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\begin{document}

\title[Classification of Compact Homogeneous Spaces]
{Classification of compact homogeneous spaces with invariant
symplectic structures}

%\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  
%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.}

\subjclass{Primary 53C15, 57S25, 53C30; Secondary 22E99, 15A75}
\keywords{Invariant structure, homogeneous space, product,
fiber bundles, symplectic manifolds, splittings, prealgebraic group,
decompositions, modification, Lie group, symplectic algebra,  
compact manifolds, uniform discrete subgroups, classifications,
locally flat parallelizable manifolds}
%\footnotetext[1]
\date{February 21, 1997}
\commby{Gregory Margulis}
\issueinfo{3}{07}{January}{1997}
\dateposted{July 29, 1997}
\pagespan{52}{54}
\PII{S 1079-6762(97)00023-1}
\copyrightinfo{1997}{American Mathematical Society}
%1991 Mathematics Subject Classification. 53C15, 57S25,
%53C30, 22E99, 15A75.
%Key words and phrases. 
%invariant structure, homogeneous space, product,
%fiber bundles, symplectic manifolds, splittings, prealgebraic group,
%decompositions, modification, Lie group, symplectic algebra,  
%compact manifolds, uniform discrete subgroups, classifications,
%locally flat parallelizable manifolds.}
% \baselineskip20pt

\begin{abstract}
We solve a longstanding problem of classification of 
compact homogeneous spaces with invariant symplectic structures. We
also give a splitting conjecture on compact homogeneous
spaces with symplectic structures (which are not necessarily invariant 
under the group action) that makes the classification of this 
kind of manifolds possible.
\end{abstract}

\maketitle


A smooth manifold $M$ equipped with a smooth transitive action of a Lie group
is called a {\em homogeneous space\/}.  If in addition $M$ is a symplectic
manifold, we refer to it as a
{\em homogeneous space with a symplectic structure\/} and, if the structure
is invariant, a {\em homogeneous space with an invariant symplectic structure\/}.

Recently there has been much progress in the area of symplectic manifolds and
group actions. However,
the {\em classical\/} problem of classifying compact 
homogeneous spaces with symplectic structure is seemingly
unreachable.

The paper \cite{DG} did give us some hope in this direction and
\cite{Gu1}, \cite{Gu2}, \cite{Hk} provide a nice picture for compact complex homogeneous
spaces with a symplectic structure. The following result can be found in
\cite{DG} (see also \cite{Hk} for a ``simpler'' proof):

%\smallskip
%  {\bf Proposition 1.} 
\begin{prop} [\cite{DG}, \cite{Hk}] 
%{\em 
Every compact homogeneous pseudo-K\"ahler
manifold is a product of a torus and a rational homogeneous space, 
both with
standard pseudo-K\"ahler structures. %\/}
\end{prop}
%\smallskip

  We note here that the proof of this result is more 
complicated
than that of
the K\"ahler case in \cite{Mt}, since in the K\"ahler case, by the compactness
of the manifold, one can easily see that a compact Lie group acts transitively
and the isotropy group is a subgroup of an orthogonal
group, i.e., both groups are reductive. 

In \cite{Gu1}, \cite{Gu2} we
observed that the method in \cite{Hk} actually works for a compact 
complex homogeneous
space with an invariant symplectic structure:

%\smallskip  {\bf Proposition 2.} {\em 
\begin{prop} Every compact complex homogeneous space with
an invariant symplectic structure is a product of a torus and a rational
homogeneous space, both with standard symplectic structures. %\/}
\end{prop}
%\smallskip

  And we also proved the following theorem: 

%\smallskip  {\bf Proposition 3.} {\em 
\begin{prop}\label{prop:3}Every compact homogeneous complex manifold
with a
$2$-cohomo\-logy class $\omega$ such that $\omega ^{n}$ is not zero
in the top cohomology is a product of a rational homogeneous space
and a
complex parallelizable solv-manifold 
with a right invariant symplectic structure on 
its universal covering. %\/}  
\end{prop}
%\smallskip

  This generalized the result of \cite{BR} for the K\"ahler case (one does not 
assume that the K\"ahler form is invariant).

  These results suggest further study in two  
directions along the lines of Proposition \ref{prop:3}. One 
is the classification of compact complex homogeneous spaces; the 
other is the classification of compact homogeneous spaces with
a symplectic structure. The first problem can be solved by the 
method in \cite{Gu3}, where we prove that every compact complex
homogeneous space with an invariant volume is a torus fiber bundle
over a product of a rational homogeneous space and a complex
parallelizable manifold. In the present paper we 
announce our result in the direction of the second problem. This is
quite analogous to the results in \cite{Gu1}, \cite{Gu2}:

%\smallskip  {\bf MAIN THEOREM.} {\em 
\begin{maint}Every compact homogeneous space with an
invariant symplectic structure is a product of a
rational homogeneous space and a torus with invariant symplectic
structures. %\/}\smallskip
\end{maint}

  In this classification, the tori which occur are not necessarily standard.
One might use such a nonstandard torus to obtain new symplectic
manifolds as in \cite{Bo} and \cite{Gu1}. 
The following conjecture arises naturally from our arguments for the 
proof of the above theorem:

%\smallskip{\bf CONJECTURE.} {\em 
\begin{conj}If $G/H$ is a compact homogeneous
space with a symplectic structure, then $G/H$ is
a product of a rational homogeneous
space and a compact locally flat parallelizable manifold with a
symplectic structure. %\/}\smallskip
\end{conj}

Here we call a manifold $N$ {\em locally flat parallelizable\/}
if $N=G/H$ for a simply connected Lie group $G$ which is diffeomorphic to
${\br}^{k}$ for some integer $k$ and $H$ is a uniform discrete subgroup.

In our future work we will attempt to prove this conjecture and to
classify the compact locally flat parallelizable
manifolds with this additional structure.

\section*{Acknowledgement} 

We thank the referee for his gracious advice on the final exposition of this
paper.
%\bigskip
  
%\hspace{5cm} {\bf References}

%\medskip

\begin{thebibliography}{GGG}

\bibitem[Bo]{Bo} 
F. A. Bogomolov,
{\em On Guan's examples of simply connected non-K\"ahler compact complex
manifolds},
Amer. J. Math. {\bf 118} (1996),
1037--1046.
\CMP{97:01}
\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[DG]{DG} 
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[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 67} (1996), 217--225.
\CMP{97:02}
\bibitem[Gu3]{Gu3} 
D. Guan,
{\em Classification of compact complex homogeneous spaces with
invariant volumes},
preprint 1996.
\bibitem[Gu4]{Gu4} 
D. Guan,
{\em Examples of compact holomorphic symplectic manifolds
which are not K\"ahlerian II},
Invent. Math. {\bf 121} (1995), 135--145.
\CMP{95:16}
\bibitem[Hk]{Hk} 
A. T. Huckleberry, 
{\em Homogeneous pseudo-K\"ahlerian manifolds: a
Hamiltonian viewpoint},
preprint, 1990.
\bibitem[Kb]{Kb} 
S. Kobayashi,
{\em Differential geometry of complex vector
bundles\/},
Iwanami Shoten Publishers and Princeton University Press, 1987.
 \MR{89e:53100}
\bibitem[Mt]{Mt} 
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[Ti]{Ti} 
J. Tits,
{\em Espaces homog\`enes complexes compacts}, 
Comm. Math. Helv.
{\bf 37}(1962), 111--120.
\MR{27:4248}
\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}

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