskip
%{\bf MAIN THEOREM 3.} {\em
\begin{mthm3}\label{mthm3}
Suppose that a compact complex
homogeneous space $M$ admits a holomorphic
fibration $\pi : M \to D$,
where $D$ is a projective rational
homogeneous space. Assume that the typical fiber
of $\pi $ is of the form $F = L/\Gamma $,
where $L$ is a connected reductive complex Lie group,
and $\Gamma $ a discrete cocompact subgroup of $L$.
Then any transitive effective complex Lie transformation group $G $,
acting on $M$ by holomorphic transforms, is reductive. %}
\end{mthm3}



%\smallskip
%{\bf Corollary.} {\em 
\begin{cor} Every compact complex homogeneous space is
a holomorphic fiber bundle whose base is a compact complex
homogeneous space of a reductive Lie group and whose typical fiber
is a complex parallelizable homogeneous space
of a nilpotent complex Lie group. %\/}
\end{cor}
%\smallskip

Having in mind a classification of all compact
complex homogeneous spaces as a goal, we hope to use the Corollary
in our future research.

%\newpage\noindent{\bf Acknowledgement. }
\section*{Acknowledgement} I thank
the referee for pointing out \cite{Ha2} to me
and many invaluable suggestions for the composition of
this paper.

%\hspace{5cm} {\bf References}
%\medskip

\begin{thebibliography}{MMM}

\bibitem[Bo]{Bo} A. Borel, 
{\em K\"ahler coset spaces of semisimple Lie groups}, 
Nat. Acad. Sci.
USA, {\bf 40} (1954), 1147--1151.
\MR{17:1108e}
\bibitem[BR]{BR} A. Borel and R. Remmert,
{\em \"Uber kompakte homogene K\"ahlersche
Mannigfaltigkeiten}, 
Math. Ann. {\bf 145} (1962), 429--439.
\MR{26:3088}
\bibitem[DG1]{DG1} J. Dorfmeister and Z. Guan,
{\em Classifications of compact homogeneous
pseudo-K\"ahler manifolds},
Comm. Math. Helv. {\bf 67} (1992), 499--513.
\MR{93i:32042}
\bibitem[DG2]{DG2} J. Dorfmeister and Z. Guan,
{\em Fine structure of reductive pseudo-K\"ahlerian
spaces}, 
Geom. Dedi. {\bf 39} (1991), 321--338.
 \MR{92h:53081}
\bibitem[DG3]{DG3} J. Dorfmeister and Z. Guan,
{\em Pseudo-K\"ahlerian homogeneous
spaces admitting a reductive transitive group of automorphisms},
Math. Z. {\bf 209} (1992), 89--100.
\MR{92k:32058}
\bibitem[DN]{DN} J. Dorfmeister and K. Nakajima, 
{\em The fundamental conjecture for
homogeneous K\"ahler manifolds},
Acta. Math. {\bf 161} (1988), 23--70.
\MR{89i:32066}
\bibitem[Gu1]{Gu1} Z. Guan, 
{\em Examples of compact holomorphic symplectic manifolds which
admit no K\"ahler structure}. 
In {\em Geometry and Analysis on Complex
Manifolds---Festschrift for Professor S. Kobayashi's 60th Birthday\/}, 
World Scientific 1994, pp. 63--74.
\bibitem[Gu2]{Gu2} D. Guan,
{\em A splitting theorem for compact complex homogeneous spaces with
a symplectic structure}, 
Geom. Dedi. {\bf 63} (1996), 217--225.
\CMP{97:02}
\bibitem[Gu3]{Gu3} D. Guan,
{\em Classification of compact complex homogeneous spaces
with invariant volumes}. 
Preprint.
\bibitem[Gu4]{Gu4} D. Guan,
{\em Classification of compact homogeneous spaces with
invariant symplectic structures}. 
Preprint.
\bibitem[Ha1]{Ha1} J. Hano, 
{\em Equivariant projective immersion of a complex coset space with
non-degenerate canonical Hermitian form}, 
Scripta Math. {\bf 29} (1971), 125--139.
\MR{51:3557}
\bibitem[Ha2]{Ha2} J. Hano, 
{\em On compact complex coset spaces of reductive Lie
groups}, 
Proceedings of AMS {\bf 15} (1964), 159--163.
\MR{28:1258} 
\bibitem[Hk]{Hk} A. T. Huckleberry, 
{\em Homogeneous pseudo-K\"ahlerian manifolds: A
Hamiltonian viewpoint}, 
Preprint, 1990.
\bibitem[HK]{HK} J. Hano and S. Kobayashi, 
{\em A fibering of a class of homogeneous complex
manifolds}, 
Trans. Amer. Math. Soc. {\bf 94} (1960), 233--243.
\MR{22:5990}
\bibitem[Kz]{Kz} J. L. Koszul, 
{\em Sur la forme  hermitienne canonique des espaces homog\`enes
complexes}, 
Canad. J. Math. {\bf 7} (1955), 562--576.
\MR{17:1109a}
\bibitem[Mt1]{Mt1} Y. Matsushima, 
{\em Sur les espaces homog\`enes k\"ahl\'eriens d'un
groupe de
Lie r\'eductif}, 
Nagoya Math. J. {\bf 11} (1957), 53--60.
\MR{19:315c}
\bibitem[Mt2]{Mt2} Y. Matsushima, 
{\em Sur certaines vari\'et\'es homog\`enes complexes},
Nagoya Math. J. {\bf 18} (1961), 1--12.
\MR{25:2147}
\bibitem[Ti]{Ti} J. Tits, 
{\em Espaces homog\`enes complexes compacts}, 
Comm. Math. Helv.
{\bf 37} (1962), 111--120.
\MR{27:4248}
\bibitem[Wa1]{Wa1} H. C. Wang, 
{\em Complex parallisable manifolds}, 
Proc. Amer. Math. Soc. {\bf 5}
(1954), 771--776.
\MR{17:531a}
\bibitem[Wa2]{Wa2} H. C. Wang, 
{\em Closed manifolds with homogeneous complex
structure}, 
Amer. J. Math. {\bf 79} (1954), 1-32.
\MR{16:518a}
\end{thebibliography}

%\begin{tabbing}
%Department of Mathematics UCB\=                               \kill
%Author's Addresses: \\
%Zhuang-Dan  Guan \\
%Department of Mathematics \\
%Princeton University\\
%Princeton, NJ 08544 U. S. A.\\
%e-mail: zguan@@math.princeton.edu\\
%Phone: (609)683-8725(H) (609)258-6466(O)\\
%Fax: (609)258-1367
%\end{tabbing}

\end{document}
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