% Compact K\"ahler manifolds with hermitian semipositive anticanonical bundle
% J.-P. Demailly, Th. Peternell, M. Schneider
% to appear in Compositio Mathematica (Ref: COMP3787)

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% Main file

\begin{opening}

\title{Compact K\"ahler manifolds with hermitian \\
semipositive anticanonical bundle}

\author{\surname{Jean-Pierre Demailly}}
\institute{Universit\'e de Grenoble I, Institut Fourier, \\
U.R.A.\ 188 du C.N.R.S., BP 74, Saint-Martin d'H\`eres, France \\
e-mail: demailly@fourier.ujf-grenoble.fr}

\author{\surname{Thomas Peternell}}
\author{\surname{Michael Schneider}}
\institute{Universit\"at Bayreuth, Mathematisches Institut \\
D-95440 Bayreuth, Deutschland \\
e-mail: thomas.peternell@uni-bayreuth.de, michael.schneider@uni-bayreuth.de} 

\date{(Received: 21 october 1994; accepted: 9 March 1995)}
\end{opening}

\runningtitle{K\"AHLER MANIFOLDS WITH SEMIPOSITIVE ANTICANONICAL BUNDLES}
\runningauthor{J.-P.\ DEMAILLY ET AL.}
        
\begin{document}

\begin{abstract}
This note states a structure theorem for compact K\"ahler manifolds
with semipositive Ricci curvature: any such manifold has a
finite \'etale covering possessing a De Rham decomposition as a
product of irreducible compact K\"ahler manifolds, each one being
either Ricci flat (torus, symplectic or Calabi-Yau manifold), or
Ricci semipositive without non trivial holomorphic forms.
Related questions and conjectures concerning the latter case are discussed.
\end{abstract}

\keywords{Compact K\"ahler manifold, semipositive Ricci curvature,
complex torus, symplectic manifold, Calabi-Yau manifold, Albanese map,
fundamental group, Bochner formula, De Rham decomposition,
Cheeger-Gromoll theorem, nef line bundle, Kodaira-Iitaka dimension,
rationally connected manifold}

\section{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 structure theorem for
Ricci-flat manifolds (due to Bogomolov [Bo74a], [Bo74b], Kobayashi [Ko81]
and Beauville [Be83]) to the Ricci semi\-positive case. 

\proclaim Structure Theorem. Let $X$ be a compact K\"ahler manifold with
$-K_X$ hermitian semipositive. Then
\begin{description}
\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 $m>0$.
\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.
\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.
\end{description}
\par

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 and
for making our work possible.

\section{Bochner Formula and Holomorphic Differential Forms}

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

\proclaim 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.
\par

\noindent{\sl 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].

\proclaim Definition. Let $Y$ be a compact complex manifold. We define
\begin{description}
\item[{\rm i)}] $\phantom{\scriptstyle+}
\kappa_+(Y)=\max \{\kappa(\det\cF): \cF \ \hbox{is a subsheaf of}\
\Omega^p_Y \hbox{ for some}\ p>0\},$
\item[{\rm ii)}]
$\kappa_{++}(Y)=\max \{\kappa(\det\cF): \cF \ \hbox{is a subsheaf of}\
\Omega^{\otimes m}_Y \hbox{ for some}\ m>0\}$.
\end{description}
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.\par

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:

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

\noindent{\sl Proof.} Assume that $\kappa_{++}(X)>0$. Then we can find an
integer $m>0$ 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 

\proclaim 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 $Y.)$\par

\noindent{\sl 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 

\section{Proof of the Structure Theorem}

We suppose here that $X$ is equipped with a K\"ahler metric $g$ such that
$\Ricci(g)\geq 0$, and we set $n=\dim_{\bc}X$.\vskip3pt

\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]).\vskip3pt

\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~$X$.\vskip3pt

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

\proclaim 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.\par

\proclaim 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,
$$\begin{array}{rl}
\kappa_+(X)=-\infty&{}\iff\kappa_{++}(X)=-\infty\\
{}&{}\iff\forall\wh X|\to X,~\forall p\geq 1,\quad
H^0(\wh X|,\Omega^p_{\widehat X})=0.
\end{array}$$
If $\kappa_+(X)=-\infty$, then $\chi(X,\cO_X)=1$ and $X$ is simply
connected.\par

\noindent{\sl Proof.} The equivalence of all three properties is a direct
consequence of the structure theorem. Now, any \'etale covering
$\wh X|\to X$ satisfies \mbox{$\kappa_+(\wh X|)=\kappa_+(X)=-\infty$},
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

\section{Related Questions for the Case $-K_X$ 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. 

\proclaim 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.\par

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 

\proclaim 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$\,?$\par

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.

\proclaim 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 $\pi_1(C)\to\GL_2(\bc)$
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 $\GL_2(\bc)$,
hence there is no invariant K\"ahler metric on the fibre.}\par

\proclaim 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 \mbox{$(-K_{X_\pp})^2=(C')^2=C^2-9=0$}.
Therefore $-K_{X_\pp}$ is always nef.\\[\parskip]\indent
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.}\par

\proclaim 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$\,?$\par

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