%_ ************************************************************************** %_ * The TeX source for AMS journal articles is the publisher's TeX code * %_ * which may contain special commands defined for the AMS production * %_ * environment. Therefore, it may not be possible to process these files * %_ * through TeX without errors. To display a typeset version of a journal * %_ * article easily, we suggest that you either view the HTML version or * %_ * retrieve the article in DVI, PostScript, or PDF format. * %_ ************************************************************************** % Author Package file for use with AMS-LaTeX 1.2 %% Modified June 12, 1997 by A. Morgoulis %% Modified June 9, 1997 by A. Morgoulis %% Modified June 7, 1997 by A. Morgoulis %\documentstyle[11pt]{article} \controldates{14-JUL-1997,14-JUL-1997,14-JUL-1997,14-JUL-1997} \documentstyle[newlfont,draft]{era-l} \newtheorem{prop}{Proposition} \newtheorem{maint}{Main Theorem}\renewcommand{\themaint}{\!\!} \newtheorem{conj}{Conjecture}\renewcommand{\theconj}{\!\!} %\newcommand{\cx}{{\bf C}} \newcommand{\g}{\Cal G} %{\cal G}} \newcommand{\ca}{\Cal A} %{\cal A}} \newcommand{\cb}{\Cal B} %{\cal B}} \newcommand{\h}{\Cal H} %{\cal H}} %\newcommand{\r}{\Cal R} %{\cal R}} \renewcommand{\r}{\Cal R} %{\cal R}} \newcommand{\s}{\Cal S} %{\cal S}} \newcommand{\n}{\Cal N} %{\cal N}} \newcommand{\cj}{\Cal J} %{\cal J}} %\newcommand{\br}{{\bf R}} \newcommand{\br}{{\bold R}} %\newcommand{\bz}{{\bf Z}} %\newcommand{\inte}{{\rm Int}} %\newcommand{\ad}{{\rm Ad}} \newcommand{\cu}{\Cal U} %{\cal U}} \newcommand{\p}{\Cal P} %{\cal P}} \newcommand{\z}{\Cal Z} %{\cal Z}} \newcommand{\e}{\Cal E} %{\cal E}} \newcommand{\cc}{\Cal C} %{\cal C}} \newcommand{\m}{\Cal M} %{\cal M}} \newcommand{\ct}{\Cal T} %{\cal T}} \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} \end{document} \endinput % Date: 16-JUN-1997 % \pagebreak: 0 \newpage: 0 \displaybreak: 0 % \eject: 0 \break: 0 \allowbreak: 0 % \clearpage: 0 \allowdisplaybreaks: 0 % $$: 0 {eqnarray}: 0 % \linebreak: 0 \newline: 0 % \mag: 0 \mbox: 0 \input: 0 % \baselineskip: 1 \rm: 2 \bf: 16 \it: 0 % \hsize: 0 \vsize: 0 % \hoffset: 0 \voffset: 0 % \parindent: 0 \parskip: 0 % \vfil: 0 \vfill: 0 \vskip: 0 % \smallskip: 10 \medskip: 1 \bigskip: 1 % \sl: 0 \def: 0 \let: 0 \renewcommand: 0 % \tolerance: 0 \pretolerance: 0 % \font: 0 \noindent: 0 % ASCII 13 (Control-M Carriage return): 0 % ASCII 10 (Control-J Linefeed): 0 % ASCII 12 (Control-L Formfeed): 0 % ASCII 0 (Control-@): 0 %