%_ ************************************************************************** %_ * 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 \controldates{26-AUG-1997,26-AUG-1997,26-AUG-1997,26-AUG-1997} \documentstyle[newlfont,draft]{era-l} %11pt]{article} \issueinfo{3}{13}{January}{1997} \dateposted{August 29, 1997} \pagespan{90}{92} \PII{S 1079-6762(97)00028-0} %{era-l} \newtheorem*{mthm1}{Main Theorem 1} \newtheorem*{mthm2}{Main Theorem 2} \newtheorem*{mthm3}{Main Theorem 3} \newtheorem*{cor}{Corollary} \newcommand{\cx}{\mathbf{C}} %{\bf C}} \newcommand{\g}{{\cal G}} \newcommand{\h}{{\cal H}} \renewcommand{\r}{{\cal R}} \newcommand{\s}{{\cal S}} \newcommand{\n}{{\cal N}} \newcommand{\cj}{{\cal J}} %\newcommand{\di}{{\rm div}} \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} \endinput 06-Aug-97 07:10:52-EST,345495;000000000000 Return-path:Received: from AXP14.AMS.ORG by AXP14.AMS.ORG (PMDF V5.1-8 #16534) id <01IM401N09LC000MGE@AXP14.AMS.ORG>; Wed, 6 Aug 1997 07:10:49 EST Date: Wed, 06 Aug 1997 07:10:46 -0400 (EDT) From: "pub-submit@ams.org " Subject: ERA/9700 To: pub-jour@MATH.AMS.ORG Reply-to: pub-submit@MATH.AMS.ORG Message-id: <01IM402VZBW2000MGE@AXP14.AMS.ORG> MIME-version: 1.0 Content-type: TEXT/PLAIN; CHARSET=US-ASCII Return-path: Received: from gate1.ams.org by AXP14.AMS.ORG (PMDF V5.1-8 #16534) with SMTP id <01IM2RP4VQ4W000GCJ@AXP14.AMS.ORG>; Tue, 5 Aug 1997 10:04:34 EST Received: from leibniz.math.psu.edu ([146.186.130.2]) by gate1.ams.org via smtpd (for axp14.ams.org [130.44.1.14]) with SMTP; Tue, 05 Aug 1997 14:00:36 +0000 (UT) Received: from gauss.math.psu.edu (aom@gauss.math.psu.edu [146.186.130.196]) by math.psu.edu (8.8.5/8.7.3) with ESMTP id KAA22095 for ; Tue, 05 Aug 1997 10:00:33 -0400 (EDT) Received: (aom@localhost) by gauss.math.psu.edu (8.8.5/8.6.9) id KAA10703 for pub-submit@MATH.AMS.ORG; Tue, 05 Aug 1997 10:00:30 -0400 (EDT) Date: Tue, 05 Aug 1997 10:00:30 -0400 (EDT) From: Alexander O Morgoulis Subject: *accepted to ERA-AMS, Volume 3, Number 1, 1997* To: pub-submit@MATH.AMS.ORG Message-id: <199708051400.KAA10703@gauss.math.psu.edu> %% Modified July 27, 1997 by A. Morgoulis %% Modified July 26, 1997 by A. Morgoulis % Date: 6-AUG-1997 % \pagebreak: 0 \newpage: 1 \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: 1 \bf: 23 \it: 0 % \hsize: 0 \vsize: 0 % \hoffset: 0 \voffset: 0 % \parindent: 0 \parskip: 0 % \vfil: 0 \vfill: 0 \vskip: 0 % \smallskip: 7 \medskip: 1 \bigskip: 0 % \sl: 0 \def: 0 \let: 0 \renewcommand: 0 % \tolerance: 0 \pretolerance: 0 % \font: 0 \noindent: 1 % ASCII 13 (Control-M Carriage return): 0 % ASCII 10 (Control-J Linefeed): 0 % ASCII 12 (Control-L Formfeed): 0 % ASCII 0 (Control-@): 0 %