\magnification=1200
\pretolerance=500 \tolerance=1000 \brokenpenalty=5000 \hsize=12.5cm
\vsize=19cm
\parskip3pt plus 1pt
\parindent=0.6cm

\font\bfseventeen=cmbx10 at 17.28pt
\font\bftwelve=cmbx10 at 12pt
\font\smallcap=cmcsc10
\font\tensans=cmss10
\font\fivesans=cmss10 at 5pt
\font\sevensans=cmss10 at 7pt
\newfam\sansfam
\textfont\sansfam=\tensans\scriptfont\sansfam=\sevensans
\scriptscriptfont\sansfam=\fivesans
\def\sans{\fam\sansfam\tensans}

\def\title#1|{\par\vskip .5cm\penalty -100 
\vbox{\noindent{\bftwelve #1}
\vskip 5pt}
\penalty 500}

\def\bibitem#1&#2&#3&#4&%
{\hangindent=1.66cm\hangafter=1
\noindent\rlap{\hbox{\bf #1}}\kern1.66cm{\rm #2}{\it #3}{\rm #4}} 

\long\def\theorem#1|#2\endtheorem{\par\vskip 5pt\noindent 
{\bf #1.}\ {\it #2}\par\vskip 5pt}

\def\today{\ifcase\month\or
January\or February\or March\or April\or May\or June\or July\or August\or
September\or October\or November\or December\fi \space\number\day,
\number\year}

\def\semidirect{\mathop{\kern2pt\vrule depth-0.3pt height4.3pt 
\kern-2pt\times}\nolimits}
\def\qed{{\hfill Q.E.D.}}
\def\wt#1|{\smash{\widetilde#1}}
\def\wh#1|{\smash{\widehat#1}}

\def\Alb{{\rm Alb}}
\def\Pic{{\rm Pic}}
\def\rank{{\rm rank}}
\def\UU{{\rm U}}
\def\SU{{\rm SU}}
\def\Sp{{\rm Sp}}
\def\pp{{\bf p}}

\def\cF{{\cal F}}
\def\cO{{\cal O}}

\def\bp{{\rm I \! \rm P}}
\def\bn{{\rm I \! \rm N}}
\def\bc{{\mathchoice {\setbox0=\hbox{$\displaystyle\rm C$}\hbox{\hbox
to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
{\setbox0=\hbox{$\textstyle\rm C$}\hbox{\hbox to0pt{\kern0.4\wd0\vrule
height0.9\ht0\hss}\box0}} {\setbox0=\hbox{$\scriptstyle\rm C$}\hbox{\hbox
to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}
{\setbox0=\hbox{$\scriptscriptstyle\rm C$}\hbox{\hbox
to0pt{\kern0.4\wd0\vrule height0.9\ht0\hss}\box0}}}} \def\bz{{\mathchoice
{\hbox{$\sans\textstyle Z\kern-0.4em Z$}} {\hbox{$\sans\textstyle
Z\kern-0.4em Z$}} {\hbox{$\sans\scriptstyle Z\kern-0.3em Z$}}
{\hbox{$\sans\scriptscriptstyle Z\kern-0.2em Z$}}}} 

\centerline{\bfseventeen Compact K\"ahler manifolds} \medskip
\centerline{\bfseventeen with hermitian semipositive} \medskip
\centerline{\bfseventeen anticanonical bundle} \medskip\medskip
\centerline{\bftwelve\hfil
J.-P.\ Demailly \hfil T.\ Peternell \hfil M.\ Schneider} 

\title 1. Main results|

This short note is a continuation of our previous work [DPS93] on compact
K\"ahler mani\-folds $X$ with semipositive Ricci curvature. Our purpose is
to state a splitting theorem describing the structure of such manifolds,
and to raise some related questions. The foundational background will be
found in papers by Lichnerowicz [Li67], [Li71], and Cheeger-Gromoll [CG71],
[CG72]. Recall that a {\it Calabi-Yau manifold} $X$ is a compact K\"ahler
manifold with $c_1(X)=0$ and finite fundamental group $\pi_1(X)$, such that
the universal covering $\wt X|$ satisfies $H^0(\wt
X|,\smash{\Omega^p_{\widetilde X}})=0$ for all $1\leq p\leq\dim X-1$.
A~{\it symplectic manifold} $X$ is a compact K\"ahler manifold admitting a
holomorphic symplectic $2$-form $\omega$ (of maximal rank everywhere); in
particular $K_X=\cO_X$. We denote here as usual
$$\Omega_X=T^\star_X,\qquad\Omega^p_X=\Lambda^pT^\star_X, \qquad
K_X=\det(T^\star_X).$$
The following structure theorem generalizes the
Bogomolov-Kobayashi-Beauville structure theorem for Ricci-flat manifolds
([Bo74a], [Bo74b], [Ko81], [Be83]) to the Ricci semi\-positive case. 

\theorem Structure Theorem|Let $X$ be a compact K\"ahler manifold with
$-K_X$ hermitian semipositive. Then
\smallskip
\item {\rm i)} The universal covering $\wt X|$ admits a holomorphic and
isometric splitting
$$\wt X|\simeq \bc^q \times \prod X_i$$
with $X_i$ being either a Calabi-Yau manifold or a symplectic mani\-fold or
a manifold with $-K_{X_i}$ semipositive and $H^0(X_i,\Omega^{\otimes
m}_{X_i})=0$ for all $m>0$. \smallskip
\item {\rm ii)} There exists a finite \'etale Galois covering $\wh X|\to X$
such that the Albanese va\-riety $\Alb(\wh X|)$ is a $q$-dimensional torus
and the Albanese map \hbox{$\alpha:\wh X|\to\Alb(\wh X|)$} is a locally
trivial holomorphic fibre bundle whose fibres are products $\prod X_i$ of the type described in {\rm i)},
all $X_i$ being simply connected.
\smallskip
\item {\rm iii)} We have $\pi_1(\wh X|)\simeq\bz^{2q}$ and $\pi_1(X)$ is an
extension of a finite group $\Gamma$ by the normal subgroup $\pi_1(\wh X|)$.
In particular there is an exact sequence
$$0\to \bz^{2q} \to \pi_1(X) \to \Gamma \to 0,$$
and the fundamental group $\pi_1(X)$ is almost abelian.\vskip0pt
\endtheorem

Recall that a line bundle $L$ is said to be hermitian semipositive if it
can be equipped with a smooth hermitian metric of semipositive curvature
form. A sufficient condition for hermitian semipositivity is that some
multiple of $L$ is spanned by global sections; on the other hand, the
hermitian semipositivity condition implies that $L$ is numerically effective
(nef) in the sense of [DPS94], which, for $X$ projective algebraic, is
equivalent to saying that $L\cdot C\geq0$ for every curve $C$ in~$X$.
Examples contained in [DPS94] show that all three conditions are different
(even for $X$ projective algebraic). By Yau's solution of the Calabi
conjecture (see [Au76], [Yau78]), a compact K\"ahler manifold $X$ has a
hermitian semipositive anticanonical bundle $-K_X$ if and only if $X$ admits
a K\"ahler metric $g$ with Ricci$(g)\geq 0$. The isometric decomposition
described in the theorem refers to such K\"ahler metrics.

In view of ``standard conjectures'' in minimal model theory it is expected
that projective manifolds $X$ with no nonzero global sections in
$H^0(X,\Omega^{\otimes m}_X)$, $m>0$, are rationally connected. We hope
that most of the above results will continue to hold under the weaker
assumption that $-K_X$ is nef instead of hermitian semipositive. However,
the technical tools needed to treat this case are still missing.

We would like to thank the DFG-Schwerpunktprogramm ``Komplexe
Mannigfaltigkeiten'' and the Institut Universitaire de France for making
our work possible.

\title 2. Bochner formula and holomorphic differential forms| 

Our starting point is the following well-known consequence of the Bochner
formula.

\theorem Lemma|Let $X$ be a compact $n$-dimensional K\"ahler manifold with
$-K_X$ hermitian semipositive. Then every section of $\Omega^{\otimes
m}_X$, $m\geq 1$ is parallel with respect to the given K\"ahler metric.
\endtheorem

\noindent {\bf Proof.} The Lemma is an easy consequence of the Bochner
formula
$$ \Delta (\parallel u \parallel^2) =\;\parallel\nabla u\parallel ^2 +\;
Q(u),$$
where $u\in H^0(X, \Omega_X^{\otimes m})$ and $Q(u)\geq m\lambda_0
\parallel u\parallel^2$. Here $\lambda_0$ is the smallest eigenvalue of the
Ricci curvature tensor. For details see for instance [Ko83].\qed 

The following definition of a modified Kodaira dimension $\kappa_+(X)$ is
taken from Campana [Ca93]. As the usual Kodaira dimension $\kappa(X)$, this
is a birational invariant of~$X$. Other similar invariants have also been
considered in [BR90] and [Ma93].

\theorem Definition|Let $Y$ be a compact complex manifold. We define
\smallskip
\item{\rm i)} $\phantom{\scriptstyle+}
\kappa_+(Y)=\max \{\kappa(\det\cF): \cF \ \hbox{is a subsheaf of}\
\Omega^p_Y \hbox{ for some}\ p>0\},$ \smallskip
\item{\rm ii)}
$\kappa_{++}(Y)=\max \{\kappa(\det\cF): \cF \ \hbox{is a subsheaf of}\
\Omega^{\otimes m}_Y \hbox{ for some}\ m>0\}$. \smallskip\noindent
Here we let as usual $\det \cF= (\Lambda^r\cF)^{**}$, where $r= \rank\cF$
and $\kappa$ is the usual Iitaka dimension of a line bundle. \endtheorem

Clearly, we have $-\infty\leq\kappa(Y)\leq\kappa_+(Y)\leq\kappa_{++}(Y)$
where $\kappa(Y)=\kappa(K_Y)$ is the usual Kodaira dimension. It would be
interesting to know whether there are precise relations between
$\kappa_+(Y)$ and $\kappa_{++}(Y)$, as well as with the weighted Kodaira
dimensions defined by Manivel [Ma93]. The above lemma implies:

\theorem Proposition|Let $X$ be a compact K\"ahler manifold with $-K_X$
hermitian semipo\-sitive. Then $\kappa_{++}(X)\leq0$. \endtheorem

\noindent{\bf Proof.} Assume $\kappa_{++}(X)>0$. Then we can find $m>0$ and
a subsheaf \hbox{$\cF\subset \Omega^{\otimes m}_X$} with $\kappa(\det \cF)
>0$. Hence there is some $\mu \in \bn$ and $s\in H^0(X,(\det\cF)^\mu)$ with $s\neq 0$. Since $\kappa(\det\cF)>0$, $s$ must have zeroes. Hence the induced section $\tilde s\in H^0(X,\Omega^{\otimes\mu rm}_X)$ has zeroes too, $r$ being the rank of $\cF$. This contradicts the previous Lemma.\qed 

\theorem Corollary|Let $X$ be a compact K\"ahler manifold with $-K_X$ hermitian
semi\-po\-si\-tive. Let $\phi:X\to Y$ be a surjective holomorphic map to a
normal compact K\"ahler space. Then $\kappa(Y)\leq 0$. $($Here
$\kappa(Y)=\kappa(\hat Y)$, where $\hat Y$ is an arbitrary
desingularization of $Y.)$
\endtheorem

\noindent {\bf Proof.} This follows from the inequalities
$0\geq\kappa_+(X)\geq\kappa_+(Y) \geq\kappa(Y)$. For the second inequality,
which is easily checked by a pulling-back argument, see [Ca93].\qed 

\title 3. Proof of the structure theorem| 

We suppose here that $X$ is equipped with a K\"ahler metric $g$ such that
${\rm Ricci}(g)\geq 0$, and we set $n=\dim_{\bc}X$.

\noindent i) Let $(\wt X|,g)\simeq\prod(X_i,g_i)$ be the De Rham
decomposition of $(\wt X|,g)$, induced by a decomposition of the holonomy
representation in irreducible representations. Since the holonomy is
contained in $\UU(n)$, all factors $(X_i,g_i)$ are K\"ahler manifolds with
irreducible holonomy and holonomy group $H_i\subset \UU(n_i)$, $n_i=\dim
X_i$. By Cheeger-Gromoll [CG71], there is possibly a flat factor
$X_0=\bc^q$ and the other factors $X_i$, $i\ge 1$, are compact. Also, the
product structure shows that $-K_{X_i}$ is hermitian semi\-positive. It
suffices to prove that $\kappa_{++}(X_i)=0$ implies that $X_i$ is a
Calabi-Yau manifold or a symplectic manifold. In view of \S$\,$2, the
condition $\kappa_{++}(X_i)=0$ means that there is a nonzero section $u\in
H^0(X_i,\Omega^{\otimes m}_{X_i})$ for some $m>0$. Since $u$ is parallel by
the lemma, it is invariant under the holonomy action, and therefore the
holonomy group $H_i$ is not the full unitary group $\UU(n_i)$ (indeed, the
trivial representation does not occur in the decomposition of
$(\bc^{n_i})^{\otimes m}$ in irreducible $\UU(n_i)$-representations, all
weights being of length $m$). By Berger's classification of holonomy groups
[Bg55] there are only two remaining possibilities, namely
$H_i=\SU(n_i)$ or $H_i=\Sp(n_i/2)$. The case $H_i=\SU(n_i)$ leads to $X_i$
being a Calabi-Yau manifold. The remaining case $H_i=\Sp(n_i/2)$ implies
that $X_i$ is symplectic (see e.g.\ [Be83]). 

\noindent ii) Set $X'=\prod_{i\geq 1}X_i$. The group of covering
transformations acts on the product $\wt X|=\bc^q\times X'$ by holomorphic
isometries of the form $x=(z,x')\mapsto(u(z),v(x'))$. At this point, the
argument is slightly more involved than in Beauville's paper [Be83],
because the group $G'$ of holomorphic isometries of $X'$ need not be finite
($X'$ may be for instance a projective space); instead, we imitate the
proof of ([CG72], Theorem 9.2) and use the fact that $X'$ and $G'={\rm
Isom}(X')$ are compact. Let $E_q=\bc^q\semidirect U(q)$ be the group of
unitary motions of $\bc^q$. Then $\pi_1(X)$ can be seen as a discrete
subgroup of $E_q\times G'$. As $G'$ is compact, the kernel of the
projection map $\pi_1(X)\to E_q$ is finite and the image of $\pi_1(X)$ in
$E_q$ is still discrete with compact quotient. This shows that there is a
subgroup $\Gamma$ of finite index in $\pi_1(X)$ which is isomorphic to a
crystallographic subgroup of $\bc^q$. By Bieberbach's theorem, the subgroup
$\Gamma_0\subset\Gamma$ of elements which are translations is a subgroup of
finite index. Taking the intersection of all conjugates of $\Gamma_0$ in
$\pi_1(X)$, we find a normal subgroup $\Gamma_1\subset\pi_1(X)$ of finite
index, acting by translations on $\bc^q$. Then $\wh X|=\wt X|/\Gamma_1$ is
a fibre bundle over the torus $\bc^q/\Gamma_1$ with $X'$ as fibre and
$\pi_1(X')=1$. Therefore $\wh X|$ is the desired finite \'etale covering
of~$X$.

\noindent iii) is an immediate consequence of ii), using the homotopy exact
sequence of a fibration.\qed

\theorem Corollary 1|Let $X$ be a compact K\"ahler manifold with $-K_X$
hermitian semi\-positive. If $\wt X|$ is indecomposable and
$\kappa_+(X)=0$, then $X$ is Ricci-flat.
\endtheorem

\theorem Corollary 2|Let $X$ be a compact K\"ahler manifold with $-K_X$
hermitian semi\-positive. Then, if $\wh X|\to X$ is an arbitrary finite
\'etale covering,
$$\eqalign{
\kappa_+(X)=-\infty &{}\iff \kappa_{++}(X)=-\infty \cr
&{} \iff \forall\wh X|\to X,~\forall p\geq 1, \quad
H^0(\wh X|, \Omega^p_{\widehat X})=0.\cr}$$
If $\kappa_+(X)=-\infty$, then $\chi(X,\cO_X)=1$ and $X$ is simply
connected.
\endtheorem

\noindent{\bf Proof.} The equivalence of all three properties is a direct
consequence of the structure theorem. Now, every \'etale covering $\wh X|
\to X$ satisfies $\kappa_+(\wh X|)=\kappa_+(X)=-\infty$, hence $\chi(\wh
X|,\cO_{\wh X|})=\chi(X,\cO_X)=1$ (by Hodge symmetry we have
$h^p(X,\cO_X)=0$ for $p\ge 1$, whilst $h^0(X,\cO_X)=1$). However, if $d$ is
the covering degree, the Riemann-Roch formula implies $\chi(\wh X|,\cO_{\wh
X|})=d\,\chi(X,\cO_X)$, hence $d=1$ and $X$ must be simply connected.\qed

\title 4. Related questions for the case -K${}_{\bf X}$ nef| 

In order to make the structure theorem more explicit, it would be necessary
to characterize more precisely the manifolds for which
$\kappa_+(X)=-\infty$. We expect these manifolds to be rationally
connected, even when $-K_X$ is just supposed to be nef. 

\theorem Conjecture|Let $X$ be a compact K\"ahler manifold such that $-K_X$
is nef and $\kappa_+(X)=-\infty$. Then $X$ is rationally connected, i.e.\
any two points of $X$ can be joined by a chain of rational curves.
\endtheorem

Campana even conjectures this to be true without assuming $-K_X$ to be nef.

Another hope we have is that a similar structure theorem might also hold in
the case $-K_X$ nef. A small part of it would be to understand better the
structure of the Albanese map. We proved in [DPS93] that the Albanese map
is surjective when $\dim X\leq 3$, and if $\dim X\leq 2$ it is well-known
that the Albanese map is a locally trivial fibration. It is thus natural to
state the following 

\theorem Problem|Let $X$ be a compact K\"ahler manifold with $-K_X$ nef. Is
the Albanese map $\alpha:X\to\Alb(X)$ a smooth locally trivial
fibration$\,?$
\endtheorem

The following simple example shows, even in the case of a locally trivial
fibration, that the structure group of transition automorphisms need not be
a group of isometries, in contrast with the case $-K_X$ hermitian
semipositive.

\theorem Example 1 {\rm(see [DPS94], Example 1.7)}|{\rm Let $C=\bc/(\bz+
\bz\tau)$ be an elliptic curve, and let $E\to C$ be the flat
rank $2$ bundle associated to the representation $\pi_1(C)\to{\rm GL}_2(\bc)$
defined by the monodromy matrices
$$\pmatrix{1&0\cr 0&1\cr},\qquad \pmatrix{1&1\cr 0&1\cr}.$$
Then the projectivized bundle $X=\bp(E)$ is a ruled surface over $C$
with $-K_X$ nef and not hermitian semipositive (cf.\ [DPS94]). In this
case, the Albanese map $X\to C$ is a locally trivial $\bp_1$-bundle,
but the monodromy group is not relatively compact in ${\rm GL}_2(\bc)$,
hence there is no invariant K\"ahler metric on the fibre.}
\endtheorem

\theorem Example 2|{\rm The following example shows that the picture is
unclear even in the case of surfaces with $\kappa_+(X)=-\infty$. Let
$\pp=(p_1,\ldots,p_9)$ be a configuration of 9 points in $\bp_2$ and let
$\pi:X_\pp\to\bp_2$ be the blow-up of $\bp_2$ with center $\pp$. Here some
of the points $p_i$ may be infinitely near: as usual, this means that the
blowing-up process is made inductively, each $p_i$ being an arbitrary point
in the blow-up of $\bp_2$ at $(p_1,\ldots,p_{i-1})$. There is always a
cubic curve $C$ containing all 9 points ($C$ is even unique if $\pp$ is
general enough). The only assumption we make is that $C$ is nonsingular,
and we let $C=\{Q(z_0,z_1,z_2)=0\}\subset\bp_2$, $\deg Q=3$. Then $C$ is an
elliptic curve and $-K_{X_\pp}=\pi^\star\cO(3)-\sum E_i$ where
$E_i=\pi^{-1}(p_i)$ are the exceptional divisors. Clearly $Q$ defines a
section of $-K_{X_\pp}$, of divisor equal to the strict transform $C'$ of
$C$, hence $-K_{X_\pp}\simeq\cO(C')$, and $(-K_{X_\pp})^2=(C')^2=C^2-9=0$.
Therefore $-K_{X_\pp}$ is always nef.

It is easy to see that $-mK_{X_\pp}$ may be generated or not by sections
according to the choice of the 9 points~$p_i$. In fact, if $p'_i$ is the
point of $C'$ lying over~$p_i$, we have
$$-K_{X_\pp|C'}=\pi^\star(\cO(3))_{|C'}\otimes\cO(-\sum p'_j)= \pi^\star\Big(\cO(3)_{|C}\otimes\cO(-\sum p_j)\Big).$$
Since $C'\simeq C$ is an elliptic curve and $-K_{X_\pp|C'}$ has degree $0$,
there are nonzero sections in $H^0(C',-mK_{X_\pp|C'})$ if and only if
$L_\pp=\cO(3)_{|C}\otimes\cO(-\sum p_j))$ is a torsion point in $\Pic^0(C)$
of order dividing~$m$. Such sections always extend to~$X_\pp$. Indeed, we
may assume that $m$ is exactly the order. Then
$\cO(-C')\otimes\cO(-mK_{X_\pp})=\cO((m-1)C')$ admits a filtration by its
subsheaves $\cO(kC')$, $0\leq k\leq m-1$, and the $H^1$ groups of the
graded pieces are $H^1(X_\pp,\cO_{X_\pp})=0$ for $k=0$ and
$$H^1(C',\cO(kC')_{|C'})=H^0(C',\cO(-kC'))=0~~\hbox{\rm for $0<k<m$}.$$
Therefore $H^1(X_\pp,\cO(-C')\otimes\cO(-mK_{X_\pp}))=0$, as desired. In
particular, $-K_{X_\pp}$ is hermitian semi\-positive as soon as $L_\pp$ is
a torsion point in $\Pic^0(C)$. In this case, there is a polynomial $R_m$
of degree $3m$ vanishing of order $m$ at all points $p_i$, such that the
rational function $R_m/Q^m$ defines an elliptic fibration
$\varphi:X_\pp\to\bp_1$; in this fibration $C$ is a multiple fibre of
multiplicity $m$ and we have $-mK_{X_\pp}=\varphi^\star\cO_{\bp_1}(1)$. An
interesting question is to understand what happens when $L_\pp$ is no
longer a torsion point in $\Pic^0(C)$ (this is precisely the situation
considered by Ogus [Og76] in order to produce a counterexample to the
formal principle for infinitesimal neighborhoods). In this situation, we
may approximate $\pp$ by a sequence of configurations $\pp_m\subset C$ such
that the corresponding line bundle $L_{\pp_m}$ is a torsion point of order
$m$ (just move a little bit $p_9$ and take a suitable $p_{9,m}\in C$ close
to~$p_9$). The sequence of fibrations $X_{\pp_m}\to\bp_1$ does not yield a
fibration $X_\pp\to\bp_1$ in the limit, but we believe that there might
exist instead a holomorphic foliation on~$X_\pp$. In this foliation, $C$
would be a closed leaf, and the generic leaf would be nonclosed and of
conformal type $\bc$ (or possibly $\bc^\star$). If indeed the foliation
exists and admits a smooth invariant transversal volume form, then
$-K_{X_\pp}$ would still be hermitian semipositive. We are thus led to the
following question.}
\endtheorem

\theorem Question|Let $X$ be compact K\"ahler manifold with $-K_X$ nef and
$X$ rationally connected. Is then $-K_X$ automatically hermitian
semipositive$\,?$ In particular, is it always the case that $\bp_2$
blown-up in 9 points of a nonsingular cubic curve has a semipositive
anticanonical bundle$\,?$
\endtheorem
\vfill\eject

\title References|

\bibitem[Au76]&Aubin, T.:& Equations du type Monge-Amp\`ere sur les
vari\'et\'es k\"ahle\-riennes compactes.& C.\ R.\ Acad.\ Sci.\ Paris
Ser.\ A {\bf 283}, 119--121 (1976)$\,$; Bull.\ Sci.\ Math.\
{\bf 102}, 63--95 (1978)&

\bibitem[Be83]&Beauville, A.:& Vari\'et\'es k\"ahleriennes dont la
premi\`ere classe de Chern est nulle.&
J.\ Diff.\ Geom.\ {\bf 18}, 775--782 (1983)& 

\bibitem[Bg55]&Berger, M.:& Sur les groupes d'holonomie des vari\'et\'es
\`a connexion affine des vari\'et\'es riemanniennes.& Bull.\ Soc.\ Math.\
France {\bf 83}, 279--330 (1955)& 

\bibitem[Bi63]&Bishop, R.:& A relation between volume, mean curvature and
diameter.& Amer.\ Math.\ Soc.\ Not.\ {\bf 10}, p.$\,$364 (1963)& 

\bibitem[Bo74a]&Bogomolov, F.A.:& On the decomposition of K\"ahler
manifolds with trivial canonical class.& Math.\ USSR Sbornik {\bf 22},
580--583 (1974)& 

\bibitem[Bo74b]&Bogomolov, F.A.:& K\"ahler manifolds with trivial canonical
class.&
Izves\-tija Akad.\ Nauk {\bf 38}, 11--21 (1974)& 

\bibitem[BR90]&Br\"uckmann, P., Rackwitz, H.-G.:& $T$-symmetrical tensor
forms on complete intersections.& Math.\ Ann.\ {\bf 288}, 627--635 (1990)& 

\bibitem[Ca93]&Campana, F.:& Fundamental group and positivity of cotangent
bundles of compact K\"ahler manifolds.& Preprint 1993&

\bibitem[CG71]&Cheeger, J., Gromoll, D.:& The splitting theorem for
manifolds of nonnegative Ricci curvature.& J.\ Diff.\ Geom.\ {\bf 6},
119--128 (1971)& 

\bibitem[CG72]&Cheeger, J., Gromoll, D.:& On the structure of complete
manifolds of nonnegative curvature.& Ann.\ Math.\ {\bf 96}, 413--443
(1972)&

\bibitem[DPS93]&Demailly, J.-P., Peternell, T., Schneider, M.:& K\"ahler
manifolds with numerically effective Ricci class.& Compositio Math.\ {\bf
89}, 217--240 (1993)& 

\bibitem[DPS94]&Demailly, J.-P., Peternell, T., Schneider, M.:& Compact
complex manifolds with numerically effective tangent bundles.& J.\ Alg.\
Geom.\ {\bf 3}, 295--345 (1994)& 

\bibitem[Ko81]&Kobayashi, S.:& Recent results in complex differential
geometry.& Jber.\ dt.\ Math.-Verein.\ {\bf 83}, 147--158 (1981)&

\bibitem[Ko83]&Kobayashi, S.:& Topics in complex differential geometry.&
In DMV Seminar, Vol.~3., Birkh\"auser 1983& 

\bibitem[Li67]&Lichnerowicz, A.:& Vari\'et\'es k\"ahleriennes et
premi\`{e}re classe de Chern.&
J.\ Diff.\ Geom.\ {\bf 1}, 195--224 (1967)& 

\bibitem[Li71]&Lichnerowicz, A.:& Vari\'et\'es K\"ahl\'eriennes \`a
premi\`ere classe de Chern non n\'egative et vari\'et\'es riemanniennes \`a
courbure de Ricci g\'en\'eralis\'ee non n\'egative.& J.\ Diff.\ Geom.\ {\bf
6}, 47--94 (1971)& 

\bibitem[Ma93]&Manivel, L.:& Birational invariants of algebraic varieties.&
Preprint Institut Fourier no.~257 (1993)& 

\bibitem[Og76]&Ogus, A.:&The formal Hodge filtration.& Invent.\ Math.\ {\bf
31}, 193--228 (1976)&

\bibitem[Yau78]&Yau, S.T.:& On the Ricci curvature of a complex K\"ahler
manifold and the complex Monge-Amp\`ere equation~I.& Comm.\ Pure and Appl.\
Math.\ {\bf 31}, 339--411 (1978)&

\vskip15pt

\parindent=0cm
(version of January 5, 1995, printed on \today) \vskip15pt

{\bf Jean-Pierre Demailly}\hfill\break
Universit\'e de Grenoble I\hfill\break
Institut Fourier, U.R.A.\ 188 du C.N.R.S., BP 74\hfill\break 38402
Saint-Martin d'H\`eres, France\hfill\break {\sl e-mail:
demailly@fourier.ujf-grenoble.fr} \vskip6pt
{\bf Thomas Peternell}\hfill\break
Universit\"at Bayreuth, Mathematisches Institut\hfill\break D-95440
Bayreuth, Deutschland\hfill\break {\sl e-mail:
thomas.peternell@uni-bayreuth.de} \vskip6pt
{\bf Michael Schneider}\hfill\break
Universit\"at Bayreuth, Mathematisches Institut\hfill\break D-95440
Bayreuth, Deutschland\hfill\break {\sl e-mail:
michael.schneider@uni-bayreuth.de} 

\end

%Complementary references of first version 

\bibitem[Ca94]&Campana, F.:& Remarques sur le rev\^{e}tement universel des
vari\'et\'es k\"ahleriennes compactes.& 

To appear in Bull.\ Soc.\ Math.\ France 1994& 

\bibitem[Gr81]&Gromov, M.:& Groups of polynomial growth and expanding
maps.&
Publ.\ Math.\ I.H.E.S.\ {\bf 53}, 53--78 (1981)& 

\bibitem[Gr91]&Gromov, M.:& K\"ahler hyperbolicity and $L^2$-Hodge theory.&
J.\ Diff.\ Geom.\ {\bf 33}, 263--292 (1991)& 

\bibitem[HK78]&Heintze, E., Karcher, H.:& A general comparison theorem with
applications to volume estimates for submanifolds.& Ann.\ Scient.\ Ec.\
Norm.\ Sup.\ $4^{{\rm e}}$ S rie, {\bf 11}, 451--470 (1978)&

\bibitem[Kl93]&Koll\'{a}r, J.:& Shafarevich maps and plurigenera of
algebraic varieties.&
Inv.\ Math.\ {\bf 113}, 177--215 (1992)& 


