 
\documentclass[12pt]{article}

\usepackage{amscd,amsmath, amssymb, fancyhdr, epsfig,url}
\setlength{\textwidth}{14cm}
\setlength{\textheight}{22.5cm}
\addtolength{\hoffset}{-4mm}
\addtolength{\voffset}{-12mm}

\numberwithin{equation}{section}

% CHANGELOG
% v. 0.1, 25.03.2013: started the file - M. V.

\newcommand{\version}{version 1.0,\ \   April 23, 2013}


\def\eqref#1{(\ref{#1})}
\newcommand{\goth}{\mathfrak}
\newcommand{\g}{{\mathfrak g}}
\newcommand{\arrow}{{\:\longrightarrow\:}}
\newcommand{\Z}{{\Bbb Z}}
\newcommand{\C}{{\Bbb C}}
\newcommand{\SLnH}{SL(n,{\Bbb H})}
\newcommand{\R}{{\Bbb R}}
\newcommand{\Q}{{\Bbb Q}}
\renewcommand{\H}{{\Bbb H}}
\newcommand{\6}{\partial}
\def\1{\sqrt{-1}\:}
\newcommand{\restrict}[1]{{\left|_{{\phantom{|}\!\!}_{#1}}\right.}}
\newcommand{\cntrct}                % contraction with a vector field
{\hspace{2pt}\raisebox{1pt}{\text{$\lrcorner$}}\hspace{2pt}}

% Taken from extarrows.sty by Chim Cut Canh
\makeatletter
\def\x@arrow{\DOTSB\Relbar}
\def\xlongequalsignfill@{\arrowfill@\x@arrow\Relbar\x@arrow}
\newcommand{\xlongequal}[2][]{%
        \ext@arrow 0099\xlongequalsignfill@{#1}{#2}}
\def\xlongrightarrowfill@{\arrowfill@\relbar\relbar\longrightarrow}
\newcommand{\xlongrightarrow}[2][]{%
        \ext@arrow 0099\xlongrightarrowfill@{#1}{#2}}
\makeatother





\renewcommand{\c}[1]{{\cal #1}}
\newcommand{\calo}{{\cal O}}
\newcommand{\cac}{{\cal C}}

% Correcting TeX...
\let\oldtilde=\tilde
%\renewcommand{\tilde}{\widetilde}
\renewcommand{\bar}{\overline}
\renewcommand{\phi}{\varphi}
\renewcommand{\epsilon}{\varepsilon}
\renewcommand{\geq}{\geqslant}
\renewcommand{\leq}{\leqslant}
\renewcommand{\max}{{\rm max}}


% Operatornames
\newcommand{\even}{{\rm even}}
\newcommand{\ev}{{\rm even}}
\newcommand{\odd}{{\rm odd}}
\newcommand{\im}{\operatorname{im}}
\newcommand{\comass}{\operatorname{\sf comass}}
\newcommand{\Av}{\operatorname{\sf Av}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\Tot}{\operatorname{Tot}}
\newcommand{\Id}{\operatorname{Id}}
\newcommand{\Cl}{\operatorname{Cl}}
\newcommand{\id}{\operatorname{\text{\sf id}}}
\newcommand{\const}{\operatorname{\text{\sf const}}}
\newcommand{\Vol}{\operatorname{Vol}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Hol}{\operatorname{Hol}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\Alt}{\operatorname{Alt}}
\newcommand{\Sec}{\operatorname{Sec}}
\newcommand{\Can}{\operatorname{Can}}
\newcommand{\Diff}{\operatorname{Diff}}
\newcommand{\Sing}{\operatorname{Sing}}
\newcommand{\codim}{\operatorname{codim}}
\newcommand{\coim}{\operatorname{coim}}
\newcommand{\coker}{\operatorname{coker}}
\newcommand{\dist}{\operatorname{dist}}
\newcommand{\diag}{\operatorname{\sf diag}}
\newcommand{\slope}{\operatorname{slope}}
\newcommand{\rk}{\operatorname{rk}}
\newcommand{\Lin}{\operatorname{Lin}}
\newcommand{\Hor}{\operatorname{Hor}}
\newcommand{\Def}{\operatorname{Def}}
\newcommand{\Tw}{\operatorname{Tw}}
\newcommand{\Tr}{\operatorname{Tr}}
\newcommand{\Spec}{\operatorname{Spec}}
\newcommand{\Spin}{\operatorname{Spin}}

\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Lemma, sublemma, corollary, proposition, theorem,             %
% definition,example defined there:                             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcounter{Mycounter}[section]
\newcounter{lemma}[section]
\setcounter{lemma}{0}
\renewcommand{\thelemma}{{Lemma \thesection.\arabic{lemma}}}
\newcommand{\lemma}{%
    \setcounter{lemma}{\value{Mycounter}}
    \refstepcounter{lemma}
    \stepcounter{Mycounter}
    {\noindent \bf \thelemma:\ }}

\newcounter{claim}[section]
\setcounter{claim}{0}
\renewcommand{\theclaim}{{Claim \thesection.\arabic{claim}}}
\newcommand{\claim}{%
    \setcounter{claim}{\value{Mycounter}}
    \refstepcounter{claim}
    \stepcounter{Mycounter}
    {\noindent \bf \theclaim:\ }}

\newcounter{sublemma}[section]
\setcounter{sublemma}{0}
\renewcommand{\thesublemma}{{Sublemma \thesection.\arabic{sublemma}}}
\newcommand{\sublemma}{%
    \setcounter{sublemma}{\value{Mycounter}}
    \refstepcounter{sublemma}
    \stepcounter{Mycounter}
    {\noindent \bf \thesublemma:\ }}

\newcounter{corollary}[section]
\setcounter{corollary}{0}
\renewcommand{\thecorollary}{{Corollary \thesection.\arabic{corollary}}}
\newcommand{\corollary}{%
    \setcounter{corollary}{\value{Mycounter}}
    \refstepcounter{corollary}
    \stepcounter{Mycounter}
    {\noindent \bf \thecorollary:\ }}

\newcounter{theorem}[section]
\setcounter{theorem}{0}
\renewcommand{\thetheorem}{{Theorem \thesection.\arabic{theorem}}}
\newcommand{\theorem}{%
    \setcounter{theorem}{\value{Mycounter}}
    \refstepcounter{theorem}
    \stepcounter{Mycounter}
    {\noindent \bf \thetheorem:\ }}

\newcounter{conjecture}[section]
\setcounter{conjecture}{0}
\renewcommand{\theconjecture}{{Conjecture \thesection.\arabic{conjecture}}}
\newcommand{\conjecture}{%
    \setcounter{conjecture}{\value{Mycounter}}
    \refstepcounter{conjecture}
    \stepcounter{Mycounter}
    {\noindent \bf \theconjecture:\ }}

\newcounter{proposition}[section]
\setcounter{proposition}{0}
\renewcommand{\theproposition}
      {{Proposition \thesection.\arabic{proposition}}}
\newcommand{\proposition}{%
    \setcounter{proposition}{\value{Mycounter}}
    \refstepcounter{proposition}
    \stepcounter{Mycounter}
    {\noindent \bf \theproposition:\ }}

\newcounter{definition}[section]
\setcounter{definition}{0}
\renewcommand{\thedefinition}
      {{Definition~\thesection.\arabic{definition}}}
\newcommand{\definition}{%
    \setcounter{definition}{\value{Mycounter}}
    \refstepcounter{definition}
    \stepcounter{Mycounter}
    {\noindent \bf \thedefinition:\ }}


\newcounter{example}[section]
\setcounter{example}{0}
\renewcommand{\theexample}{{Example \thesection.\arabic{example}}}
\newcommand{\example}{%
    \setcounter{example}{\value{Mycounter}}
    \refstepcounter{example}
    \stepcounter{Mycounter}
    {\noindent \bf \theexample:\ }}

\newcounter{remark}[section]
\setcounter{remark}{0}
\renewcommand{\theremark}{{Remark \thesection.\arabic{remark}}}
\newcommand{\remark}{%
    \setcounter{remark}{\value{Mycounter}}
    \refstepcounter{remark}
    \stepcounter{Mycounter}
    {\noindent \bf \theremark:\ }}


\newcounter{problem}[section]
\setcounter{problem}{0}
\renewcommand{\theproblem}{{Problem \thesection.\arabic{problem}}}
\newcommand{\problem}{%
    \setcounter{problem}{\value{Mycounter}}
    \refstepcounter{problem}
    \stepcounter{Mycounter}
    {\noindent \bf \theproblem:\ }}


\newcounter{question}[section]
\setcounter{question}{0}
\renewcommand{\thequestion}{{Question \thesection.\arabic{question}}}
\newcommand{\question}{%
    \setcounter{question}{\value{Mycounter}}
    \refstepcounter{question}
    \stepcounter{Mycounter}
    {\noindent \bf \thequestion:\ }}

\renewcommand{\theequation}{\thesection.\arabic{equation}}

\makeatletter


\setlength{\headheight}{15pt} \pagestyle{fancy}  \cfoot{-- \thepage \ -- } \rfoot{\tiny \sc\version}
\renewcommand{\leftmark}{{\scriptsize K\"ahler threefolds having no subvarieties}}
\renewcommand{\rightmark}{{\small 
F.\ Campana, J.-P.\ Demailly, M.\ Verbitsky}}


\@addtoreset{equation}{section} \@addtoreset{footnote}{section} \makeatother


\def\blacksquare{\hbox{\vrule width 5pt height 5pt depth 0pt}}
\def\endproof{\blacksquare}
\def\shortdash{\mbox{\vrule width 4.5pt height 0.55ex depth -0.5ex}}

\addtolength{\topmargin}{-5mm} \addtolength{\textheight}{10mm}
%\addtolength{\oddsidemargin}{-10mm}
%\addtolength{\textwidth}{20mm}
%DIF PREAMBLE EXTENSION ADDED BY LATEXDIFF
%DIF UNDERLINE PREAMBLE %DIF PREAMBLE
\RequirePackage[normalem]{ulem} %DIF PREAMBLE
\RequirePackage{color}\definecolor{RED}{rgb}{1,0,0}\definecolor{BLUE}{rgb}{0,0,1} %DIF PREAMBLE
\providecommand{\DIFadd}[1]{{\protect\color{blue}\uwave{#1}}} %DIF PREAMBLE
\providecommand{\DIFdel}[1]{{\protect\color{red}\sout{#1}}}                      %DIF PREAMBLE
%DIF SAFE PREAMBLE %DIF PREAMBLE
\providecommand{\DIFaddbegin}{} %DIF PREAMBLE
\providecommand{\DIFaddend}{} %DIF PREAMBLE
\providecommand{\DIFdelbegin}{} %DIF PREAMBLE
\providecommand{\DIFdelend}{} %DIF PREAMBLE
%DIF FLOATSAFE PREAMBLE %DIF PREAMBLE
\providecommand{\DIFaddFL}[1]{\DIFadd{#1}} %DIF PREAMBLE
\providecommand{\DIFdelFL}[1]{\DIFdel{#1}} %DIF PREAMBLE
\providecommand{\DIFaddbeginFL}{} %DIF PREAMBLE
\providecommand{\DIFaddendFL}{} %DIF PREAMBLE
\providecommand{\DIFdelbeginFL}{} %DIF PREAMBLE
\providecommand{\DIFdelendFL}{} %DIF PREAMBLE
%DIF END PREAMBLE EXTENSION ADDED BY LATEXDIFF

\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\LARGE\bf
Compact K\"ahler 3-manifolds without non-trivial subvarieties\\[4mm]
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Fr\'ed\'eric Campana, Jean-Pierre Demailly, Misha Verbitsky\footnote{Misha Verbitsky is
partially supported  by RFBR grants
 12-01-00944-a,  
AG Laboratory NRI-HSE, RF government grant,
ag. 11.G34.31.0023, and NRI-HSE 
Academic Fund Program in 2013-2014, research grant
12-01-0179.
 }

\end{center}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\small \hspace{0.10\linewidth}
\begin{minipage}[t]{0.85\linewidth}
{\bf Abstract} \\
We prove that any compact K\"ahler 3-dimensional manifold
which has no non-trivial complex subvarieties is a torus. This is a very special case of a general conjecture \DIFaddbegin \DIFadd{on the structure of `simple manifolds'}\DIFaddend , central in the bimeromorphic classification of compact K\"ahler manifolds. The proof follows from the Brunella pseudo-effectivity theorem, \DIFaddbegin \DIFadd{combined with }\DIFaddend fundamental results of Siu and \DIFdelbegin \DIFdel{Demailly }\DIFdelend \DIFaddbegin \DIFadd{of the second author }\DIFaddend on the Lelong numbers of closed positive $(1,1)$-currents, and \DIFdelbegin \DIFdel{on the Takegoshi and Demailly-Peternell-Schneider }\DIFdelend \DIFaddbegin \DIFadd{with a }\DIFaddend version of the hard Lefschetz theorem for pseudo-effective line bundles, \DIFdelbegin \DIFdel{together with special }\DIFdelend \DIFaddbegin \DIFadd{due to Takegoshi and Demailly-Peternell-Schneider. Special }\DIFaddend features of the Riemann-Roch formula in dimension $3$ \DIFaddbegin \DIFadd{are also used}\DIFaddend .
\end{minipage}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%\tableofcontents


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Introduction}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\subsection{K\"ahler manifolds without non-trivial subvarieties}


We consider here connected compact K\"ahler manifolds $X$ of (complex) dimension $n>1$. An irreducible compact analytic subset $Z$ of $X$ will be said to be a `subvariety' of $X$. It is said to be `non-trivial' if its (complex) dimension is neither $0$, nor $n$.

The bimeromorphic classification of compact K\" ahler manifolds can be reduced, by means of suitable functorial\footnote{In the category of connected compact K\"ahler manifolds, morphisms being the dominant rational maps with connected fibres.} fibrations\footnote{Relative algebraic reductions, and relative Albanese maps, see \cite{_Ca80_}, \cite{_Ca85_}, \cite{_F_}.}, to the following two extreme particular cases: either $X$ is projective, or $X$ is `simple', which means that its `general'\footnote{`general' means: in the intersection of countably many dense Zariski open subsets.}) point $x$ of $X$ is not contained in any non-trivial subvariety of $X$.

The present text is concerned with the bimeromorphic classification of such `simple' $X$. Its difficulty, in contrast to the projective case, is not due to the
abundance and complexity of the examples, but to their (expected) scarceness
and `simple structure', which makes essentially all usual invariants of the classification vanish.

For example, if $X$ is `simple', its algebraic dimension\footnote{Recall that $a(X)\in \{0,...,n\}$ is the transcendence degree of the field of global meromorphic functions on $X$, $a(X)=n$ if and only if $X$ is projective.} $a(X)$ vanishes, which implies, by \cite{_U_}, theorem 9.3, that its Albanese map is surjective and has connected fibres. Because the fibres must then have dimension either $0$ or $n$, we get: either $q=0$, or $X$ is bimeromorphic to its Albanese torus. Thus only the case where $q = 0$ needs to be considered.

The known\footnote{It is shown in \cite{_KV1_} that a stable bundle which does not degenerate to a direct sum of stable bundles on the generic deformation of a Hilbert scheme of $K3$ has a compact moduli space; this may lead to new examples of hyperk\"ahler manifolds. The `rigidity' of the category of coherent sheaves on the general members of these families is established in \cite{_V02_}, \cite{_V03_}.}
examples of `simple' compact K\"ahler manifolds are (up to bimeromorphic equivalence) `general' complex tori, $K3$ surfaces, or the `general' member of the deformation families of hyperk\"ahler manifolds\footnote{Recall that $X$ is `hyperk\"ahler' it is K\"ahler and admits a holomorphic symplectic form. It is said to be `irreducible' if, moreover, $\pi_1(X)$ is finite, and $h^{2,0} (X) = 1$.}. `Simple' surfaces are thus classified: either Tori ($q > 0$), or $K3$ (q = 0). But when $n\geq 3$, the classification is open. The situation is however expected to be similar to the
surface case in higher dimensions (see Conjecture 1.1), where the only known
examples are constructed out of surfaces which are either $K3$, or tori.

\DIFaddbegin \

\DIFaddend The following was formulated in \cite{_Ca05_}, as question 1.4.

\

\begin{conjecture} Let $X$ be a `simple' compact K\"ahler manifold. Then:
\begin{itemize}
\item[\bf 1.] either $X$ has a finite \'etale cover bimeromorphic to a complex torus, or $H^0(X,\Omega^2_X)$ is generated by some $\sigma$ which is `generically symplectic' (i.e. $n=2m$ is even, and $\wedge^m \sigma\neq 0$). 

We should thus have: $\kappa(X)=0$. If $\dim(X)$ is odd, then $X$ should be bimeromorphic to a complex torus, possibly after some finite \'etale cover. Such a cover will be needed, as shown by the Kummer quotient by the~$-1$ involution.

\item[\bf 2.] When $X$ does not contain any nontrivial subvariety, $X$ should be either a complex torus, or an irreducible hyperk\"ahler manifold\footnote{Notice that a general deformation of a Hilbert scheme of a $K3$ surface has no subvarieties \cite{_V98_}, while a general deformation of a generalized Kummer variety has some subvarieties, partially classified in \cite{_KV2_}.}.

Thus $K_X$ should be trivial in this case, the two cases being told by $q>0$, or $q=0$.

\item[\bf 3.] If $X$ is `generically symplectic', then $\pi_1(X)$ should be finite, of cardinality at most $2^m$, where $2m=n$.
\end{itemize}
\end{conjecture} 

\
The assertion 3 follows from \cite{_Ca85_}, if $\chi({\cal O}_X)\neq 0$ for $X$ `simple' and `generically symplectic'.

\

In the sequel we prove \DIFdelbegin \DIFdel{that }\DIFdelend the second assertion of the conjecture for $n=3$.

\

This conjecture can be motivated by the conjectural existence of minimal models in the bimeromorphic category of connected compact K\" ahler manifolds. Indeed, if such a theory exists, and if $X$ is simple, we have $a(X)=0$, which brings $\kappa(X)\leq 0$. The possibility $\kappa(X)=-\infty$ is excluded, since $X$ should then be uniruled. This implies $\kappa(X)=0$.

In this case, a K\"ahler minimal model is a map $X\dasharrow X'$  such that $X'$ has terminal singularities and $K_{X'}\equiv 0$). A (conjectural) Bogomolov-type decomposition (\cite{_bogomolov_}, \cite{_Beauville_c1=0_}) for manifolds with terminal singularities and $K_{X'}\equiv 0$ could be used to represent a finite cover of $X'$ as a product. However $X$ is simple, hence either $X'$ is covered by a simple torus (with finite locus of ramification), or $X'$ carries a symplectic holomorphic $2$-form, establishing the conjecture above.

\hfill

\theorem\label{_Main_Theorem_}
Let $X$ be a compact K\"ahler 3-dimensional manifold.
Assume that $X$ has no non-trivial subvariety.
Then $X$ is a torus.


\


The proof will be given as \DIFdelbegin \DIFdel{corollary }\DIFdelend \ref{mth}. Although it is an easy combination of known results, it seems worth to present it, because it indicates that the above conjecture might be accessible in dimension $3$ with the actually existing techniques (see \DIFdelbegin \DIFdel{Remark }\DIFdelend \ref{finalrk}). Let us start with an easy remark:

\

\begin{lemma}\label{khi} Let $X$ be a `simple' compact K\"ahler threefold. If $\chi({\cal O}_X)=0$, then $X$ is bimeromorphic to its Albanese torus.
\end{lemma} 

\

\noindent
{\bf Proof:} By the above remarks, it is sufficient to show that $q>0$. But 
\DIFdelbegin \DIFdel{$0=\chi(\cal{O}$$_X)=1-q+h^{2,0}-h^{3,0}$.}\DIFdelend \DIFaddbegin \begin{DOLLARDOLLAR}\DIFadd{0=\chi(}{\cal \DIFadd{O}}\DIFadd{_X)=1-q+h^{2,0}-h^{3,0}.}$$
\DIFaddend By Kodaira's theorem, $h^{2,0}>0$ since $X$ is not projective, and $h^{3,0}\leq 1$ since $a(X)=0$. Thus $0=1-q+h^{2,0}-h^{3,0}\geq 1-q+1-1=1-q$, and $q>0$\DIFaddbegin \DIFadd{. }\DIFaddend \endproof

\

The proof of \ref{_Main_Theorem_} is thus reduced to showing that \DIFdelbegin \DIFdel{$\chi(\cal{O}\end{DOLLARDOLLAR}_X)=0$}\DIFdelend \DIFaddbegin \DIFadd{$\chi(\cal{O}$$_X)=0$}\DIFaddend , which is the objective of the following sections.

\

Finally, let us mention that compact (connected) complex manifolds, not necessarily K\"ahler, without non-trivial subvarieties are also quite important in model theory, giving examples of so-called `trivial' Zariski geometries (\cite{_Moo_}, \cite{_MP_} \cite{_MMT_}). Our arguments below show that compact three-dimensional complex manifolds without non-trivial subvarieties are tori, provided their canonical bundle are pseudo-effective. Brunella's theorem is indeed the single step of our proof requiring the K\"ahler hypothesis.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Pseudoeffective currents and line bundles}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We refer to \DIFdelbegin %DIFDELCMD < \cite{_Demailly:Reg_} %%%
\DIFdelend \DIFaddbegin \cite{_Demailly:ecole_} \DIFaddend for the basic notions and properties of currents on complex manifolds, and only recall briefly some of the facts needed here.

\

Let $X$ be a compact, $n$-dimensional complex manifold, and $L$ a holomorphic line bundle on $X$. Assume that $L$ is equipped with a singular hermitian metric $h$ with local weights $e^{-\varphi}$ which are locally integrable. Its curvature current is then $\Theta(L,h):=i\partial \overline{\partial }\varphi$, and its first Chern class $c_1(L)$ is represented by the cohomology class \DIFdelbegin \DIFdel{$\frac{1}{2\pi}.[\Theta(L,h)]\in H^{1,1}(X)\subset H^2(X,\Bbb R)$}\DIFdelend \DIFaddbegin \DIFadd{$\frac{1}{2\pi}\;[\Theta(L,h)]\in H^{1,1}(X)\subset H^2(X,\Bbb R)$}\DIFaddend .

\

\definition A positive, closed (1,1)-current $\Theta$ is
{\bf nef} (resp.\ pseudo-effective) if it is a limit of positive, closed $(1,1)$-forms (resp. if it is positive, or equivalently, if the functions $\varphi$ are plurisubharmonic). If $\Theta$ is nef, and $\int_M [\Theta]^n >0$, $\Theta$ is said to be {\bf big}\footnote{A more general notion exists, assuming $\Theta$ pseudo-effective only.}. A line bundle $L$ on $X$ is said to be nef (resp. pseudo-effective, resp. big) if it admits a singular metric $h$ such that $\Theta(L,h)$ has the same property.

\

\begin{theorem}\label{siu} (\cite{_Siu2_})  Let $X$ be a compact complex manifold. Assume that $X$ carries a nef and big line bundle $L$. Then $X$ is bimeromorphic to a projective manifold (using sections of $L^{\otimes m}$, for $m>0$ sufficiently large). In particular, $X$ is not simple.
\end{theorem}

\

This result has been generalized \DIFdelbegin \DIFdel{by Demailly by means of his }\DIFdelend \DIFaddbegin \DIFadd{in }\cite{_Demailly:Morse_}\DIFadd{, using }\DIFaddend holomorphic Morse inequalities\DIFdelbegin \DIFdel{(see }%DIFDELCMD < \cite{_Demailly:Morse_}%%%
\DIFdel{)}\DIFdelend . This improved version is however not used here.


\


\definition
The {\bf Lelong number} $\nu_x(\Theta(L,h))=\nu(\varphi,x)$ 
of the $(1,1)$-current $\Theta(L,h)$ at $x\in X$, is defined as \DIFdelbegin \DIFdel{$lim$ $ inf_{z\to x}\frac{\varphi(z)}{\vert z-x\vert}$}\DIFdelend \DIFaddbegin \DIFadd{$\liminf_{z\to x}\frac{\varphi(z)}{\vert z-x\vert}$}\DIFaddend , for a local metric on $X$ near $x$.


\hfill

\definition
 For a positive real number $c>0$, {\bf the Lelong set} $F_c$
 of a (1,1)-current $\eta$ is the set of points 
 $x\in M$ with $\nu(\eta,x) \geq c$.
By \DIFdelbegin \DIFdel{Siu's theorem }\DIFdelend \DIFaddbegin \DIFadd{a well-known theorem of Siu }\DIFaddend (\cite{_Siu:1974_}\DIFdelbegin \DIFdel{, with a considerably simplified proof using Demailly's regularization of currents and the Ohsawa-Takegoshi extension theorem}\DIFdelend ) any Lelong set of a positive, closed current is a complex analytic subvariety of $M$
\DIFdelbegin \DIFdel{.
}\DIFdelend \DIFaddbegin \DIFadd{(the proof of this difficult result has been considerably simplified, using regularization of currents and the Ohsawa-Takegoshi extension theorem, see }\cite{_Demailly:Reg_}\DIFadd{.)
}\DIFaddend 

\

\begin{theorem}\label{nef}(\cite{_Demailly:Reg_}) Let $X$ be a compact complex manifold. Let $L$ be a pseudo-effective holomorphic line bundle on $X$, with singular hermitian metric $h$ with positive curvature current $\Theta(L,h)$. Assume that the Lelong sets of $\Theta(L,h)$ are all zero-dimensional. Then $L$ is nef, and big \DIFdelbegin \DIFdel{unless all Lelong numbers vanish}\DIFdelend \DIFaddbegin \DIFadd{as soon as there is at least one nonzero Lelong number}\DIFaddend .\end{theorem}

\

\noindent
{\bf Proof:} The first assertion is (\cite{_Demailly:Reg_}, corollary 6.4). The second \DIFdelbegin \DIFdel{results from theorem }%DIFDELCMD < \cite{_Siu2_} %%%
\DIFdel{applied to }\DIFdelend \DIFaddbegin \DIFadd{follows from (}\DIFaddend \cite{_Demailly:Reg_}, corollary 7.6\DIFaddbegin \DIFadd{)}\DIFaddend . See also \cite{_Verbitsky:parabolic_}, theorem 3.12.
\endproof

\

\begin{corollary}\label{zeroLelong} If $X$ is a compact K\"ahler manifold without nontrivial subvarieties, and with pseudo-effective canonical bundle, then $K_X$ is nef, and for any singular hermitian metric $h$ on $K_X, m>0$ with positive curvature current, the Lelong number vanish at any point. In particular, the associated multiplier ideal sheaves on $K_X^{\otimes m}$ are all trivial for any $m>0$ (i.e: \DIFdelbegin \DIFdel{$e^{-m.\varphi}$ }\DIFdelend \DIFaddbegin \DIFadd{$e^{-m\,\varphi}$ }\DIFaddend is integrable for any $m>0$).
\end{corollary}


\section{Hard Lefschetz theorem for the cohomology of pseudo-effective line bundles}

We recall the version of the Hard Lefschetz theorem which is going to be used here:

\

\theorem\label{THL}(\cite{_Takegoshi_}, \cite{_Demailly_Peternell_Schneider:ps-eff_}) Let $(X, \omega)$ be a compact K\"ahler manifold, of dimension $n$ with K\"ahler form $\omega$, let
$K_X$ be its canonical bundle,
and $L$ be a pseudo-effective holomorphic line bundle on $X$ equipped with a singular
Hermitian metric $h$. Assume that the 
curvature $\Theta$ of $(L,h)$ is a positive current on $X$,
and denote by ${\cal I}(h)$ the corresponding multiplier ideal sheaf.
Then the wedge multiplication operator $\eta \arrow \omega^i \wedge \eta$
induces a surjective map
\[
H^0(X,\Omega^{n-i}_X\otimes  L\otimes {\cal I}(h))\stackrel {\omega^i \wedge
\cdot}\arrow H^i(X,K  \otimes  L\otimes {\cal I}(h)).
\]
Here $\omega$ is considered as an element in $H^1(X,\Omega^1_X)$,
and multiplication by $\omega$ maps $H^k(X,\Omega^{n-l}_X\otimes F)$
to $H^{k+1}(X,\Omega^{n-l+1}_X\otimes F)$.
\endproof

\

This theorem was obtained under various hypothesis during the decade 1990: see
\cite{_Enoki:semipositive_}, and
\cite{_Mourougane_}. The most general form given here is due to \cite{_Demailly_Peternell_Schneider:ps-eff_}, Theorem 2.1.1. It was proved in \cite{_Takegoshi_} when $L$ nef.

\hfill

\begin{corollary}\label{a=0} Let $X$ be a compact K\"ahler manifold of dimension $n>1$ without non-trivial subvariety. Assume that $K_X$ is pseudo-effective. Then $h^i(X,m.K_X)\leq \big(^n _i\big)$, for any $i\geq 0$, and the polynomial $P(m):=\chi(X,m.K_X)$ is constant, equal to \DIFdelbegin \DIFdel{$\chi(\cal{O}\begin{DOLLARDOLLAR}_X)$}\DIFdelend \DIFaddbegin \DIFadd{$\chi(\cal{O}$$_X)$}\DIFaddend .
\end{corollary}

\

\noindent
{\bf Proof:} Since $n>1$ and $X$ has no non-trivial subavrieties, the algebraic imension of $X$ vanishes: $a(X)=0$. Therefore $h^0(X,E)\leq \hbox{\rm rank}(E)$, for any holomorphic vector bundle $E$ on $X$. Combining with the Hard Lefschetz theorem (Theorem \ref{THL}), this gives the first claim of \ref{a=0}. The second claim is clear because a polynomial function $P(m)$ which remains bounded when $m\to+\infty$ is necessarily constant.
\endproof

\

\begin{corollary}\label{n=3} Let $X$ be a compact K\"ahler manifold of dimension $3$ without non-trivial subvariety. Assume that $K_X$ is pseudo-effective. The  polynomial $P(m):=\chi(X,m.K_X)$ is constant, equal to $\chi({\cal O}_X)=0$.
\end{corollary}

\

\noindent
{\bf Proof:} The intersection number $K_X^3$ vanishes (either by \ref{}, or because it is the leading term of $P(m)$, up to the factor $3!$). The Riemann-Roch formula then gives: $P(m):=\frac{(1-2n)}{24}.c_1(X).c_2(X)$. The boundedness of $P(m)$ then implies that \DIFdelbegin \DIFdel{$24.\chi(\cal{O}\end{DOLLARDOLLAR}_X)=c_1(X).c_2(X)=0$}\DIFdelend \DIFaddbegin \DIFadd{$24.\chi(\cal{O}$$_X)=c_1(X).c_2(X)=0$}\DIFaddend . \endproof

\

\begin{remark} Arguments close to some of the ones presented here have been already used in the proof \cite{_Demailly_Peternell_Schneider:ps-eff_}, theorem 2.7.3: if $X$ is a compact K\"ahler manifold with pseudo-effective canonical bundle admitting a metric with weights $\varphi$ having analytic singularities and positive curvature current, then either $H^0(X, \Omega^i_X\otimes (m.K_X))\neq 0$ for infinitely many $m>0$ and some $i\geq 0$, or \DIFdelbegin \DIFdel{$\chi(X, \cal{O}\begin{DOLLARDOLLAR}_X)= 0$}\DIFdelend \DIFaddbegin \DIFadd{$\chi(X, \cal{O}$$_X)= 0$}\DIFaddend .
\end{remark}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Brunella's pseudo-effectivity criterion.}

The rest of our arguments is based on the following strong (and very difficult)
theorem by Brunella.

\

\begin{theorem}\label{Bru}(\cite{_Brunella:pseudoeff_}) Let $X$ be a compact K\"ahler manifold with a $1$-dimensional holomorphic foliation $F$ given by a nonzero morphism of vector bundle $L\to T_X$, where $L$ is a line bundle on $X$, and $T_X$ is its holomorphic tangent bundle. If $L^{-1}$ is not pseudo-effective, the closures of the leaves of $F$ are rational curves (and $X$ is thus uniruled).
 \end{theorem}

\

The following corollary had already been observed in \cite{_HPR_}, proposition 4.2.

\

\begin{corollary} If $X$ is an $n$-dimensional compact K\"ahler manifold with $H^0(X,\Omega^{n-1}_X)\neq 0$. Then $K_X$ is \DIFdelbegin \DIFdel{pseudo effective}\DIFdelend \DIFaddbegin \DIFadd{pseudo-effective}\DIFaddend . 
\end{corollary}

\

\noindent
{\bf Proof:} $\Omega^{n-1}_X$ is canonically isomorphic to $K_X\otimes T_X$. Any nonzero section thus provides a nonzero map $K_X^{-1}\to T_X$, and an associated foliation \endproof

\

\begin{corollary}\label{s+n=3} If $X$ is a $3$-dimensional non-projective compact K\"ahler manifold. Then: either $K_X$ is \DIFdelbegin \DIFdel{pseudo effective}\DIFdelend \DIFaddbegin \DIFadd{pseudo-effective}\DIFaddend , or $X$ is uniruled. If $X$ is `simple', $K_X$ is \DIFdelbegin \DIFdel{pseudoeffective}\DIFdelend \DIFaddbegin \DIFadd{pseudo-effective}\DIFaddend .
\end{corollary}

\

\noindent
{\bf Proof:}  Kodaira's theorem implies
that any compact K\"ahler manifold with $H^{2,0}(X)=0$
is projective. Thus $H^{2,0}(X)\neq 0$, and the preceding corollary applies (since $2=(n-1)$, here) and gives the first claim. 
If $X$ is uniruled, it is not simple, hence the second assertion. \endproof

\

By combining \ref{s+n=3}, \ref{n=3}, and \ref{khi}, we get our main result:

\

\begin{corollary}\label{mth} If $X$ is a $3$-dimensional compact K\"ahler manifold without non-trivial subvariety, then $X$ is a complex torus.\end{corollary}

\


In higher dimensions, we can replace the assumption that $K_X$ was pseudo-effective from our preceding corollary \ref{a=0} by the existence of a holomorphic $(n-1)$-form:

\

\begin{corollary}\label{} Let $X$ be a compact K\"ahler manifold of dimension $n>1$ without non-trivial subvariety. Assume that $H^0(X,\Omega^{n-1}_X)\neq 0$. Then $h^i(X,m.K_X)\leq \big(^n _i\big)$, for any $i\geq 0$, and the polynomial $P(m):=\chi(X,m.K_X)$ is constant, equal to $\chi({\cal O}_X)$\footnote{The intersection numbers \DIFdelbegin \DIFdel{$K_X^j.Todd_{n-j}(X)$ }\DIFdelend \DIFaddbegin \DIFadd{$K_X^j\cdot\hbox{\rm Todd}_{n-j}(X)$ }\DIFaddend thus all vanish when $j>0$, as expected since $K_X$ should then be trivial.}.\end{corollary}

\

\begin{remark}\label{finalrk} In order to show that if $X$ is compact K\"ahler and `simple', of dimension $3$, then $X$ is bimeromorphic to a torus, it were sufficient either:

\begin{itemize}

\item[\bf A.] show that \DIFdelbegin \DIFdel{$\chi(\cal{O}\end{DOLLARDOLLAR}_X)=0$}\DIFdelend \DIFaddbegin \DIFadd{$\chi(\cal{O}$$_X)=0$}\DIFaddend , by a suitable control of the multiplier ideal sheaves on $K_X^{\otimes m}, m>0$, or proceed in the following two steps:

\item[\bf B1.] Assume first that $K_X$ is nef: $X$ should then contain no divisor, but possibly curves. One might then show that \DIFdelbegin \DIFdel{$\chi(\cal{O}$$_X)=0$ }\DIFdelend \DIFaddbegin \DIFadd{$\chi(\cal{O}$$_X)=0$ }\DIFaddend through a suitable control  of the multiplier ideal sheaves of $K_X^{\otimes m}, m>0$, which are concentrated on curves and points.

\item[\bf B2.] In the remaining case where $K_X$ is pseudo-effective, but not nef, by \cite{_Demailly_Peternell_Schneider:ps-eff_}, theorem 5.4, and \cite{P1}, \cite{P2}, one has a Mori contraction. Showing that its image can be chosen to be smooth and K\"ahler would permit to conclude by induction on the rank of $H^2(X,\Bbb R)$.

\end{itemize}
\end{remark}







\hfill

{\bf Acknowledgements:} We are grateful to the Tata Institute
of Fundamental research in Mumbai, India, where this paper was conceived, and especially to I. Biswas and A.J Parameswaran, organizers  of our stay there.

%We would like also to thank Jean-Pierre Demailly for giving us explanations and references for \ref{nef}.


\hfill

{\small
\begin{thebibliography}{AV1}

\bibitem[Bea]{_Beauville_c1=0_} A. Beauville. {\em Vari\'et\'es K\"ahleriennes dont la premi\`ere classe de Chern est nulle}. J.\ Differential Geom.\ 18 (1983), no. 4, 755–782.

\bibitem[Bo1]{_bogomolov_} F.A. Bogomolov. {\em On the decomposition of K¨hler manifolds with trivial canonical class.} Mat. USSR-Sb. 22 (1974), no. 4, 580–583.

\bibitem[Br]{_Brunella:pseudoeff_} M. Brunella. {\em Uniformization of foliations by curves}.  Holomorphic dynamical systems,  \DIFdelbegin \DIFdel{105???163}\DIFdelend \DIFaddbegin \DIFadd{105-163}\DIFaddend , Lecture Notes in Math., 1998, Springer, Berlin, 2010. Also arXiv.math/0802.4432. 


\bibitem[C1]{_Ca80_} Fr\'ed\'eric Campana. {\em R\'eduction alg\'ebrique d'un morphisme faiblement K\" ahl\'erien propre}. Math. Ann. 256 (1980), 157-189.

\bibitem[C2]{_Ca85_} Fr\'ed\'eric Campana. {\em R\'eduction d'Albanese d'un morphisme propre et faiblement K\" ahl\'erien et applications (I, II)}. Comp. Math. 54 (1985), 373-416.

\bibitem[C3]{_Ca05_} Fr\'ed\'eric Campana. {\em Isotrivialit\'e de certaines familles K\" ahl\'eriennes de vari\'et\'es non projectives}. Math. Z. 252 (2006), 147-156 . Also arXiv.math/ 0408148.



\bibitem[C4]{_Ca85_} Fr\'ed\'eric Campana. {\em Fundamental group and positivity of cotangent bundles of compact
K\" ahler manifolds}. J. Algebraic Geom., 4(3):487?502, 1995.


\bibitem[D1]{_Demailly:Morse_}
J.-P. Demailly,
{\em Holomorphic Morse inequalities},
Several complex variables and complex geometry,
Part 2 (Santa Cruz, CA, 1989), 93--114,
Proc. Sympos. Pure Math., 52, Part 2,
Amer. Math. Soc., Providence, RI, 1991.


\bibitem[D2]{_Demailly:Reg_} 
Demailly, Jean-Pierre,
{\em Regularization of closed positive currents 
and Intersection Theory,} J. Alg. Geom. {\bf 1} (1992) 361-409


\bibitem[D3]{_Demailly:ecole_}
Demailly, Jean-Pierre,
{\em Analytic methods in algebraic geometry,} Lecture Notes,
\'Ecole d'\'et\'e de Mathematiques de Grenoble ``G\'eom\'etrie
des vari\'et\'es projectives complexes : programme du
mod\`ele minimal'' (June-July 2007)

%\bibitem[D2]{_Demailly:Chern_Reg_}
%Demailly, Jean-Pierre,
%{\em Regularization of closed positive 
%currents of type (1,1) by the flow of a 
%Chern connection,} Actes du Colloque en l'honneur de P. Dolbeault (Juin 1992),  
%Aspects of Mathematics, Vol. E 26, Vieweg, (1994) 105-126

%\bibitem[D3]{_Demailly:Trento_}
%Demailly, Jean-Pierre,
%{\em Monge-Amp\`ere operators, Lelong numbers and intersection
%theory,} Complex Analysis and Geometry, Univ. Series in
%Math., edited by V. Ancona and A. Silva, Plenum Press,
%New-York (1993) 


%\bibitem[DP]{_Demailly_Paun_} Demailly, J.-P., Paun, M., {\em Numerical characterization of the K\"ahler cone of a compact K\"ahler manifold}, math.AG/0105176 also in Annals of Mathematics, {\bf 159} (2004), 1247-1274.



\bibitem[DPS]{_Demailly_Peternell_Schneider:ps-eff_}
Jean-Pierre Demailly, 
Thomas Peternell, Michael Schneider,
{\em Pseudo-effective line bundles on compact K\"ahhler
manifolds}, International Journal of Math. {\bf 6} (2001),
pp. 689-741.


\bibitem[E]{_Enoki:semipositive_}
Enoki, Ichiro, {\em Strong-Lefshetz-type theorem for
semi-positive line bundles over compact K\"ahler manifolds},
Geometry and global analysis (Sendai, 1993),  211--212,
Tohoku Univ., Sendai, 1993

\bibitem[F]{_F_} A. Fujiki. {\em On the structure of compact complex manifolds in C}.  Algebraic varieties and analytic varieties (Tokyo, 1981),  231???302, Adv. Stud. Pure Math., 1, North-Holland, Amsterdam, 1983. 

\bibitem[HPR]{_HPR_} A. H\" oring, T. Peternell, I. Radloff. {\em Uniformisation in dimension four: toward a conjecture of Iitaka}. arXiv/math. 1103.5392. 

\bibitem[KV1]{_KV1_} M.\ Verbitsky. {\em Hyperholomorphic sheaves and new examples of hyperk\"ahler manifolds}. alg-geom 9712012, a chapter in a book ``Hyperk\"ahler manifolds'' (joint with D.~Kaledin), Int'l Press, Boston, 2001.

\bibitem[KV2]{_KV2_} D. Kaledin, M. Verbitsky. {\em Partial resolutions of Hilbert type, Dynkin diagrams, and generalized Kummer varieties.} 33 pages, arXiv:math/9812078

\bibitem[Moo]{_Moo_} R. Moosa, {\em The model theory of compact complex spaces}, Logic Colloquium '01, volume 20 of Lect. Notes Log., pages 317-349, Assoc. Symbol. Logic, 2005.

\bibitem[MMT]{_MMT_} R. Moosa, R. Moraru, M. Toma, {\em An essentially saturated surface not of Kaehler-type}, Bull. Lond. Math. Soc., volume 40 (2008), number 5, 845–854.

\bibitem[MP]{_MP_} R. Moosa, A. Pillay, {\em Model theory and Kaehler geometry}, in: Model Theory with Applications to Algebra and Analysis Volume 1 (Eds. Z. Chatzidakis, D. Macpherson, A. Pillay, A. Wilkie), London Mathematical Society Lecture Note Series Nr 349, Cambridge University Press 2008, 167–195.

\bibitem[Mou]{_Mourougane_} 
Mourougane, Ch., {\em Th\'eor\`emes d'annulation 
g\'en\'eriques pour les fibr\'es vectoriels
semi-n\'egatifs}, Bull. Soc. Math. Fr. 127 (1999) 115--133.

\bibitem[P1]{P1} Peternell, Th.: {\em Towards a Mori theory on compact K\"ahler 3-folds, II}. Math. Ann. 311
(1998), 729-764.


\bibitem[P2]{P2}  Peternell, Th.: {\em Towards a Mori theory on compact K\"ahler 3-folds, III}. Bull. Soc. Math. France. 129 (2001), 339-356.



\bibitem[Si1]{_Siu:1974_}
Y.T. Siu.,
{\em Analyticity of sets associated to Lelong numbers and the extension of
  closed positive currents}, Invent. Math., 27 (1974), 53-156.


\bibitem[Si2]{_Siu2_} Y.T Siu. {\em A vanishing theorem for semi-positive line bundles over non-K\" ahler manifolds.} J. Diff. Geom. 19 (1984), 431-452.


\bibitem[U]{_U_} K. Ueno. {\em Classification theory of compact analytic spaces and algebraic varieties}. LNM 439. Springer 1975.

\bibitem[T]{_Takegoshi_} 
Takegoshi, K.,
{\em On cohomology groups of nef line bundles tensorized with multiplier ideal
sheaves on compact K\"ahler manifolds}, 
Osaka J. Math. 34 (1997) 783--802.

\bibitem[V98]{_V98_}Verbitsky, M., {\em Trianalytic subvarieties of the Hilbert scheme of points on a $K3$ surface}, Geom. Funct. Anal. 8 (1998), no. 4, 732–782

\bibitem[V02]{_V02_} Misha Verbitsky, {\em Coherent sheaves on general $K3$ surfaces and tori}, arXiv:math/0205210, Pure Appl. Math. Q. 4 (2008), no. 3, part 2, 651–714.

\bibitem[V03]{_V03_} Verbitsky, M., {\em Coherent sheaves on generic compact tori}, math.AG/0310329, CRM Proc. and Lecture Notices vol. 38 (2004), 229-249.

\bibitem[V09]{_Verbitsky:parabolic_}
Misha Verbitsky, 
{\em Parabolic nef currents on hyperkaehler manifolds},
arXiv:0907.4217, 22 pages. 

\end{thebibliography}
}

\hfill

\hfill


\small{

\noindent
{\sc Fr\'ed\'eric Campana}
{\sc Institut Elie Cartan\\
Universit\'e Henri Poincar\'e\\
BP 239\\
F-54506. Vandoeuvre-les-Nancy C\'edex\\
et: Institut Universitaire de France\\
\tt frederic.campana@univ-lorraine.fr}\\

\noindent
{\sc Jean-Pierre Demailly}\\
{\sc Acad\'emie des Sciences, et:\\
Universit\'e de Grenoble I\\
Institut Fourier, Laboratoire de Math\'ematiques\\
UMR 5582 du CNRS, BP 74\\
100 rue des Maths, F-38402 Saint-Martin d'H\`eres,\\
\tt jean-pierre.demailly@ujf-grenoble.fr}\\

\noindent
{\sc Misha Verbitsky}\\
{\sc  Laboratory of Algebraic Geometry, SU-HSE,\\
7 Vavilova Str. Moscow, Russia, 117312}\\
{\tt verbit@maths.gla.ac.uk, \ \  verbit@mccme.ru\\}


 }


\end{document}
