% K\"ahler manifolds with numerically effective Ricci class
% J.P. Demailly, Th. Peternell, M. Schneider
% TeX file

\magnification=1200
\input ams.tex
\input broc.mac
\parindent=0.833cm

\def\than{\tha\noindent}
\def\rem#1|{\remarque\noindent{\petcap#1}|}
\def\proof{\noindent{\sl{Proof.}\ }}
\def\Ricci{{\rm Ricci}}
\def\Alb{{\rm Alb}}
\def\Aut{{\rm Aut}}
\def\Vol{{\rm Vol}}
\def\Pic{{\rm Pic}}
\def\alb{{\rm alb}}
\def\deg{{\rm deg}\,}
\def\rank{{\rm rank}}
\def\diam{{\rm diam}\,}
\def\oc{{\cal O}}
\def\gr{{\cal G}}
\def\hr{{\cal H}}
\def\ir{{\cal I}}
\def\lr{{\cal L}}
\def\rr{{\cal R}}
\def\ort{\hbox{~\vrule width4.0pt height0.4pt depth0pt
                \vrule width0.4pt height6.0pt depth0pt~$\,$}}
\def\stimes{\mathop{\kern0.7pt
\vrule height 0.4pt depth 0pt width 5pt\kern-5pt
\vrule height 5.4pt depth -5pt width 5pt\kern-5pt
\vrule height 5.4pt depth 0pt width 0.4pt\kern4.6pt
\vrule height 5.4pt depth 0pt width 0.4pt\kern-6.5pt
\raise0.3pt\hbox{$\times$}\kern-0.7pt}\nolimits}
\def\ovp{\overline\partial}                
\let\la=\longrightarrow
\let\ov=\overline
\let\wt=\widetilde

\centerline{\bftwelve K\"AHLER MANIFOLDS WITH}
\vskip4pt
\centerline{\bftwelve NUMERICALLY EFFECTIVE RICCI CLASS}

\vskip3pt
{\noindent\hangindent0.6cm\hangafter-1
{\bf Jean-Pierre Demailly${}^\star$, Thomas Peternell${}^{\star\star}$,
Michael Schneider${}^{\star\star}$

}}

{\noindent\hangindent0.6cm\hangafter-4{\sl
\llap{${}^\star~$}Universit\'e de Grenoble I\hfill
${}^{\star\star}$ Universit\"at Bayreuth\kern0.6cm\break
Institut Fourier, BP 74\hfill Mathematisches Institut\kern0.6cm\break
U.R.A. 188 du C.N.R.S.\hfill Postfach 10 12 51\kern0.6cm\break
38402 Saint-Martin d'H\`eres, France\hfill D-8580 Bayreuth,
Deutschland\kern0.6cm

}}

\vskip15pt
\titre Introduction|
  
Compact K\"ahler manifolds with semipositive Ricci curvature have been
investigated by various authors. S.~Kobayashi [Ko61] first proved the
simple connectedness of Fano manifolds, namely manifolds with positive
Ricci curvature or equivalently, with ample anticanonical line bundle
$-K_X$. Later on, generalizing results of Y.~Matsushima [Ma69],
A.~Lichnerowicz [Li71,$\,$72] proved the following interesting
fibration theorem: if $X$ is a compact K\"ahler manifold with
semipositive Ricci class, then $X$ is a smooth fibration over its
Albanese torus and there is a group of analytic automorphisms of $X$
lying over the group of torus translations (see also Section~2 for another
proof of these facts based on the solution of Calabi's conjecture and
on Bochner's technique). Finally, there were extensive works in the last
decades to study the structure and classification of Ricci flat
K\"ahler manifolds, see e.g. [Ca57], [Bo74a,b], [Be83] and [Kr86]$\,$;
of special interest for physicists is the subclass of so-called
Calabi-Yau manifolds, i.e. Ricci flat compact K\"ahler manifolds with
finite fundamental group, which appear as a natural generalization of
K3 surfaces.

To make things precise, one says that $X$ has semipositive Ricci class
$c_1(X)$ if $c_1(X)$ contains a smooth semipositive closed $(1,1)$-form,
or equivalently if $-K_X$ carries a smooth hermitian metric with
semipositive curvature. By the Aubin-Calabi-Yau theorem, this is equivalent
to $X$ having a K\"ahler metric with semipositive Ricci curvature.
On the other hand, recent developments of algebraic geometry
(especially those related to Mori's minimal model program) have
shown the importance of the notion of numerical effectivity,
which generalizes hermitian semipositivity but is much more flexible.
It would thus be important to extend the above mentioned results to the
case where $-K_X$ is numerically effective. The purpose of this paper is
to contribute to the following two conjectures.

\than Conjecture 1|Let $X$ be a compact K\"ahler manifold with numerically
effective anticanonical bundle $-K_X$. Then the fundamental group
$\pi_1(X)$ has polynomial growth.\fintha

\than Conjecture 2|Let $X$ be a compact K\"ahler manifold with $-K_X$
numerically effective. Then the Albanese map $\alpha : X \to \Alb(X)$
is surjective.\fintha

Before we state the results, let us recall the definition of a numerically 
effective line bundle $L$ on a compact complex manifold (see [DPS91] for more
details). The abbreviation ``nef'' will be used for ``numerically effective''.

\than Definition|Let $X$ be a compact complex manifold with a fixed
hermitian metric~$\omega$. A holomorphic line bundle $L$ over $X$ is
$\underline{\sl nef}$ if for every $\varepsilon>0$ there exists a
smooth hermitian metric $h_\varepsilon$ on $L$ such that the curvature
satisfies
  $$\Theta_{h_\varepsilon} \ge-\varepsilon\omega.$$\fintha

Of course this notion does not depend on the choice of $\omega$.
If $X$ is projective, $L$~is nef precisely if $L\cdot C \ge 0$ for all curves
$C \subset X$. Our main contribution to Conjecture~1 is

\than Theorem 1|Let $X$ be a compact K\"ahler manifold with $-K_X$ nef. Then
$\pi_1(X)$ is a group of subexponential growth.\fintha

The main tool to prove this result is the solution of the Calabi
conjecture by Aubin~[Au76] and Yau~[Y77], combined with volume bounds for
geodesic balls due to Bishop [Bi63] and Gage [Ga80] (see Section~1 for
details). In fact, the volume of a geodesic ball of radius $R$ in the
universal covering of $X$ essentially counts the number of words of
$\pi_1(X)$ of length ${}\le R$. The difficulty is that we have to deal
with a sequence of metrics with Ricci curvature closer and closer to being
semipositive, but nevertheless slightly negative in some points, and
moreover the diameter of $X$ need not remain uniformly bounded; this
difficulty is solved by observing that a large fraction of the volume
of $X$ remains at bounded distance without being disconnected (Lemma~1.3).
A by-product of our proof is that Conjecture~1 holds in the semipositive
case. This was in fact already known since a long time in the context of
riemannian manifolds (cf. e.g.~[HK79]); our method is then nothing more
than the usual riemannian geometry proof combined with the Aubin-Calabi-Yau
theorem. In this way we get:

\than Theorem 2|Let $X$ be a compact K\"ahler manifold with $-K_X$ hermitian
semipositive. Then $\pi_1(X)$ has polynomial growth of degree ${}\le 2
\dim X$, in particular \hbox{$h^1 (X,\oc_X) \le \dim X$}.\fintha 

Note that there are simple examples of compact K\"ahler manifolds $X$
with $-K_X$ nef but not hermitian semipositive, e.g. some ruled
surfaces over elliptic curves (see examples 1.7 and 3.5 in [DPS91]).
Also, to give a more precise idea of what Conjecture~1 means, let us
recall Gromov's well-known result [Gr81]$\,$: a finitely generated
group has polynomial growth if and only if it contains a nilpotent
subgroup of finite index. Much more might be perhaps expected in the
present situation:

\than Question|Let $X$ be a compact K\"ahler manifold with $-K_X$ nef.
Does there exist a finite \'etale covering $\wt X$ of $X$ such the Albanese
map $\wt X\to\Alb(\wt X)$ induces an isomorphism of fundamental groups$\,$?
\fintha

If this would be the case, $\pi_1(X)$ would always be an extension of a
finite group by a free abelian group of even rank.
Concerning Conjecture~2, the following result will be proved in Section~2:

\than Theorem 3|Let $X$ be a $n$-dimensional compact K\"ahler manifold
such that $-K_X$ is nef. Then
\item{\rm(i)} The Albanese map $\alpha : X \to \Alb(X)$ is surjective
as soon as the Albanese dimension $p=\dim\alpha(X)$ is $0$, $1$ or $n$,
and also for $p=n-1$ if $X$ is projective.
\item{\rm(ii)} If $X$ is projective and if the generic fiber $F$ of $\alpha$
has $-K_F$ big, then $\alpha$ is surjective.\smallskip
\fintha

The case $p=1$ in (i) is a straightforward consequence of Theorem~1, as
pointed out to us by F.~Campana, if one observes that the growth of
the fundamental group of a curve of genus ${}\ge2$ is of exponential type.
The other interesting case $p=n-1$ is obtained as a consequence of point (ii),
which is itself a rather simple consequence of the Kawamata-Viehweg vanishing
theorem. Theorem~3 settles Conjecture~2 for
projective $3$-folds. In that case, we can also obtain a direct
algebraic proof of the Albanese surjectivity in most cases by an examination
of the structure of Mori contractions of~$X$. When the contraction is
not a modification, we give the description of the fibration structure
of~$X$. This is done in Section~3.

To conclude let us mention that the first theorem was used in
the classification of compact K\"ahler manifolds with nef tangent
bundles [DPS91] in a crucial way.

\noindent{\sl Acknowledgement:}\/ Our collaboration has been made
possible by PROCOPE and the DFG Schwerpunktprogramm ``Komplexe
Mannigfaltigkeiten''. We would like to thank Prof. F.A.~Bogomolov,
F.~Campana and P.~Gauduchon for very useful discussions.\vskip20pt

\titre 1. Subexponential growth of the fundamental group|

If $G$ is a finitely generated group with generators $g_1,\ldots,g_p$,
we denote by $N(k)$ the number of elements $\gamma\in G$ which can be
written as words
   $$\gamma=g_{i_1}^{\varepsilon_1}\ldots g_{i_k}^{\varepsilon_k},~~~~
                                  \varepsilon_j=0,~1~~\hbox{\rm or}~~-1$$
of length ${}\le k$ in terms of the generators. The group $G$ is said
to have {\sl subexponential growth} if for every $\varepsilon>0$ there is
a constant $C(\varepsilon)$ such that
   $$N(k) \le  C(\varepsilon)e^{\varepsilon k}~~~~\hbox{\rm for}~~k\ge 0.$$
This notion is independent of the choice of generators. In the free
group with  two generators, we have
   $$N(k)=1+4(1+3+3^2+...+3^{k-1})=2\cdot 3^k-1.$$
It follows immediately that a group with subexponential growth cannot
contain a non abelian free subgroup. The main goal of this section is to
prove

\than Theorem 1.1|Let $X$ be a compact K\"ahler manifold such that
$K_X^{-1}$ is nef. Then $\pi_1 (X)$ has subexponential growth.\fintha

\proof The first step consists in the construction of suitable K\"ahler
metrics by means of the Aubin-Calabi-Yau theorem. Let $\omega$ be a
fixed K\"ahler metric on~$X$. Since $K_X^{-1}$ is nef, for every
$\varepsilon>0$ there exists a smooth hermitian metric $h_\varepsilon$
on $K_X^{-1}$ such that
   $$ u_\varepsilon=\Theta_{h_\varepsilon} (K_X^{-1}) \ge
                                                 -\varepsilon\omega.$$
By [Au76] and [Y77,$\,$78] there exists a unique K\"ahler metric
$\omega_\varepsilon$ in the cohomology class $\{\omega\}$ such that
   $$\Ricci(\omega_\varepsilon)=-\varepsilon\omega_\varepsilon+
                            \varepsilon\omega+u_\varepsilon.\leqno(+)$$
In fact $u_\varepsilon$ belongs to the Ricci class $c_1 (K_X^{-1}) =
c_1 (X)$, hence so does the right hand side
$-\varepsilon\omega_\varepsilon+\varepsilon\omega+u_\varepsilon$.
In particular there exists a function $f_\varepsilon$ such that 
   $$u_\varepsilon=\Ricci(\omega)+i\partial\ovp f_\varepsilon.$$
If we set $\omega_\varepsilon=\omega+i\partial\ovp\varphi$ (where
$\varphi$ depends on $\varepsilon$), equation $(+)$ is equivalent to the 
Monge-Amp\`ere equation
   $$ {(\omega+i\partial\ovp\varphi)^n\over\omega^n}=
                        e^{\varepsilon\varphi-f_\varepsilon} \leqno(++)$$
because
   $$\eqalign{
   i\partial\ovp \log (\omega+i\partial\ovp\varphi)^n/\omega^n
   &=\Ricci(\omega)-\Ricci(\omega_\varepsilon)\cr
   &=\varepsilon(\omega_\varepsilon-\omega)+\Ricci(\omega)-u_\varepsilon\cr
   &=i\partial\ovp (\varepsilon\varphi-f_\varepsilon).\cr}$$
It follows from the results of [Au76] that (++) has a unique
solution~$\varphi$, thanks to the fact the right hand side of (++) is
increasing in~$\varphi$. Since $u_\varepsilon \ge-\varepsilon\omega$,
equation (+) implies in particular that $\Ricci(\omega_\varepsilon) \ge
-\varepsilon\omega_\varepsilon$.

We now recall a well-known differential geometric technique used to get
bounds for~$N(k)$ (this technique has been explained to us in a very
efficient way by S.~Gallot). Let $(M,g)$ be a compact Riemannian $m$-fold
and let $E \subset \wt M$ be a fundamental domain for the 
action of $\pi_1 (M)$ on the universal covering $\wt M$. Fix $a\in E$
and set $\beta=\diam E$. Since $\pi_1(M)$ acts isometrically on $\wt M$
with respect to the pull-back metric~$\wt g$, we infer that
   $$E_k=\bigcup_{\gamma\in\pi_1(M),~{\rm length}(\gamma)\le k} \gamma (E)$$
has volume equal to $N(k)\,\Vol(M)$ and is contained in the geodesic ball
$B(a,\alpha k+\beta)$, where $\alpha$ is the maximum of the length of loops
representing the generators~$g_j$. Therefore
   $$ N(k) \le {\Vol\big(B(a,\alpha k+\beta)\big)\over \Vol(M)}
                                                   \leqno(\star)$$
and it is enough to bound the volume of geodesic balls in~$\wt M$. For this
we use the following fundamental inequality due to R.~Bishop [Bi63],
Heintze-Karcher [HK78] and M.~Gage [Ga80].

\than Lemma 1.2|Let
  $$ \Phi : T_{\wt M,\,a} \to \wt M,~~~~\Phi (\zeta)=\exp_a (\zeta) $$
be the $($geodesic$)$ exponential map. Denote by 
  $$ \Phi^\star dV_g =a(t,\zeta)\,dt\, d\sigma (\zeta)$$
the expression of the volume element in spherical coordinates with
$t\in\rb_+$ and $\zeta\in S_a (1)={}$ unit sphere in~$T_{\wt M,\,a}$. Suppose
that $a(t,\zeta)$ does not vanish for $t\in{}]0,\tau (\zeta)[\,$, with
$\tau (\zeta)=+\infty$ or $a(\tau (\zeta),\zeta)=0$. Then $b(t,\zeta)=
a(t,\zeta)^{1/(m-1)}$ satisfies on $]0,\tau (\zeta)[$ the inequality
   $${\partial^2\over\partial t^2}\,b(t,\zeta)+
     {1\over m-1}\,\Ricci_g (v(t,\zeta),v(t,\zeta))\,b(t,\zeta) \le 0$$
where
   $$v(t,\zeta)={d\over dt}\,\exp_a (t\zeta) \in  S_{\Phi(t\zeta)}(1)
                    \subset T_{\wt M,\,\Phi(t\zeta)}.~~~\finpr$$\fintha

If $\Ricci_g \ge-\varepsilon g$, we infer in particular
   $${\partial^2b\over\partial t^2}-{\varepsilon\over m-1}\,b \le 0$$
and therefore $b(t,\zeta) \le \alpha^{-1}\sinh(\alpha t)$ with $\alpha =
\sqrt{\varepsilon/(m-1)}$ (to check this, observe that $b(t,\zeta)=t+o(t)$
at $0$ and that $\sinh(\alpha t)\,\partial b/\partial t-\alpha
\cosh(\alpha t)\,b$ has a negative derivative). Now, every point $x\in
B(a,r)$ can be joined to $a$ by a minimal geodesic arc of length
${}< r$. Such a geodesic arc cannot contain any focal point (i.e. any
critical value of $\Phi$), except possibly at the end point~$x$.
It~follows that $B(a,r)$ is the image by $\Phi$ of the open set 
   $$\Omega(r)=\big\{(t,\zeta)\in[0,r[{}\times S_a (1)\,;\,
                                                t<\tau(\zeta)\big\}.$$
Therefore
   $$\Vol_g(B(a,r)) \le \int_{\Omega (r)}\Phi^\star dV_g=\int_{\Omega (r)}
                          b(t,\zeta)^{m-1} dt\, d\sigma (\zeta).$$
As $\alpha^{-1} \sinh(\alpha t) \le t\,e^{\alpha t}$, we get
   $$\Vol_g (B(a,r)) \le \int_{S_a(1)} d\sigma (\zeta) \int_0^r t^{m-1}
   e^{(m-1)\alpha t}dt\le v_mr^m e^{\sqrt{(m-1)\varepsilon}\,r}\leqno(\star\star)$$
where $v_m$  is the volume of the unit ball in $\rb^m$.\finpr
\vskip8pt

In our application, the difficulty is that the metric
$g=\omega_\varepsilon$ varies with $\varepsilon$ as well as the
constants $\alpha =\alpha_\varepsilon$, $\beta =\beta_\varepsilon$
in~$(\star)$, and $\alpha_\varepsilon \sqrt{(m-1)\varepsilon}$ need not
converge to $0$ as $\varepsilon$ tends to~$0$. We overcome this
difficulty by the following lemma, which shows that an arbitrary large
fraction of the volume of $\wt X$ remains at bounded distance without
being disconnected.

\than Lemma 1.3|Let $U_1$, $U_2$ be compact subsets of~$\wt X$. Then for every
$\delta>0$, there are closed subsets $U_{1,\varepsilon,\delta} \subset U_1$
and $U_{2,\varepsilon,\delta} \subset U_2$ with $\Vol_\omega (U_j \ssm
U_{j,\varepsilon,\delta})<\delta$, such that any two points
$x_1\in U_{1,\varepsilon,\delta}$, $x_2\in U_{2,\varepsilon,\delta}$ can be
joined by a path of length ${}\le C\,\delta^{-1/2}$ with
respect to $\omega_\varepsilon$, where $C$ is a constant independent of
$\varepsilon$ and~$\delta$.\fintha

\proof The basic observation is that
   $$\int_X \omega_\varepsilon \wedge \omega^{n-1}=\int_X \omega^n$$
does not depend on $\varepsilon$, therefore $\Vert\omega_\varepsilon
\Vert_{L^1(X)}$ is uniformly bounded. First suppose that $U_1 =U_2 =K$ where
$K$ is a compact convex set in some coordinate 
open set $\Omega$ of~$\wt X$. We simply join $x_1\in K$, $x_2\in K$ by the
segment $[x_1 ,x_2] \subset K$ and compute the average
$\omega_\varepsilon$-length of this segment when $x_1$, $x_2$ vary
(the average being computed in $L^2$ norm with respect to the Lebesgue
measure of $\Omega$). By Fubini and the Cauchy-Schwarz inequality we get
   $$\eqalign{
    \int_{K\times K}&\Big(\int_0^1 \sqrt{\omega_\varepsilon
             \big((1-t)x_1+tx_2\big)\cdot(x_2-x_1)}\,dt\Big)^2 dx_1\,dx_2\cr
 &{}\le\Vert x_2-x_1\Vert\int_0^1 dt\int_{K\times K}
   \big\Vert\omega_\varepsilon\big((1-t)x_1+tx_2\big)\big\Vert\,dx_1\,dx_2\cr
 &{}\le 2^{2n}\diam K\cdot\Vol(K)\cdot\Vert\omega_\varepsilon\Vert_{L^1(K)}
         \le C_1\cr}$$
where $C_1$ is independent of $\varepsilon\,$; the last inequality is
obtained by integrating first with respect to $y=(1-t)x_1$ when
$t\le{1\over 2}$, resp. $y=tx_2$  when $t\ge{1\over 2}$ (observe that 
$dx_j\le 2^{2n}dy$ in both cases).

It follows that the set $S$ of pairs $(x_1 ,x_2)\in K\times K$ such that
${\rm length}_{\omega_\varepsilon}([x_1 ,x_2 ])$ exceeds $(C_1/ \delta)^{1/2}$
has measure ${}<\delta$ in $K\times K$. By Fubini, the set $Q$ of points
$x_1\in K$ such that the slice $S(x_1)=\{x_2\in K\,;\,(x_1 ,x_2)\in S\}$
has volume $\Vol(S(x_1))\ge{1\over 2} \Vol(K)$ itself has volume
$\Vol(Q)<2\delta/\Vol(K)$. Now for $x_1,x_2\in K\ssm Q$ we have by
definition $\Vol(S(x_j))<{1\over 2} \Vol(K)$, therefore
   $$K\ssm \big( S(x_1) \cup \big(K\ssm S(x_2)\big) \ne \emptyset.$$
If $y$ is a point in this set, then $(x_1,y)\notin S$ and $(x_2,y)\notin S$,
hence
   $$\hbox{\rm length}_{\omega_\varepsilon}\big([x_1,y] \cup [y,x_2 ]\big)
                       \le  2(C_1/ \delta)^{1/2}.$$
By continuity, a similar estimate still holds for any two points
$x_1,x_2\in\ov{K\ssm Q}$, with some $y\in K$. When $U_1 =U_2 =K$, the
lemma is thus proved with $U_{j ,\varepsilon, \delta}=\ov{K\ssm Q}\,$:
note that
   $$\Vol_\omega(U_j\ssm U_{j,\varepsilon,\delta})\le\Vol_\omega(Q)
               \le C_2\Vol(Q)<2C_2\delta/\Vol(K)$$
and replace $\delta$ by $c\delta$ with $c=\Vol(K)/(2C_2)$ to get the
desired bound $\delta$ for the volume of $U_j\ssm U_{j,\varepsilon,\delta}$.

If $U_1$, $U_2$  are isomorphic to compact convex subsets in $\cb^n$,
we find a chain of such sets $V_1,\ldots,V_N$ with $V_1 =U_1$, $V_N
=U_2$ and $V_j^\circ\cap V_{j+1}^\circ\ne\emptyset$. By the first case,
there exists for each $j=1,\ldots,N$ a subset $V_{j,\varepsilon,\delta}
\subset V_j$ with $\Vol_\omega(V_j \ssm V_{j,\varepsilon,\delta})<\delta$
such that any pair of points in $V_{j,\varepsilon,\delta}$ can be joined
by a path of length ${}\le C_3\delta^{-1/2}$. If we take $\delta<{1\over 2}
\Vol_\omega(V_j\cap V_{j+1})$ for every~$j$, then 
$(V_j \ssm V_{j,\varepsilon,\delta}) \cup (V_{j+1}\ssm V_{j+1,\varepsilon,
\delta})$ cannot contain $V_j\cap V_{j+1}$ and therefore 
$V_{j,\varepsilon,\delta}\cap V_{j+1,\varepsilon,\delta}\ne\emptyset$.
This implies that any $x\in U_{1,\varepsilon,\delta}:= V_{1,\varepsilon,
\delta}$ can be joined to any $y\in U_{2,\varepsilon,\delta} :=
V_{N,\varepsilon,\delta}$ by a piecewise linear path of length
${}\le NC_3\delta^{-1/2}$. The case where $U_1$, $U_2$ are arbitrary
is obtained by covering these sets with finitely many 
compact convex coordinate patches.\finpr
\vskip8pt

We take $U$ to be a compact set containing the fundamental domain~$E$,
so large that $U^\circ\cap g_j (U^\circ)\ne \emptyset$ for each
generator~$g_j$. We apply Lemma~1.3 with $U_1 =U_2 =U$ and 
$\delta>0$ fixed such that
    $$\delta <{1\over 2} \Vol_\omega (E),~~~~
                 \delta <{1\over 2} \Vol_\omega\big(U\cap g_j (U)\big).$$
We get $U_{\varepsilon,\delta}\subset U$ with $\Vol_\omega(U\ssm
U_{\varepsilon,\delta})<\delta$ and $\diam_{\omega_\varepsilon}
(U_{\varepsilon,\delta})\le C\delta^{-1/2}$. The ine\-qualities on volumes
imply that $\Vol_\omega(U_{\varepsilon,\delta}\cap E)\ge{1\over 2}
\Vol_\omega (E)$ and $U_{\varepsilon,\delta}\cap g_j (U_{\varepsilon,\delta})
\ne \emptyset$ for every~$j$ (note that all $g_j$ preserve volumes).
It is then clear that
   $$W_{k,\varepsilon,\delta}:=\bigcup_{\gamma\in\pi_1(X),~
               {\rm length}(\gamma)\le k} \gamma (U_{\varepsilon,\delta})$$
satisfies
   $$\eqalign{
      \Vol_\omega(W_{k,\varepsilon,\delta})&{}\ge N(k)\,\Vol_\omega
      (U_{\varepsilon,\delta}\cap E)\ge N(k)\,{1\over 2}\,\Vol_\omega(E)~~~~
      \hbox{\rm and}\cr
      \diam_{\omega_\varepsilon}(W_{k,\varepsilon,\delta})
                &{}\le k\,\diam_{\omega_\varepsilon} U_{\varepsilon,\delta}
                                      \le kC\delta^{-1/2}.\cr}$$
Since $m=\dim_\rb X=2n$, inequality $(\star\star)$ implies
   $$\Vol_{\omega_\varepsilon}(W_{k,\varepsilon,\delta})
              \le \Vol_{\omega_\varepsilon}\big(B(a,kC\delta^{-1/2})\big)
              \le C_4 k^{2n} e^{C_5 \sqrt{\varepsilon}\,k}.$$
Now $X$ is compact, so there is a constant $C(\varepsilon)>0$ such that
$\omega^n\le C(\varepsilon)\,\omega_\varepsilon^n$. We~conclude that
   $$N(k) \le {2\,\Vol_\omega(W_{k,\varepsilon,\delta})\over \Vol_\omega(E)}
          \le C_6 C(\varepsilon)k^{2n}e^{C_5 \sqrt{\varepsilon}\,k}.$$
The proof of Theorem 1.1 is complete.\finpr

\rem Remark 1.4|In the non-K\"ahler case, one might try instead to use
hermitian metrics $\omega_\varepsilon$ in the same conformal class
as~$\omega$, such that $\int_X \omega_\varepsilon^n=\int_X \omega^n$ and
$\Theta_{\omega_\varepsilon}(K_X^{-1})=u_\varepsilon \ge-\varepsilon\omega$.
Then Lemma 1.3~still holds. The major difficulty is that the first Chern form
$\Theta_{\omega_\varepsilon}(K_X^{-1})$ differs from the Riemannian Ricci
tensor $\Ricci(\omega_\varepsilon)$ and there is no known analogue of
Inequality~1.2 in that case. The fact that we control
$\Theta_{\omega_\varepsilon}(K_X^{-1})$ by $-\varepsilon\omega$ instead of
$-\varepsilon\omega_\varepsilon$ could be also a source of difficulties.

\rem Remark 1.5|It is well known and easy to check that equation (++) implies
    $$C(\varepsilon)=\max{\omega^n\over\omega_\varepsilon^n}
           \le\exp\Big(\max_X f_\varepsilon-\min_X f_\varepsilon\Big).$$
In fact, this follows from the observation that $i\partial\ovp\varphi
\ge0$ at a minimum point, thus $(\omega+i\partial\ovp\varphi)^n/
\omega^n\ge 1$ and (++) implies $\varepsilon\min\varphi\ge\min f_\varepsilon$.
Similarly we have $\varepsilon\max\varphi\le\max f_\varepsilon$.
Since $f_\varepsilon$ is a potential of $\Theta_{h_\varepsilon}(K_X^{-1})-
\Ricci(\omega)$ and converges to an almost plurisubharmonic function as
$\varepsilon$ tends to~$0$, it is reasonable to expect that $C(\varepsilon)$
has polynomial growth in $\varepsilon^{-1}\,$; this would imply that
$\pi_1(X)$ has polynomial growth by taking $\varepsilon =k^{-2}$. When
$K_X^{-1}$ has a semipositive metric, we can even take $\varepsilon =0$
and find a metric $\omega_0$ with $\Ricci(\omega_0)=u_0\ge 0$. This
gives:

\than Theorem 1.6|If $X$ is K\"ahler and $K_X^{-1}$ is hermitian semipositive
$($e.g. if K$_X^{-m}$ is generated by sections for some $m)$ then $\pi_1 (X)$
has polynomial growth of degree ${}\le 2n$. In particular
   $$q(X)=h^1 (X,\oc)={1\over 2}\,\hbox{\rm rank}_\zb\,H_1 (X,\zb) \le n.$$
\fintha

\rem Remark 1.7|If $X$ is a Fano manifold, i.e. if $K_X^{-1}$ is ample,
the above techniques can be used to show that $X$ is simply connected,
as observed long ago by S.~Kobayashi [Ko61]. In fact the
Aubin-Calabi-Yau theorem provides a K\"ahler metric $\omega$ with
$\Ricci(\omega)=u>0$, say $u\ge\varepsilon\omega$. Then Lemma~1.2
implies $\partial^2 b/\partial t^2+\smash{{\varepsilon\over
2n-1}}b(t,\zeta)\le 0$, thus $b(t,\zeta)\le\alpha^{-1}\sin(\alpha t)$
with $\alpha=\sqrt{\varepsilon/(2n-1)}$. In~particular $\tau
(\zeta)\le\pi/ \alpha$, hence the universal covering $\wt X$ is compact
of diameter ${}\le \pi/ \alpha$ and $\pi_1 (X)$ is  finite (all this
was already settled by S.~Myers [My41] for arbitrary Riemannian
manifolds). The Hirzebruch-Riemann-Roch formula implies
   $$\chi (\wt X,\oc_{\wt X})=k\,\chi (X,\oc_X)~~~~\hbox{\rm with}~~
                k=\#\,\pi_1 (X).$$
Moreover, the Kodaira vanishing theorem applied to the ample line bundle
$L=K_{\wt X}^{-1}$ gives
   $$H^q (\wt X,\oc_{\wt X})=H^q (\wt X,K_{\wt X}\otimes L)=0~~~~
                       \hbox{\rm for}~~q\ge 1,$$
hence $\chi(\wt X,\oc_{\wt X})=h^0(\wt X,\oc_{\wt X})=1$ and~$k=1$.\finpr
\vskip20pt

\titre 2. Surjectivity of the Albanese map|

Let $X$ be a compact K\"ahler manifold and let $q(X)=h^1 (X,\oc_X)$ be
its irregularity. Recall that the Albanese map is a holomorphic map
from $X$ to the Albanese torus
   $$\Alb(X):=H^0 (X,\Omega_X)^\star/ H_1 (X,\zb),~~~~\dim \Alb(X)=q(X),$$
defined by
   $$\alpha (x)(h) := \int_{x_0}^x h,~~~~h \in  H^0 (X,\Omega^1_X )\,;$$
the path from $x_0$ to $x$ in the integral is taken modulo an arbitrary
loop at~$x_0$, i.e. modulo $H_1 (X,\zb)$. We first reprove
Lichnerowicz' fibration theorem [Li71] by a simpler method based on the
Bochner technique (of course Lichnerowicz had somehow to circumvent the
Aubin-Calabi-Yau theorem, which was not available at that time). Our
starting point is the following basic formula.

\than Formula 2.1|Let $\#$ be the conjugate linear $C^\infty$-isomorphism
$T_X\to\Omega^1_X$, $v\mapsto i\ov v\ort\omega$, given by a K\"ahler
metric~$\omega$. Denote also by $\#:\Lambda^pT_X\to\Omega^p_X$ the induced
$C^\infty$ isomorphism from $p$-vectors to $p$-forms. Then for an arbitrary
smooth section $v$ of $\Lambda^p T_X$ we have
   $$\int_X \Vert\ovp(\# v)\Vert^2dV_\omega
     =\int_X\Vert\ovp v\Vert^2dV_\omega+\int_X\langle\rr(v),v\rangle\,
                                                              dV_\omega$$
where $dV_\omega$ is the K\"ahler element of volume and $\rr$ is the
hermitian operator
   $$v=\sum_{|I|=p}v_I{\partial\over\partial z_I}\longmapsto
       \rr(v)=\sum_{|I|=p}\Big(\sum_{k\in I}\rho_k\Big)v_I
                 {\partial\over\partial z_I}$$
associated to the Ricci curvature form: $\rho_k$ denotes the eigenvalues
of $\Ricci(\omega)$ in an $\omega$-orthonormal frame
$(\partial/\partial z_k)$.\fintha

\proof We first make a pointwise calculation of
$\ovp^\star\ovp v$ and $\ovp^\star\ovp(\# v)$ in a normal coordinate
system for the K\"ahler metric~$\omega$. In such coordinates we can
write
   $$\omega=i\sum_{1\le m\le n}dz_m\wedge d\ov z_m-
     i\sum_{1\le j,k,\ell,m\le n}c_{jk\ell m}z_j\ov z_k dz_\ell\wedge
     d\ov z_m+O(|z|^3)$$
where $(c_{jk\ell m})$ is the curvature tensor of $T_X$ with respect
to~$\omega$. The K\"ahler property shows that we have the symmetry
relations $c_{jk\ell m}=c_{\ell kjm}=c_{jm\ell k}$, and the Ricci
tensor $R=\sum R_{\ell m}dz_\ell\wedge d\ov z_m$ is obtained as the
trace: $R_{\ell m}=\sum_j c_{jj\ell m}$. Since~$\omega$
is tangent of order $2$ to a flat metric at the center $x_0$ of the chart,
we easily see that the first order operator $\ovp^\star$ has the same
formal expression at $x_0$ as in the case of the flat metric on $\cb^n\,$: 
if~$w$ if a smooth $(0,q)$-form with values in a holomorphic vector bundle
$E$ trivialized locally by a holomorphic frame $(e_\lambda)$ such that
$(e_\lambda(x_0))$ is orthonormal and $De_\lambda(x_0)=0$, we have at
$x_0$ the formula
   $$w=\sum_{\lambda,\,|J|=q}w_{\lambda,J}\,e_\lambda\otimes d\ov z_J,~~~~
     \ovp^\star w=-\sum_{\lambda,\,|J|=q,k}{\partial w_{\lambda,J}\over
      \partial z_k}\,e_\lambda\otimes\Big({\partial\over\partial\ov z_k}
      \ort d\ov z_J\Big).$$
This applies of course to the case of sections of $\Lambda^pT_X$ or
$\Omega^p_X$ expressed in terms of the frames $\partial/\partial z_I$
and $dz_I$, $|I|=p$. From this, we immediately find that for any smooth
sections $v=\sum v_I\,\partial/\partial z_I$ and $w=\sum w_I\,dz_I$ we have
$$\ovp^\star\ovp v=-\sum_{I,k}{\partial^2v_I\over\partial z_k\partial\ov z_k}
\,{\partial\over\partial z_I},~~~~
\ovp^\star\ovp w=-\sum_{I,k}{\partial^2w_I\over\partial z_k\partial\ov z_k}
\,dz_I$$
at the point~$x_0$. Now, we get
  $$\eqalign{
     \#{\partial\over\partial z_m}&=i{\partial\over\partial\ov z_m}\ort\omega
      =dz_m-\sum_{j,k,\ell}c_{jk\ell m}z_j\ov z_kdz_\ell+O(|z|^3),\cr
 \# v&=\sum_I\ov v_Idz_I-\sum_{I,j,k,\ell,m}\ov v_Ic_{jk\ell m}z_j\ov z_k
   dz_\ell\wedge\Big({\partial\over\partial z_m}\ort dz_I\Big)+O(|z|^3).\cr}$$
Computing $\ovp^\star\ovp(\# v)$ at $x_0$ we obtain
  $$\eqalign{\ovp^\star\ovp(\# v)&=
      -\sum_{I,k}{\partial^2\ov v_I\over\partial z_k\partial\ov z_k}dz_I
      +\sum_{I,k,\ell,m}\ov v_Ic_{kk\ell m}dz_\ell\wedge
             \Big({\partial\over\partial z_m}\ort dz_I\Big)\cr
      &=\#(\ovp^\star\ovp v)+\sum_{I,\ell,m}\ov v_IR_{\ell m}dz_\ell\wedge
             \Big({\partial\over\partial z_m}\ort dz_I\Big)
       =\#(\ovp^\star\ovp v)+\#\rr(v).\cr}$$
Formula 2.1 then follows from this identity by writing
  $$\int_X\Vert\ovp(\#v)\Vert^2dV_\omega=
    \int_X\langle\ovp^\star\ovp(\#v),\#v\rangle\,dV_\omega=
    \int_X\langle\ovp^\star\ovp v+\rr(v),v\rangle\,dV_\omega.\finpr$$
\vskip3pt
We easily deduce from this the fibration theorem of Lichnerowicz
[Li71,$\,$72], as well as analogous results of [Li67] in the case
$\Ricci(\omega)\le 0$.
    
\than Theorem 2.2|Let $(X,\omega)$ be a compact K\"ahler manifold. Consider
the natural contraction pairing
   $$\Psi:H^0(X,\Lambda^pT_X)\times H^0(X,\Omega_X^p)\la\cb,~~~~
      0\le p\le n=\dim X.$$
\item{\rm(i)} If $\Ricci(\omega)\ge 0$, then $\Psi$ has zero kernel in
$H^0(X,\Omega^p_X)$. In that case, the Albanese map $\alpha:X\to\Alb(X)$ is
a submersion and every holomorphic vector field of $\Alb(X)$ admits a lifting
to~$X$. Therefore, there is a group of analytic automorphisms of $X$
lying over the group of translations of $\Alb(X)$.
\smallskip
\item{\rm(ii)} If $\Ricci(\omega)\le 0$, then $\Psi$ has zero kernel in
$H^0(X,\Lambda^pT_X)$. In that case the identity component $\Aut(X)^\circ$
of $\Aut(X)$ is abelian and leaves invariant all global holomorphic
$p$-forms or $p$-vector fields.
\smallskip\fintha

\proof Let $v$ be a smooth section of $\Lambda^pT_X$ and let $w=\#v$ be
the associated smooth $(p,0)$-form. By definition of $\#$ we have
$v\ort w=\Vert v\Vert^2$. Now, when $\Ricci(\omega)\ge 0$, Formula 2.1
shows that $\int_X\Vert\ovp w\Vert^2dV_\omega\ge\int_X\Vert\ovp v\Vert^2
dV_\omega$, thus $v$ is holomorphic as soon as $w$ is.
Therefore we get an injective conjugate linear map
   $$\#^{-1}:H^0(X,\Omega_X^p)\longrightarrow H^0(X,\Lambda^pT_X)$$
with the property that $(\#^{-1}w)\ort w$ is a non zero constant for $w\ne0$.
This shows that the kernel of $\Psi$ in $H^0(X,\Omega^p_X)$ is zero.
On the other hand, when $\Ricci(\omega)\le 0$, the inequality is reversed
and we get an injection
   $$\#:H^0(X,\Lambda^pT_X)\longrightarrow H^0(X,\Omega_X^p).$$
Hence in that case the kernel of $\Psi$ in $H^0(X,\Lambda^pT_X)$ is zero.
\smallskip
\noindent{\bf(i)} By the above with $p=1$, every holomorphic $1$-form $h$
which is not identically zero does not vanish at all, because there is a
vector field $v$ such that $v\ort h=1$. Let $(h_1,\ldots,h_q)$
be a basis of $H^0(X,\Omega^1_X)$. Then for each point $x\in X$ the
$1$-forms $h_1(x),\ldots, h_q(x)$ must be independent in~$T^\star_{X,x}$.
In the basis of $T_{\Alb(X)}$ provided by the $h_j$'s, we have
$d\alpha(x)=(h_1(x),\ldots h_q(x))$ and so $d\alpha(x)$ is surjective.
Now, there are holomorphic vector fields $v_1,\ldots,v_q$ on $X$ such that
$v_i\ort h_j=\delta_{ij}$. These vector fields clearly generate a subgroup $G$
of $\Aut(X)^\circ$ which lies over the group of translations of $\Alb(X)$.
\smallskip
\noindent{\bf(ii)} Let $v_1,\ldots,v_q$ be a basis of the Lie algebra
of~$\Aut(X)^\circ$. Then all Lie brackets $[v_i,v_j]$ vanish, because
we have
   $$[v_i,v_j]\ort h=v_i\cdot(v_j\ort h)-v_j\cdot(v_i\ort h)=0$$
for every holomorphic $1$-form~$h$ (just observe that $v_i\ort h$ and
$v_j\ort h$ are constant functions). Thus $\Aut(X)^\circ$ is
abelian. Moreover, for any holomorphic $p$-form $w$, the Lie derivative
$\lr_{v_i}(w)$ vanishes$\,$:
   $$\lr_{v_i}(w)=d(v_i\ort w)+v_i\ort(dw)=0,$$
because all holomorphic forms on a compact K\"ahler manifold are $d$-closed.
Hence $w$ is invariant under $\Aut(X)^\circ$. By duality, we easily
conclude that the holomorphic $p$-vectors are also kept invariant.\finpr
\vskip7pt

We now discuss conjecture 2 for compact K\"ahler manifolds $X$ with
$-K_X$ being only nef. The proof of the following theorem has been
communicated to us by F.~Campana.  

\than Theorem 2.3|Let $X$ be a compact K\"ahler manifold with $-K_X$~nef.
Let $\alpha:X\longrightarrow\Alb(X)$ be the Albanese map.
If $\dim\alpha(X)=1$, then $\alpha$ is surjective.
\fintha

\proof $\alpha(X)$ is a smooth curve $C$. Assume that $C$ has genus $g\ge 2$.
Then $\pi_1(C)$ has exponential growth; in fact it contains a free group
with $2g-1$ generators. Because of the exact sequence $\pi_1(X)
\buildo \alpha_\star\over\longrightarrow\pi_1(C)\longrightarrow\pi_0(F)$,
the image of $\alpha_\star$ has finite index in $\pi_1(C)$, hence
$\pi_1(X)$ is of exponential growth, contradicting Theorem~1.1.\finpr
\vskip7pt

First suppose $\dim\alpha(X)=\dim X$. If $\alpha(X)\ne\Alb(X)$,
there would be at least two independent sections of $K_X$
coming from $H^0(\Omega^n_{\Alb(X)})\,$; since $-K_X$ is nef, these
sections would not vanish and so $K_X=\oc_X$, contradiction. The
next interesting case is $\dim\alpha(X)=\dim X-1$, which we treat next.
We first prove a more general statement.  

\than Theorem 2.4|Let $X$ be a compact K\"ahler manifold with $-K_X$
nef. Then there is no holomorphic surjective map $\varphi:X\longrightarrow
Y$ to a projective variety $Y$ with $\kappa(Y)>0$ such that $-K_F$ is big
for the general fiber $F$ of $\varphi$.
\fintha

By definition the Kodaira dimension $\kappa(Y)$ is the Kodaira dimension
of a desingularisation.  

\proof Assume there is a map $\varphi$ as above. We may assume that
$Y$ is normal by passing to the normalization, and moreover that the fibers
are connected by taking the Stein factorization.
Choose a very ample divisor $H$ on~$Y$. Letting $m=\dim Y$, we pick a
curve
$$C= H_1\cap\ldots\cap H_{m-1}$$
with $H_i\in |H|$ in general position. Then $C$ is smooth as well as
$X_C= \varphi^{-1}(C)$ by Bertini's lemma. Moreover 
$$c_1(\omega_Y)\cdot C>0, \leqno(1)$$
since $C\cap{\rm Sing}(Y)= \emptyset$ and since $\omega_Y =
\pi_\star(\omega_{\wt Y})$ on $Y\ssm{\rm Sing}(Y)$ for every desingularisation
$\pi:\wt Y\longrightarrow Y$. 

Let $f=\varphi_{|\,X_C}$. We claim:
$$\omega_{X_C/C}^{-1}~~~\hbox{\rm is big and nef.}\leqno(2)$$  
In fact,
   $$\omega_{X_C/C}^{-1}={\omega_{X/Y}^{-1}}_{|X_C}=
     {\omega_X^{-1}}_{|X_C}\otimes f^\star\big({\omega_Y}_{|C}\big)$$
which is nef because of (1). Letting $p=\dim X_C$ and taking $p$-th powers,
we observe that $c_1(f^\star{\omega_Y}_{|C})\ge c_1(\oc(F))$ by (1),
$F$ being a generic fibre, thus
   $$c_1(\omega_{X_C/C}^{-1})^p\ge
     c_1\big({\omega_X^{-1}}_{|X_C}\big)^{p-1}\cdot c_1\big(f^\star
     {\omega_Y}_{|C}\big)\ge
     c_1\big({\omega_X^{-1}}_{|X_C}\big)^{p-1}\cdot F=
     c_1(\omega_F^{-1})^{p-1},$$
which is positive by our assumption that $-K_F$ is big. Now we can apply
Kawamata-Viehweg's vanishing theorem [Ka82,$\,$Vi82] to obtain  
   $$H^1\big(X_C,\omega_{X_C}\otimes\omega_{X_C/C}^{-1}\big)=0$$
But $\omega_{X_C}\otimes\omega_{X_C/C}^{-1}=f^\star(\omega_C)$, so via
the Leray spectral sequence we conclude $H^1(C,\omega_C)=0$, which is
absurd.\finpr

\than Corollary 2.5|Let $X$ be a $n$-dimensional projective $($or Moishezon$)$
mani\-fold with $-K_X$ nef. Assume that the Albanese map $\alpha$ has
$(n-1)$-dimensional image. Then $\alpha$ is surjective.\fintha

\proof If $\alpha$ is not surjective, the image $Y=\alpha(X)$ automatically
has $\kappa(Y)>0$ since we get at least two independent holomorphic forms
of maximum degree from the Albanese torus. We may thus assume $\kappa(Y)>0$.
Let $F$ be the general fiber of $\alpha$, which is a curve. Since
$-K_F=-{K_X}_{|F}$ is nef, $F$ is rational or elliptic. In the first case,
$\alpha$ is surjective by the previous theorem. If $F$ is elliptic,
then $\kappa(X)=0$, so $h^0(X,mK_X)\ne 0$ for some $m$ and consequently
$mK_X=\oc_X$. Therefore $\alpha$ is onto by Theorem~2.2~(i).\finpr 

The last part of the proof shows more generally that 
conjecture 2 holds if $\kappa(X)=0$. A different proof of Corollary 2.5
has been given by F. Campana.
\vskip20pt

\titre 3. Threefolds whose anticanonical bundles are nef|

In this section we want to study the structure of projective 3-folds
$X$ with $- K_X$ nef in more detail. In particular we prove Conjecture~2
in dimension~3 with purely algebraic methods, except in one very special
case. In fact, we prove that the Albanese map is surjective except possibly
when all extremal contractions of $X$ are of type (B), defined in
Proposition 3.3$\,$(2). For the structure of surfaces with $-K_X$ nef we
refer to [CP91]. 

Let always $X$ denote a smooth projective $3$-fold with $- K_X$ nef and
let \hbox{$\alpha:X\longrightarrow \Alb(X)$} be the Albanese map. By the last
words of Section~2, the structure of $X$ is clear if $\kappa(X)=0\,$;
so we will assume $\kappa(X)=-\infty\,$; note that $K_X$ is not nef in
this case. Then there exists an extremal ray on $X$ ([Mo82], [KMM87]);
let \hbox{$\varphi:X\longrightarrow W$} be the associated contraction. We want
to analyze the structure of~$\varphi$.  

\than Proposition 3.1|If $\dim W\le 2$, $\alpha$ is surjective. More precisely: 
\medskip
\item{\rm(1)} If $W$ is a point, $X$ is Fano with $b_2=1$, in particular
$X$ is simply connected.
\medskip
\item{\rm(2)} If W is a $($smooth$)$ curve, then $g(W)\le 1$. In case $g(W)=1$,
we have $\alpha=\varphi\,$; if $g(W)=0$, we have $q(X)=0$.
In all cases $\varphi$ has the structure of a del Pezzo fibration.
\medskip
\item{\rm(3)} If $W$ is a $($smooth$)$ surface, then either 
\smallskip
\itemitem{\rm a)} $\varphi$ is a $\pb_1$-bundle and $-K_W$ is nef
\smallskip
\itemitem{\rm b)} $\varphi$ is a proper conic bundle with discriminant
locus $\Delta$ such that $-(4K_W+\Delta)$ is nef, and $q(W)\le 1$.
\smallskip\fintha

\proof (1) If $\dim W=0$, then $q(X)=0$, hence our claim is obvious.

\noindent (2) Let $\dim W=1$. Since $R^q \varphi_\star (\oc_X)=0$ for
$q>0$, either $\varphi$ is the Albanese map and then we must show that
$q(W)=1$ or $q(W)=0$. So assume $q(W) \ge  2$. Then the canonical
bundle $K_W$ is ample.  Let $K_{X/W}$ be the relative canonical bundle, so
   $$K_{X/W}=K_X-\varphi^\star (K_W).$$
Since the Picard number $\rho (X)=\rho (W)+1=2$ (see e.g. [KMM87]),
and since $-K_X$ is nef and $\varphi$-ample ([KMM87]), $-K_{X/W}$ is
ample. Hence by Kodaira vanishing:
   $$H^1 \big(X,\oc_X (-K_{X/W}) \otimes \oc_X (K_X)\big)=0,$$
so $H^1 \big(X,\varphi^\star (\oc_W (K_W)\big)=0$ and
$H^1 \big(W,\oc_W (K_W)\big)=0$, which is absurd.

\noindent (3)  Now assume $\dim W=2$. Then $W$ is smooth and $\varphi$ is
a $\pb_1$-bundle or a conic bundle ([Mo82]). Since $q(X)=q(W)$, we
have a diagram
 $$\matrix{~X&\buildo{\displaystyle\alpha}\over\la&\Alb(X)\vphantom{\Big)}\cr
             \llap{\hbox{$\varphi~$}}\big\downarrow&&
                  \big\downarrow\rlap{\hbox{$~\gamma$}}\cr
            W&\buildo \smash{{\displaystyle\beta}}\over\la &\Alb(W)\cr}$$
with $\beta$ being the Albanese map of $W$ and $\gamma$ being finite.

\noindent(3a) Assume $\varphi$ to be a $\pb_1$-bundle. We will prove that
$-K_W$ is nef, hence $\beta$ is onto and so is $\alpha$. Take any
irreducible curve $C \subset  W$. Since $\varphi^{-1}(C)=\pb (E_C)$ with
a rank 2-bundle $E_C$ on $C$, we have (after possibly passing to the
normalization of~$C$):
   $$-K_{X\,|\,\pb(E_C)}=-K_{\pb (E_C)}+\varphi^\star (N_{C/W})$$
by the adjunction formula. Since
   $$-K_{\pb (E_C)}=\oc_{\pb\Big(E_C\otimes{\det E_C^\star\over 2}
                              \otimes{(-K_C)\over 2}\Big)} (2),$$
we have
   $$-K_{X\,|\,\pb(E_C)}=\oc_{\pb\Big(E_C\otimes{\det E_C^\star\over 2}
                              \otimes{(-K_{W/C})\over 2}\Big)} (2).$$
Since $-K_X$ is nef, we conclude that
   $$E_C\otimes{\det E_C^\star\over 2}
                       \otimes{(-K_{W/C})\over 2}~~~\hbox{\rm is nef.}$$
Now $c_1\big(E_C\otimes{\det E_C^\star\over 2}\big)=0$, hence $-K_{W/C}$
must be nef and $-K_W$ itself is nef.

\noindent (3b) Next assume $\varphi$ to be a proper conic bundle. Let
$\Delta \subset W$ be the discriminant locus,~i.e. 
   $$\Delta =\big\{w \in W\,;\, X_w~\hbox{\rm not smooth}\big\}.$$
From the well-known formula (see e.g. [Mi81]) 
   $$K_X^2\cdot\varphi^{-1}(C)=-(4K_W+\Delta)\cdot C$$
for every curve $C\subset W$, we deduce from the nefness of $-K_X$
that $-(4K_W+\Delta)$ is nef.

Now we conclude by means of the following:

\than Lemma 3.2|Let $W$ be a smooth projective surface, $\Delta \subset  W$
be a $($possibly reducible$)$ curve. Assume that $-(4K_W+\Delta)$ is nef.
Then $q(W) \le 1$.\fintha

\proof Obviously $\kappa (W)=-\infty$. We can easily reduce the problem
to the case of $W$ being minimal. If $W \ne \pb_2$, then $W$ is ruled over
a curve~$C$. Now it is an easy exercise using [Ha77, V.2] to prove
that $g(C) \le 1$.\finpr

\than Proposition 3.3|Assume $\dim W=3$. Let $E$ be the exceptional set
of~$\varphi$.
\item{$(1)$} If $\dim \varphi (E)=0$, then $-K_W$ is big and nef and
$q(X)=0$.
\item{$(2)$} If $\dim \varphi (E)=1$, then $W$ is smooth, $\varphi$ is
the blow-up of the smooth curve $C_0 =\varphi (E)$ and $-K_W$ is again
nef except for the following special cases:
\itemitem{} $C_0$ is rational and moreover either
\itemitem{\rm(A)} $N_{C_0/W} \simeq \oc (-2) \oplus \oc (-2)$~~ or
\itemitem{\rm(B)} $N_{C_0/W} \simeq \oc (-1) \oplus \oc (-2)$.
\fintha

\proof By [Mo82] $E$ is always an irreducible divisor and if
$\dim \varphi (E)=1$,~ $W$ is smooth and $\varphi$ is the blow-up of a
smooth curve. We may always assume $K_X^3=0$, otherwise $q(X)=0$ by
Kawamata-Viehweg vanishing.

\noindent(1)  We have the following formula of $\qb$-divisors:
   $$ K_X =\varphi^\star (K_W)+\vartheta E$$
with some $\vartheta \in \qb_+$ ([Mo82], in fact either $E \simeq \pb_2$
or $E$ is a normal quadric and $\vartheta =2,1$ or $1/2$). Hence $-K_W$
is nef. Furthermore:
  $$K_X^3=K_W^3+\vartheta^3 E^3$$
and since $E^3>0$ ($E$ has always negative normal bundle, [Mo82]),
we conclude from $K_X^3=0$ that $K_W^3<0$, so $-K_W$ is big and nef
(observe that $W$ might be singular). Now a ``singular'' version of the
Kawamata-Viehweg vanishing theorem ([KMM87, 1.2.5, 1.2.6] applied
to $D=0$) yields
   $$H^1 (W,\oc_W)=0.$$
Since $R^q\varphi_\star (\oc_X)=0$ for $q>0$, we get $q(X)=0$.

\noindent(2) From the formula $K_X=\varphi^\star (K_W)+E$, we
immediately see that 
   $$-K_W\cdot C \ge  0$$
for every curve $C \ne C_0$.

Let $N_{C_0/W}=N$ denote the normal bundle of~$C_0$. Let $V=N^\star
\otimes\lr$ with $\lr \in \Pic(C_0)$ be its normalization, i.e.
$H^0 (V) \ne 0$, but $H^0 (V\otimes\gr)=0$ for all line bundles $\gr$ with
$\deg\gr<0$. Let $\mu=\deg \lr$. Then $E$ can be written as
$E=\pb (N^\star)=\pb (V)$. The ``tautological'' line bundle $\oc_{\pb (V)}(1)$
has a ``canonical'' section $C_1$ satisfying $C_1^2=-e=c_1 (V)$
(see [Ha77, V.2]). In this terminology
   $$(K_X\cdot C_1)=(K_W\cdot C_0)+(E\cdot C_1).$$
Let $F$ be a fiber of $\varphi_{|\,E}$. Write for numerical equivalence
  $$-K_{X\,|\,E} \equiv aC_1+bF.$$
Since $(K_X\cdot F)=-1$, we have $a=1$. Moreover
  $$N_{E\,|\,X} \equiv-C_1+\mu F$$
just by definition of $\mu$ and by the fact that
$N^\star_{E\,|\,X}=\oc_{\pb(N^\star)}(1)$. Hence
  $$(K_W\cdot C_0)=(K_X\cdot C_1)-(E\cdot C_1)=-b-\mu\,;$$
so $-K_W$ is nef if
$$b+\mu \ge  0.  \leqno(\star)$$
Letting $g$ be the genus of $C_0$, we have $K_E^2=8(1-g)\,$; on the other
hand we compute by adjunction:
   $$K_E^2 =(K_X+E )_{|\,E}^2=\big(-2C_1+(\mu-b)F\big)^2.$$
Thus $8(1-g)=4b-4e-4\mu$, and consequently 
   $$ b+\mu =2b-e+2(g-1). \leqno(\star\star)$$
Since $-K_{X\,|\,E}$ is nef, we have $K_X^2\cdot E \ge 0$, which is
equivalent to $b \ge e/2$. Combining this with $(\star\star)$ shows that
$b+\mu \ge 0$ at least if $g \ge 1$. Therefore,\break
if $g \ge 1$, $-K_W$ is nef.

Now assume $g=0$. Then $(\star$) is equivalent to
  $$ b \ge {e\over 2}+1. \leqno(\star')$$
Again by nefness of $-K_{X\,|\,E}$ we get
  $$ 0 \le  (-K_X\cdot C_1)=(C_1+b\,F\cdot C_1)=b-e,$$
so $b \ge  e$. This settles already the case $e \ge 2$.

Since $e \ge 0$, we are left with $e=0$ and $e=1$ and additionally
$e \le b<{e\over 2}+1$. This leads to (A) and (B).\finpr

\rem Remark 3.4|Assume that $-K_X$ is big and nef. Then $X$ is ``almost
Fano'' in the following sense. By the Base Point Free Theorem [KMM87]
we have a surjective map $\varphi:X\longrightarrow Y$, given by the
base point free system $|-mK_X|$ for some suitable $m$, to a normal
projective variety $Y$. This variety $Y$ carries an ample line bundle
$L$ such that $\varphi^\star(L)=-mK_X$. The map $\varphi$ being a
modification, we conclude that $L=-mK_Y$. Thus $Y$ is
$\qb$-Gorenstein with at most canonical singularities and the
$\qb$-Cartier divisor $-K_Y$ is ample. We say that $Y$ is ``$\qb$-Fano''.
In particular $Y$ has irregularity $0$ ($\qb$-Fano are even expected to
be simply connected and so $X$ would be simply connected).\finpr
\vskip7pt

We are now interested in those $X$ which are not ``almost Fano'', i.e.
such that $(-K_X)^3= 0$.  

\than Proposition 3.5|Assume $-K_X$ nef and $K_X^3=0$. If $\varphi$
is a contraction of type {\rm(B)}, then $q(X)=0$ and moreover $X$ is
birational to a $\qb$-Fano variety. In particular $\Alb(X)=0$.\fintha

\proof Let $\varphi:X\longrightarrow W$ be the contraction, which is
the blow-up of $C_0\subset W$ such that $C_0\simeq\pb_1$ and
   $$N_{C_0/W}\simeq\oc(-1) \oplus \oc(-2).$$
Let $E\subset X$ be the exceptional divisor. We have
   $$E\simeq\pb(N_{C_0/W}^\star)=\pb\big(\oc(1)\oplus\oc(2)\big)=\Sigma_1$$
($\Sigma_1={}$ Hirzebruch surface of index~$1$). Let $C\subset E$
be the unique section with $C^2=-1$. Let $\pi:X^+\la X$ be the
blow-up of~$C$. Since $N_{C/E}=\oc(-1)$ and
   $$N_{E/X}=\oc_{\pb(N_{C_0/X}^\star)}(-1)=\oc_{\pb(\oc\oplus\oc(-1))}(-1)
     \otimes\varphi^\star\oc(-2)=\oc(-C-2\,F_\varphi),$$
we get ${N_{E/X}}_{\big|C}=\oc(-1)$ and obtain from
   $$0\la N_{C/E}\la N_{C/X}\la {N_{E/X}}_{\big|C}\la 0$$
that $N_{C/X}=\oc(-1)\oplus\oc(-1)$. Hence the exceptional divisor
$D=\pi^{-1}(C)$ is $\pb_1\times\pb_1$ and
$N_{D/X^+}=\oc_{\pb(N_{C/X}^\star)}(-1)=\oc(-1)\stimes\oc(-1)$. Therefore
$D$ can be blown down along the other ruling. Let $\sigma:X^+\la X^-$ be
this blowing down.

\than Claim 3.6|The anticanonical divisor $-K_{X^-}$ is nef.\fintha

\proof Let $A^-\subset X^-$ be an arbitrary curve not in the center
of~$\sigma$, $A^+$ the strict transform in $X^+$ and $A$ the image in~$X$.
As $K_{X^+}=\sigma^\star(K_{X^-})+D$, we have
   $$(-K_{X^+}\cdot A^+)=(-K_{X^-}\cdot A^-)-(D\cdot A^+)$$
and   
   $$(-K_{X^+}\cdot A^+)=(-K_X\cdot A)-(D\cdot A^+).$$
Hence $(-K_{X^-}\cdot A^-)=(-K_X\cdot A)\ge 0$. Since the center $B$ of
$\sigma$ is rational with $N_{B/X^+}=\oc(-1)\oplus\oc(-1)$, we have
$K_{X^-}\cdot B=0$ and hence  $-K_{X^-}$ is nef.\finpr
\vskip7pt
Let $E^+$ be the strict transform of $E$ in $X^+$ and $E^-=\sigma(E^+)$.
We have $E=\pb\big(\oc(1)\oplus\oc(2)\big)=\Sigma_1$, $E^+\simeq E$, and
$E^-\simeq\pb_2$ because the $(-1)$-curve of $E^+$ gets contracted
by~$\sigma$.

\than Claim 3.7|We have $N_{E^-/X^-}=\oc(-2)$.\fintha

\proof We first compute $E^3$, $(E^+)^3$ and $(E^-)^3$. We have
 $$E^3=c_1(N_{E/X})^2=(-C-2\,F_\varphi)_{|E}^2=C^2+4 C\cdot F_\varphi=-1+4=3.$$
As $\pi$ is the blow-up of a curve in $E$, we get $\pi^\star(E)=E^++D$. Hence
   $$(E^+)^3=E^3-3\,\pi^\star(E)^2\cdot D+3\,\pi^\star(E)\cdot D^2-D^3.$$
Moreover
   $$\eqalign{
     D^3&=c_1(N_{D/X^+})^2=2,~~~~E^+\cdot D^2=(\pb_1\times\{0\})\cdot D=-1,\cr
     \pi^\star(E)\cdot D^2&=(E^++D)\cdot D^2=1,~~~~(E^+)^3=3-0+3-2=4\,;\cr}$$
note that $\pi^\star(E)^2\cdot D=0$ since $\pi$ projects $D$ to a curve.
We finally have $\sigma^\star(E^-)=E^+$ because $\sigma$ is a blowing down
along ruling lines of $D$ which intersect $E^+$ only in one point. Therefore
$(E^-)^3=(E^+)^3=4$. We must have $N_{E^-/X^-}=\oc(k)$ for some $k<0$
(since $E^-$ is exceptional) and
  $$(E^-)^3=c_1(N_{E^-/X^-})^2=k^2,$$
so $k=-2$ as desired.\finpr
\vskip7pt
Let $\psi:X^-\la Z$ be the blowing down of $E^-$. Then $Z$ has only one
rational singularity which is in fact terminal. The nefness of $-K_Z$
follows from the nefness of $-K_{X^-}$. A well-known calculation
(see [Mo82]) yields
   $$K_{X^-}=\psi^\star(K_Z)+{1\over 2}E^-,$$
hence
   $$0\le(-K_{X^-})^3=(-K_Z)^3-{1\over 8}(E^-)^3=(-K_Z)^3-{1\over 2}.$$
Therefore $(-K_Z)^3\ge 1/2$ and $-K_Z$ is big, i.e. $Z$ is birational to a
$\qb$-Fano manifold (see Remark~3.4). The singular Kawamata-Viehweg
vanishing theorem (cf. [KMM], 1.2.6) applied to $-K_Z$ gives
   $$H^1(Z,\oc_Z)=0,$$
therefore $q(X)=q(Z)=0$.\finpr   
\vskip7pt

We have thus shown that case (B) does not occur when $q(X)>0$. Therefore,
putting everything together, we have proved:

\than Theorem 3.8|Let $X$ be a smooth projective $3$-fold with
$-K_X$~nef and $\kappa(X)=-\infty$.
Let $\varphi:X\longrightarrow W$ be the contraction of an extremal
ray. Then $- K_W$ is nef except possibly for the following cases:
\smallskip
\item{\rm(a)} $\varphi$ is the blow-up of a smooth rational curve $C$ such that 
   $$N_{C/W} \simeq\cases{
        \oc(-1) \oplus \oc(-2)&or \cr
        \oc(-2) \oplus \oc(-2).\cr}$$
In the first case $X$ has irregularity $0$ and is birational to a $\qb$-Fano
variety.
\smallskip        
\item{\rm(b)} $\varphi$ is a proper conic bundle over a surface $W$ with
$-(4K_W+\Delta)$ nef, $\Delta$ being the discriminant locus.\smallskip
\fintha

\rem Remark 3.9|Let $X$ be a smooth projective $3$-fold with $-K_X$~nef.
The above algebraic considerations again show that the Albanese map
\hbox{$\alpha:X\longrightarrow\Alb(X)$} is surjective, except possibly if
all contractions $X\la W$ are of type~(A) or if this situation occurs
after finitely many blowing-downs.
\vskip5pt

\proof We may assume $K_X$ not nef. Let $\varphi:X\longrightarrow W$ be
the contraction of an extremal ray. If $\dim W\le 2$, $\alpha$ is
already surjective by Prop.~3.1. If $\dim W=3$, $\varphi$~must be
either the blow-up of a point, hence $(-K_W)^3>0$ and $q(X)=q(W)=0$
(except possibly for case~(A)), or $\varphi$ is the blow-up of a smooth
curve and $-K_W$ is again nef with $W$ smooth. Then we proceed by
induction on $b_2(W)$.\finpr
  
\rem Remarks 3.10|\smallskip
\noindent(1) In 3.8$\,$(a) consider the morphism $\psi=\Phi_{|-mK_X|}$ with
suitable~$m$. In case $N_{C/W}=\oc(-1)\oplus\oc(-2)$, $\psi$ contracts
the exceptional curve of $E\simeq\Sigma_1$, in the other case $\varphi$
contracts all curves in $E\simeq\pb_1\times\pb_1$ which are ruling lines
non contracted by $\varphi$. It would be interesting to know whether
3.8$\,$(a) can really occur. 
\smallskip
\noindent(2) If $\varphi:X\longrightarrow W$ is a proper conic
bundle with $-K_X$ nef, then $(-K_W\cdot C)\ge 0$ if $C\not\subset\Delta$
or $C\subset\Delta$ but a multiple of $C$ moves. So $- K_W$ is ``almost
nef''. It would be interesting to have a rough classification of conic
bundles $X$ with $-K_X$ nef.
\smallskip
\noindent(3) For contractions of type (A), we have in fact the following
additional information:

\than Proposition 3.11|Assume that $-K_X$ is nef, $K_X^3=0$ and that
$\varphi:X\la W$ is of type {\rm(A)}. Then $K_X^2=0$.\fintha

\proof Let $\psi : W \to W'$ be the blow-down of $C_0=\varphi (E)$.
Let $\sigma=\psi\circ\varphi$. Let $N^1 (Z)=\Pic(Z)
\otimes_\zb\rb/{\equiv}$ for any~$Z$. Then $\sigma^\star (N^1 (W'))$ is a
linear subspace of codimension $2$ in~$N^1(X)$, in fact $\psi^\star (N^1
(W'))$ is of codimension $1$ in~$N^1(W)$, and $\varphi^\star (N^1 (W))$ is
of codimension $1$ in~$N^1 (X)$, as one checks immediately from Mori theory.

Assume $K_X^2 \ne 0$. Then $(K_X^2)^\perp=\big\{L \in N^1(X)\,;\,
L\cdot K_X^2=0\big\}$ is of codimension $1$ in $N^1 (X)$. Hence:
   $$(K_X^2)^\perp=\sigma^\star (N^1 (W')) \oplus \rb\cdot K_X$$
because $K_X^3=0$. Since $K_X^2\cdot E=0$, $E$ is in $(K_X^2)^\perp$, so
   $$E=\mu K_X+\sigma^\star (H)$$
with $H \in  N^1 (W')$. Cutting by a fiber of $\varphi$ yields $\mu=1$.
Since $K_X=\varphi^\star (K_W)+E$, we conclude $\varphi^\star (K_W) =
-\sigma^\star (H)$, i.e. $K_W =-\psi^\star (H)$, which is false.\finpr

\vskip30pt

\centerline{\bf References}
\vskip10pt\eightpoint
\lettre KMM87|
\article Au76|Aubin, T|Equations du type Monge-Amp\`ere sur les vari\'et\'es
 k\"ahl\'eriennes compactes|C.R. Acad. Sci. Paris Ser. A {\bf
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\article Be83|Beauville, A|Vari�t�s K\"ahl\'eriennes dont la premi�re
 classe de Chern est nulle|J. Diff. Geom.|18|1983|755-782|
\article Bi63|Bishop, R|A relation between volume, mean curvature and 
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\article Bo74a|Bogomolov, F.A|On the decomposition of K\"ahler
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\article Bo74b|Bogomolov, F.A|K\"ahler manifolds with trivial canonical
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\divers Ca57|Calabi, E|On K\"ahler manifolds with vanishing canonical
 class|Alg. geometry and topology, Symposium in honor of S.~Lefschetz,
 Princeton Univ. Press, Princeton ({\oldstyle 1957}), 78-89|
\divers CP91|Campana, F., Peternell, Th|On the second exterior power of 
 tangent bundles of $3$-folds|Preprint {\oldstyle 1991}|
\divers DPS91|Demailly, J.P., Peternell, Th., Schneider, M|Compact complex 
 manifolds with numerically effective tangent bundles|Preprint
 {\oldstyle 1991}|
\article Ga80|Gage, M.E|Upper bounds for the first eigenvalue of the Laplace-
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\article Gr81|Gromov, M|Groups of polynomial growth and expanding maps,
 Appendix by J.~Tits|Publ. I.H.E.S|53|1981|53-78|
\livre Ha77|Hartshorne, R|Algebraic Geometry|Graduate Texts in Math.,
 Springer, Berlin|1977|
\article 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|11|1978|451-470|
\article Ka82|Kawamata, Y|A generalization of Kodaira-Ramanujam's vanishing
 theorem|Math. Ann.|261|1982|43-46|
\article KMM87|Kawamata, Y., Matsuki, K., Matsuda, K|Introduction to the
 minimal model program|Adv. Studies Pure Math.|10|1987|283-360|
\article Ko61|Kobayashi, S|On compact K\"ahler manifolds with positive definite 
 Ricci tensor|Ann. Math|74|1961|570-574|
\article Kr86|Koll\'ar, J|Higher direct images of dualizing sheaves|Ann.
 Math.|123|1986|11-42|
\article Li67|Lichnerowicz, A|Vari\'et\'es K\"ahl\'eriennes et premi\`ere
 classe de Chern|J. Diff. Geom.|1|1967|195-224|
\article 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.|6|1971|47-94|
\divers Li72|Lichnerowicz, A|Vari\'et\'es K\"ahl\'eriennes \`a premi\`ere
 classe de Chern non n\'egative et situation analogue dans le cas
 riemannien|Ist. Naz. Alta Mat., Rome, Symposia Math., vol.~10, Academic
 Press, New-York({\oldstyle 1972}), 3-18|
\article Ma69|Matsushima, Y|Recent results on holomorphic vector fields|J.
 Diff. Geom.|3|1969|477-480|
\article Mi81|Miyanishi, M|Algebraic methods in the theory of algebraic 
 threefolds|Adv. Studies in Pure Math.|1|1983|69-99|
\article Mo82|Mori, S|Threefolds whose canonical bundles are not
 numerically effective|Ann. Math.|116|1982|133-176|
\article My41|Myers, S.B|Riemannian manifolds with positive mean
 curvature|Duke Math.~J.|8|1941|401-404|
\article Vi82|Viehweg, E|Vanishing theorems|J. Reine Angew.
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\article Y77|Yau, S.T|Calabi's conjecture and some new results in algebraic
 geometry|Proc. Nat. Acad. Sci. USA|74|1977|1789-1790|
\article Y78|Yau, S.T|On the Ricci curvature of a complex K\"ahler
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Math.|31|1978|339-411|
\vskip25pt
\tenpoint\noindent
(July 16, 1992)

\end

% K\"ahler manifolds with semipositive Ricci curvature

\titre 3. Threefolds whose anticanonical bundles are nef|

\noindent Theorem 2.1 leads us to the following

\than Conjecture 3.1|Let $X$ be a compact K\"ahler manifold with $-K_X$~nef.
Then the Albanese map $\alpha : X \to  A(X)$ is surjective. In particular
$q(X) \le  \dim X$.\fintha

Concerning this conjecture we prove

\than Theorem 3.2|Let $X$ be a projective manifold of dimension $n \le 3$.
Assume $-K_X$ to be nef. Then the Albanese map $\alpha : X \to  A(X)$ is
surjective, with the following possible very special exception:
$K_X^2=0$, the Albanese dimension $\alb(X)=1$ and every 
contraction $\varphi : X \to W$ of an extremal ray is the blow-up of a
smooth rational curve $C_0 \subset W$ with normal bundle
$N_{C_0/W}=\oc (-2) \oplus \oc (-2)$.\fintha

The rest of this section gives a proof of (3.2). Note first that (3.2)
for surfaces is an easy exercise (left to the reader; see [CP91]).
So we now assume $\dim X=3$. We always let $\alpha:X\to
A(X)$ be the Albanese of~$X$, $Y=\alpha (X)$ be its image and let
$\alb(X)=\dim Y$ be the Albanese dimension of~$X$. If~$K_X$ is  nef,
too, then $K_X \equiv 0$ and hence (3.2) follows from Lemma~2.2~(ii).
So  we  will assume from now on that $K_X$  is not nef.  Observe also
that we can assume $\kappa (X)=-\infty\,$: otherwise we would have a
non-zero section $s \in H^0 (X,\oc_X (mK_X))$ and since $-K_X$ is nef,
$s$~cannot have zeroes, thus $mK_X =\oc_X$. Observe also that only
the case  $q(X) \ge 2$ is interesting, the claim (3.2) being obvious
for $q(X) \le 1$. Since $K_X$ is not nef, there is an extremal ray on
$X$ ([Mo82], [KMM87]); let $\varphi : X \to  W$ be the associated
contraction. What we do first is to analyze the structure of~$\varphi$.

\than Proposition 3.3|If $\dim W \le 2$,~ $\alpha$ is surjective.\fintha

\proof (1) If $\dim W=0$, then $q(X)=0$, hence our claim is obvious.

\noindent (2) Let $\dim W=1$. Since $R^q \varphi_\star (\oc_X)=0$ for
$q>0$, either $\varphi$ is the Albanese map and then we must show that
$q(W)=1$ or $q(W)=0$. So assume $q(W) \ge  2$. Then the canonical
bundle $K_W$ is ample.  Let $K_{X/W}$ be the relative canonical bundle, so
   $$K_{X/W}=K_X-\varphi^\star (K_W).$$
Since the Picard number $\rho (X)=\rho (W)+1=2$ (see e.g. [KMM87]),
and since $-K_X$ is nef and $\varphi$-ample ([KMM87]), $-K_{X/W}$ is
ample. Hence by Kodaira vanishing:
   $$H^1 \big(X,\oc_X (-K_{X/W}) \otimes \oc_X (K_X)\big)=0,$$
so $H^1 \big(X,\varphi^\star (\oc_W (K_W)\big)=0$ and
$H^1 \big(W,\oc_W (K_W)\big)=0$, which is absurd.

\noindent (3)  Now assume $\dim W=2$. Then $W$ is smooth and $\varphi$ is
a $\pb_1$-bundle or a conic bundle ([Mo82]). Since $q(X)=q(W)$, we
have a diagram
 $$\matrix{~X&\buildo{\displaystyle\alpha}\over\la&A(X)\vphantom{\Big)}\cr
             \llap{\hbox{$\varphi~$}}\big\downarrow&&
                  \big\downarrow\rlap{\hbox{$~\gamma$}}\cr
            W&\buildo \smash{{\displaystyle\beta}}\over\la &A(W)\cr}$$
with $\beta$ being the Albanese map of $W$ and $\gamma$ being finite.

\noindent(3a) Assume $\varphi$ to be a $\pb_1$-bundle. We will prove that
$-K_W$ is nef, hence $\beta$ is onto and so is $\alpha$. Take any
irreducible curve $C \subset  W$. Since $\varphi^{-1}(C)=\pb (E_C)$ with
a rank 2-bundle $E_C$ on $C$, we have (after possibly passing to the
normalization of~$C$):
   $$-K_{X\,|\,\pb(E_C)}=-K_{\pb (E_C)}+\varphi^\star (N_{C/W})$$
by the adjunction formula. Since
   $$-K_{\pb (E_C)}=\oc_{\pb\Big(E_C\otimes{\det E_C^\star\over 2}
                              \otimes{(-K_C)\over 2}\Big)} (2),$$
we have
   $$-K_{X\,|\,\pb(E_C)}=\oc_{\pb\Big(E_C\otimes{\det E_C^\star\over 2}
                              \otimes{(-K_{W/C})\over 2}\Big)} (2).$$
Since $-K_X$ is nef, we conclude that
   $$E_C\otimes{\det E_C^\star\over 2}
                       \otimes{(-K_{W/C})\over 2}~~~\hbox{\rm is nef.}$$
Now $c_1\big(E_C\otimes{\det E_C^\star\over 2}\big)=0$, hence $-K_{W/C}$
must be nef and $-K_W$ itself is nef.

\noindent (3b) Next assume $\varphi$ to be a proper conic bundle. Let
$\Delta \subset W$ be the discriminant locus,~i.e. 
   $$\Delta =\big\{w \in W\,;\, X_w~\hbox{\rm not smooth}\big\}.$$
From the well-known formula (see e.g. [Mi81]) 
   $$K_X^2\cdot\varphi^{-1}(C)=-(4K_W+\Delta)\cdot C$$
for every curve $C\subset W$, we deduce from the nefness of $-K_X$
that $-(4K_W+\Delta)$ is nef.

Now we conclude by means of the following

\than Lemma 3.4|Let $W$ be a smooth projective surface, $\Delta \subset  W$
be a $($possibly reducible$)$ curve. Assume that $-(4K_W+\Delta)$ is nef.
Then $q(W) \le 1$.\fintha

\proof Obviously $\kappa (W)=-\infty$. We can easily reduce the problem
to the case of $W$ being minimal. If $W \ne \pb_2$, then $W$ is ruled over
a curve~$C$. Now it is an easy exercise using [Ha77, V.2] to prove
that $g(C) \le 1$.\finpr

\than Proposition 3.5|Assume $\dim W=3$. Let $E$ be the exceptional set
of~$\varphi$.
\item{$(1)$} If $\dim \varphi (E)=0$, then $-K_W$ is big and nef and
$q(X)=0$.
\item{$(2)$} If $\dim \varphi (E)=1$, then $W$ is smooth, $\varphi$ is
the blow-up of the smooth curve $C_0 =\varphi (E)$ and $-K_W$ is again
nef except for the following special cases:
\itemitem{} $C_0$ is rational and moreover either
\itemitem{\rm(A)} $N_{C_0/W} \simeq \oc (-2) \oplus \oc (-2)$~~ or
\itemitem{\rm(B)} $N_{C_0/W} \simeq \oc (-1) \oplus \oc (-2)$.
\fintha

\proof By [Mo82] $E$ is always an irreducible divisor and if
$\dim \varphi (E)=1$,~ $W$ is smooth and $\varphi$ is the blow-up of a
smooth curve. We may always assume $K_X^3=0$, otherwise $q(X)=0$ by
Kawamata-Viehweg vanishing.

\noindent(1)  We have the following formula of $\qb$-divisors:
   $$ K_X =\varphi^\star (K_W)+\vartheta E$$
with some $\vartheta \in \qb_+$ ([Mo82], in fact either $E \simeq \pb_2$
or $E$ is a normal quadric and $\vartheta =2,1$ or $1/2$). Hence $-K_W$
is nef. Furthermore:
  $$K_X^3=K_W^3+\vartheta^3 E^3$$
and since $E^3>0$ ($E$ has always negative normal bundle, [Mo82]),
we conclude from $K_X^3=0$ that $K_W^3<0$, so $-K_W$ is big and nef
(observe that $W$ might be singular). Now a ``singular'' version of the
Kawamata-Viehweg vanishing theorem ([KMM87,1.2.5,1.2.6] applied
to $D=0$) yields
   $$H^1 (W,\oc_W)=0.$$
Since $R^q\varphi_\star (\oc_X)=0$ for $q>0$, we get $q(X)=0$.

\noindent(2) From the formula $K_X=\varphi^\star (K_W)+E$, we
immediately see that 
   $$-K_W\cdot C \ge 0$$
for every curve $C \ne C_0$.

Let $N_{C_0/W}=N$ denote the normal bundle of~$C_0$. Let $V=N^\star
\otimes\lr$ with $\lr \in \Pic(C_0)$ be its normalization, i.e.
$H^0 (V) \ne 0$, but $H^0 (V\otimes\gr)=0$ for all line bundles $\gr$ with
$\deg\gr<0$. Let $\mu=\deg \lr$. Then $E$ can be written as
$E=\pb(N^\star)=\pb(V)$. The ``tautological'' line bundle $\oc_{\pb (V)}(1)$
has a ``canonical'' section $C_1$ satisfying $C_1^2=-e=c_1 (V)$
(see [Ha77, V.2]). In this terminology
   $$(K_X\cdot C_1)=(K_W\cdot C_0)+(E\cdot C_1).$$
Let $F$ be a fiber of $\varphi_{|\,E}$. Write for numerical equivalence
  $$-K_{X\,|\,E} \equiv aC_1+bF.$$
Since $(K_X\cdot F)=-1$, we have $a=1$. Moreover
  $$N_{E\,|\,X} \equiv-C_1+\mu F$$
just by definition of $\mu$ and by the fact that
$N^\star_{E\,|\,X}=\oc_{\pb(N^\star)}(1)$. Hence
  $$(K_W\cdot C_0)=(K_X\cdot C_1)-(E\cdot C_1)=-b-\mu\,;$$
so $-K_W$ is nef if
$$b+\mu \ge  0.  \leqno(\star)$$
Letting $g$ be the genus of $C_0$, we have $K_E^2=8(1-g)\,$; on the other
hand we compute by adjunction:
   $$K_E^2 =(K_X+E )_{|\,E}^2=\big(-2C_1+(\mu-b)F\big)^2.$$
Thus $8(1-g)=4b-4e-4\mu$, and consequently 
   $$ b+\mu =2b-e+2(g-1). \leqno(\star\star)$$
Since $-K_{X\,|\,E}$ is nef, we have $K_X^2\cdot E \ge 0$, which is
equivalent to $b \ge e/2$. Combining this with $(\star\star)$ shows that
$b+\mu \ge 0$ at least if $g \ge 1$. Therefore,\break
if $g \ge 1$, $-K_W$ is nef.

Now assume $g=0$. Then $(\star$) is equivalent to
  $$ b \ge {e\over 2}+1. \leqno(\star')$$
Again by nefness of $-K_{X\,|\,E}$ we get
  $$ 0 \le  (-K_X\cdot C_1)=(C_1+b\,F\cdot C_1)=b-e,$$
so $b \ge  e$. This settles already the case $e \ge 2$.

Since $e \ge 0$, we are left with $e=0$ and $e=1$ and additionally
$e \le b<{e\over 2}+1$. This leads to (A) and (B).\finpr

If $C_0$ is of type (A) (resp. (B)) then $(K_W\cdot C_0)=2$ (resp.~1)
and $-K_W$ is not nef. Although we don't have an example where this
phenomenon occurs, we cannot see any reason to rule it out. Anyway
these special cases create a lot of difficulties for proving (3.2).

\than Notation 3.6|We say that a smooth projective $3$-fold $X$ is of
exceptional type if $\underline{\smash{\hbox{\sl every}}}$ extremal
contraction $\varphi : X \to W$ is the blow-up of a smooth rational
curve in $W$ whose normal bundle is of type {\rm(A)} or {\rm(B)}.
\fintha

From (3.4) and (3.5) we deduce by induction on $b_2 (X)\,$:

\than Conclusion 3.7|In order to prove theorem 3.2 it is sufficient to
prove it for $3$-folds of exceptional type $($with $-K_X$ nef\/$)$.\fintha

It might be interesting to note (implicitly proved):

\than Corollary 3.8|Let $X$ be a smooth projective $3$-fold with $-K_X$~nef.
Let $\varphi : X \to W$ be a contraction of an extremal ray. Assume that
$\varphi$ is not of type\break {\rm(A)} or {\rm(B)}. Then:
\item{\rm(1)} if $\varphi$ is not a conic bundle, $-K_W$ is nef $(W$
              may be singular$)\,;$
\item{\rm(2)} if $\varphi$ is a conic bundle with discriminant locus~$\Delta$,
              then $-(4K_W+\Delta)$ is nef, and in particular
              $(-K_W\cdot C) \ge 0$ if
\itemitem{\rm(a)} $C \not\subset \Delta$~~ or
\itemitem{\rm(b)} $C \subset \Delta$~~ but a multiple of $C$ moves.
\fintha

Returning to $3$-folds of exceptional type, we first prove, generalizing an 
argument in~(3.3):

\than Lemma 3.9|Let $X$ be a smooth projective $3$-fold with $-K_X$ nef
$($we may assume $X$ to be of exceptional type$)$. Assume $\alb(X)=1$. Let
$F$ be a general fiber of $\alpha$ and assume $K_X^2\cdot F \ne 0$. Then
$\alpha$ is onto.\fintha

\proof We know that $Y=\alpha (X)$ is a smooth curve. We must show
that $g(Y) \le 1$. Assuming the contrary, $K_Y$ is ample. Hence
$-K_{X/Y}=-K_X+\alpha^\star (K_Y)$ is nef. By our assumption and the
nefness of $-K_X$, we have $K_X^2\cdot F>0$, hence $K_X^2\cdot
\alpha^\star (K_Y)>0$. Since $(-K_{X/Y})^3 =(-K_X)^3+3\,K_X^2\cdot
\alpha^\star (K_Y)$ we see that $-K_{X/Y}$ is big, too. Hence
   $$ 0=H^1 (-K_{X/Y} \otimes  K_X)=H^1 \big(\alpha^\star (K_Y)\big) $$
by Kawamata-Viehweg vanishing. In a moment we will verify
  $$ R^1\alpha_\star (\oc_X)=0. \leqno(\star)$$
Granted this, we conclude $H^1 (K_Y)=0$, contradiction.

So it remains to prove $(\star)$. Since $K_F=K_{X\,|\,F}$, $-K_F$ is big
and nef, hence $q(F)=0$. So $R^1\alpha_\star (\oc_X)$ is a torsion sheaf
by standard theorems. But since $q(X)=q(Y)$, $\alpha$ being the Albanese,
and since on the other hand
   $$ q(X)=q(Y)+h^0 (R^1\alpha_\star (\oc_X))$$
(use the Leray spectral sequence), it follows that
$h^0 (R^1\alpha_\star (\oc_X))=0$, hence $R^1\alpha_\star (\oc_X))=0$.\finpr

\than Corollary 3.10|Assume $X$ of exceptional type, $-K_X$ nef,
$\alb(X)=1$. If~some contraction $\varphi$ of $X$ is of type {\rm(B)},
then $\alpha$ is onto.\fintha

\proof Take $\varphi : X \to W$ of type (B). Let $E$ be the exceptional
divisor. Then we compute immediately:
   $$ K_X^2\cdot E=3.$$
By (3.9) it is sufficient to prove $K_X^2\cdot F \ne 0$ for a fiber $F$
of $\alpha$ (all fibers are equivalent). Let $y=\alpha (E)$ and
$F=\alpha^{-1}(y)$. Then $F=\mu E+\sum\mu_i F_i$ with $\mu ,\mu_i>0$
and all $F_i$ irreducible. Since $-K_X$ is nef, $K_X^2\cdot F_i \ge 0$ for
all~$i$. Hence $K_X^2\cdot E>0$ yields $K_X^2\cdot F>0$.\finpr

In case (A) we unfortunately have $K_X^2\cdot E=0$, so we cannot conclude.
In~this case we even have

\than Lemma 3.11|Assume $X$ to be of exceptional type, $-K_X$ nef,
$\alb(X)=1$ and all contractions of type {\rm(A)}. Then $K_X^2=0$.
\fintha

\proof Consider $\varphi : X \to W$ as usual, let $\psi : W \to W'$ be the
blow-down of $C_0=\varphi (E)$. Let $\sigma=\psi\circ\varphi$. Let
$N^1 (Z)=\Pic(Z) \otimes_\zb\rb/{\equiv}$ for any~$Z$.
Then $\sigma^\star (N^1 (W'))$ is a linear subspace of codimension $2$
in~$N^1(X)$, in fact $\psi^\star (N^1 (W'))$ is of codimension $1$ in~$N^1(W)$,
and $\varphi^\star (N^1 (W))$ is of codimension $1$ in~$N^1 (X)$, as one checks
immediately.

Assume $K_X \ne 0$. Then $(K_X^2)^\perp=\big\{L \in N^1(X)\,;\,
L\cdot K_X^2=0\big\}$ is of codimension $1$ in $N^1 (X)$. Hence:
   $$(K_X^2)^\perp=\sigma^\star (N^1 (W')) \oplus \rb\cdot K_X$$
because $K_X^3=0$. Since $K_X^2\cdot E=0$, $E$ is in $(K_X^2)^\perp$, so
   $$E=\mu K_X+\sigma^\star (H)$$
with $H \in  N^1 (W')$. Cutting by a fiber of $\varphi$ yields $\mu=1$.
Since $K_X=\varphi^\star (K_W)+E$, we conclude $\varphi^\star (K_W) =
-\sigma^\star (H)$, i.e. $K_W =-\psi^\star (H)$, which is false.\finpr

\rem Remark 3.12|Let $X$ be as in (3.11). Take a general smooth fiber $F$
of~$\alpha$. Then $-K_F$ is nef and $K_F^2=K_X^2\cdot F=0$. These
surfaces are easily classified (cf. [CP91]): either

\noindent 1) $F$ is a ruled surface $\pb (E)$ over an elliptic curve $C$
with $E$ a rank $2$-bundle of\break
\phantom{1) {}}the form
   $$ 0  \to \oc  \to  E  \to L \to  0$$
\phantom{1) {}}with $\deg L=0$ or $1\,$;

\noindent 2) $F$ is $\pb_2$ blown up in $9$ points in
``almost general position'' in the sense of\break
\phantom{1) {}}Demazure [Dem80].

\noindent In (3.13) we will rule out case (1) while (2) will remain
unsettled; this is the possible exception mentioned in (3.2).

\than Proposition 3.13|Let $X$ be as in (3.11) and assume that the general
smooth fiber $F$ of $\alpha$ is as in $3.12\,(1)$. Then $\alpha$ is onto.
\fintha

\proof We first claim that $R^1\alpha_\star (\oc_X)$ is a line bundle on the
smooth curve~$Y$. Clearly the coherent sheaf $R^1\alpha_\star (\oc_X)$ has
(generic) rank~$1$. Since $q(X)=q(Y)$, $\alpha$~being the Albanese,
it follows from the Leray spectral sequence that
   $$ H^0 (Y,R^1\alpha_\star (\oc_X))=0.$$
So $R^1\alpha_\star (\oc_X)$ is torsion free, hence locally free.

Now let $L\in \Pic(Y)$, $\deg L=1$. Let
   $$G=-K_X \otimes \alpha^\star (L).$$
Then $G$ is nef and $G^2 \ne 0$, since $G^2=-2\,K_X\cdot F$. So by
Kawamata-Viehweg vanishing:
   $$H^2 (G \otimes  K_X)=0,~~~~\hbox{\rm i.e.}~~~H^2 (\alpha^\star (L))=0.$$
By Leray's spectral sequence, we deduce
   $$\leqalignno{
         H^1 (Y, L \otimes  R^1\alpha_\star (\oc_X)) &= 0, &(\star)\cr
         H^0 (Y, L \otimes  R^2\alpha_\star (\oc_X)) &= 0. &\cr}$$
In particular $R^2\alpha_\star (\oc_X)=0$, since this sheaf vanishes
generically.

Now we claim:
   $$ \deg R^1\alpha_\star (\oc_X) \le  0. \leqno(\star\star) $$
In fact, again by Leray's spectral sequence
   $$ H^2 (X,\oc_X) \simeq  H^1 \big(Y,R^1\alpha_\star (\oc_X)\big).$$
Hence
   $$ \chi (\oc_X)=\chi (\oc_Y)-\chi\big(R^1\alpha_\star (\oc_X)\big).$$
In (3.14) below we will construct a bimeromorphic map $\rho:X \to Z$ where
$Z$ has at most terminal singularities, a map $\tau : Z \to Z'$ onto a
smooth surface which is generically a $\pb_1$-bundle and in fact an
extremal contraction and a holomorphic map $\beta'' : Z' \to Y$ factorizing
over the minimal model $Z' \to Z''$. Then $\beta'' : Z'' \to Y$ is an
elliptic surface, and $\alpha=\beta'\circ \tau \circ \rho$.
Hence $\chi (\oc_X)=\chi (\oc_Z'')$. Since $\chi (\oc_Z'') \ge 0$
(see [BPV84]), we have $\chi (\oc_X) \ge 0$ and by $\chi (\oc_Y)=1-g$,
$g=g(Y)$ the genus, we obtain $\chi (R^1\alpha_\star (\oc_X))\le 1-g$.
This implies $(\star\star)$ by Riemann-Roch.

Let $H=L\otimes R^1\alpha_\star(\oc_X)\,$; then $\deg H\le1$ by~$(\star\star)$.
From $(*)$ we see that $\chi (H) \ge 0$, on the other hand
$\chi (H)=1-g+\deg H$, so $1-g+\deg H \ge 0$, which in turn gives
$g \le 2$ and $g=2$ if $\deg H=1$. So we are left with the case
   $$g(Y)=2,~~~~\deg R^1\alpha_* (\oc_X)=0,$$
which must be ruled out.

For the minimal surface $\beta '' : Z'' \to Y$ constructed in (3.14) below
this means also
   $$\deg R^1\beta''_* (\oc_Z'')=0,$$
as easily checked, hence by [BPV84] $\,Z''$ cannot have any rational curve.
By~(3.15), we thus obtain a contradiction.\vskip8pt

\noindent (3.14) Let $X$ be a smooth projective $3$-fold with $-K_X$ nef
as usual. We will perform Mori's program to construct minimal models
[Mo88]. We start with $\varphi : X \to W$ where $\varphi$ may be
assumed to be a modification. Since $K_W$ is not nef, we can find a
contraction $\varphi_1 : W \to W_1$, etc. Successively we obtain
  $$X\buildo\varphi\over\la W \buildo\varphi_1\over\la W_1
     \buildo\varphi_2\over\la\cdots\cdots\buildo\varphi_r\over\la W_r,$$
where $\varphi_i$ is either a contraction of an extremal ray or a
(directed) flip, i.e. $\varphi_i$ is of the form
   $$\eqalign{
        W_i&\buildo{\displaystyle\varphi_i}\over
        {\relbar\joinrel\relbar\joinrel\relbar\joinrel\la} W_{i+1}\cr
         \varphi_i'&\searrow\qquad\swarrow\varphi_i^+\cr
                   &\quad~~ W_i'\cr}$$
where $\varphi_i'$ is the contraction of an extremal ray contracting
exactly finitely many rational curves and where $\varphi_i^+$ is a
modification again contracting exactly finitely many rational curves
(and such that $K_{W_{i+1}}$ is $\varphi_i^+$-ample). Moreover all
$W_i$ are normal projective varieties with at most terminal
singularities and finally $\dim W_r \le 2$ (because $\kappa (X)=-\infty$
and $X$ is uniruled).

We put $Z=W_{r-1}$, $Z'=W_r$, $\tau=\varphi_r$. Then $Z'$ is a smooth
surface. Let $\gamma : Z' \to Z''$ be its minimal model and
$\beta'' : Z'' \to Y$ the Albanese reduction of~$Z''$. Since $H^1 (\oc_X)
\simeq H^1 (\oc_{Z'})$ by the properties of Mori contractions and flips, we 
have $Y=\wt Y$ and
   $$ \alpha=\beta'' \circ \gamma \circ \tau \circ \rho, $$
where $\rho : X \to  W_{r-1}=Z$ is the composition of $\varphi$ and the
$\varphi_i$, $i \le r-1$.\finpr\vskip8pt

\noindent (3.15) We keep the situation of (3.14). From the description
of the flips the following statement is immediately clear. 

\noindent(3.15.a) There are finitely many rational curves $C_1 ,\ldots,C_s$
and finitely many other curves $C'_1,\ldots,C'_r \subset Z$ such that the
following holds. If $C \subset Z$ is an irreducible curve with
   $$ C \cap C_i =\emptyset,~~1 \le  i \le  s~~~\hbox{\rm and}~~~
      C \ne C'_j,~~ 1 \le  j \le r,$$
then $(-K_Z\cdot C) \ge 0$.

In fact, the $C_i$ come from the ``bad'' curves of the flips and the
$C'_j$ are images of the exceptional divisors of ``ordinary'' extremal
contractions.

\noindent (3.15.b) Assume now additionally that $Z''$ is an elliptic
surface without any rational curve, as in (3.13). Then we find
``big families'' of curves $C'' \subset Z''$ whose strict transforms
$C' \subset  Z'$ do not meet $\tau(C_i)$ and which are different from
$\tau(C_j)$ for every~$i,j$. Hence $-K_{Z\,|\,\tau^{-1}(C')}$ is nef,
and if additionally (as in (3.13)) $\tau$ is a \hbox{$\pb_1$-bundle}
except over possibly a finite set in~$Z'$, then we can conclude
  $$ (-K_{Z'} \cdot C') \ge  0.$$
Hence $(-K_{Z''}\cdot C'') \ge 0$, which implies $(-K_{Z''}\cdot C'')=0$,
$Z''$ being minimal. But this is only possible if $C''$ is contracted by the 
elliptic fibration, contradiction to the general choice of~$C''$.\finpr
\vskip8pt

The last case to be treated is

\than Proposition 3.16|Let $X$ be a smooth $3$-fold with $-K_X$~nef.
Assume $\alb(X)=2$. Then $\alpha$ is onto.\fintha

\proof Assume that $Y=\alpha (X)$ is a proper subset of~$A(X)$. Again
we make use of (3.14) and take over all notations. Then $Z''$ is generically
finite over~$Y$, hence $Z''$ has only finitely many rational curves which
are all contracted by $\beta'' : Z'' \to Y$. Now use (3.15.a) and similar
arguments as in (3.15.b) to get a contradiction (we don't need here the
assumption of $X$ being of exceptional type).\finpr

Combining the results of this section, Theorem 3.2 is completely proved.
\vskip30pt
\centerline{\bf References}
\vskip10pt\eightpoint
\lettre KMM87|
\article AN55|Akizuki, Y., Nakano, S|Note on Kodaira-Spencer's proof of 
 Lefschetz theorems|Proc. Japan Acad.|30|1955|266-272|
\livre BPV84|Barth, W., Peters, C., Van de Ven, A|Compact Complex
 Surfaces|Erg. der Math. 3. Folge, Band 4, Springer, Berlin|1984|
\article Ko81|Kobayashi, S|Recent results in complex differential
 geometry|Jahresber. Deutsche Math. Ver.|83|1981|147-158|
\article Mi81|Miyanishi, M|Algebraic methods in the theory of algebraic 
 threefolds|Adv. Studies in Pure Math.|1|1983|69-99|
\article Mo88|Mori, S|Flip theorem and the existence of minimal models for 
 3-folds|Journal Amer. Math. Soc.|1|1988|117-253|
\vskip25pt
\tenpoint\noindent
(January 30, 1992)

\end

