% Compact complex manifolds with numerically effective tangent bundles
% J.P. Demailly, Th. Peternell, M. Schneider
% TeX file

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%% BeginText

\centerline{\bftwelve COMPACT COMPLEX MANIFOLDS WITH}\vskip4pt
\centerline{\bftwelve NUMERICALLY EFFECTIVE TANGENT BUNDLES}

\vskip10pt
{\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

}}

\vskip30pt
\noindent{\bf Contents}\vskip8pt
{\eightpoint\noindent
0. Introduction\dotfill p. 2\break
\phantom{0. {}}\hfill\break
1. Basic properties of nef line bundles\dotfill p. 5\break
\phantom{0. {}}1.A. Nef line bundles\dotfill p. 5\break
\phantom{0. {}}1.B. Nef vector bundles\dotfill p. 11\break
\phantom{0. {}}\hfill\break
2. Inequalities for Chern classes\dotfill p. 18\break
\phantom{0. {}}\hfill\break
3. Compact manifolds with nef tangent bundles. Structure of the Albanese
   map\dotfill p. 22\break
\phantom{0. {}}3.A. Some examples\dotfill p. 22\break
\phantom{0. {}}3.B. General properties\dotfill p. 23\break
\phantom{0. {}}3.C. Structure of the Albanese map (K\"ahler case)
\dotfill p. 25\break
\phantom{0. {}}3.D. Numerical flatness of the Albanese map
\dotfill p. 32\break
\phantom{0. {}}\hfill\break
4. Moishezon manifolds with nef tangent bundles\dotfill p. 36\break
\phantom{0. {}}\hfill\break
5. Two structure theorems\dotfill p. 37\break
\phantom{0. {}}\hfill\break
6. Surfaces with nef tangent bundles\dotfill p. 40\break
\phantom{0. {}}\hfill\break
7. K\"ahler 3-folds with nef tangent bundles\dotfill p. 44\break
\phantom{0. {}}\hfill\break
References\dotfill p. 45
}\vfill
\noindent
{\bf Key words:} Compact complex manifold, numerically effective vector
bundle, Chern curvature tensor, Chern classes, Albanese map, Fano variety,
Kodaira classification of surfaces, Mori theory, Frankel conjecture.
\vskip15pt
\noindent{\bf A.M.S. Classification:} 32J25, 32L15, 14J15
\vfill\eject

$~$
\titre 0. Introduction|

The compact Riemann surfaces which are not of general type are those
which are of semipositive curvature. In higher dimensions the
situation is much more subtle and it has been a long standing
conjecture due to Frankel to characterize \Pn\ as the only compact
K\"ahler manifold with positive holomorphic bisectional curvature.
Hartshorne extended Frankel's conjecture and weakened the assumption to
the case of ample tangent bundle. In his famous paper~[Mo79] Mori
solved the Hartshorne conjecture by using characteristic p methods.
Around the same time Siu and Yau~[SY80] gave an (analytic) proof of the
Frankel conjecture. Combining algebraic and analytic tools Mok~[Mk88]
classified all compact K\"ahler manifolds with semipositive
holomorphic bisectional curvature.

In algebraic geometry a powerful and flexible notion of semipositivity
is  {\sl numerical effectivity} ("nefness"). A line bundle $L$ on a
projective manifold $X$ is said to be {\sl nef} if $L\cdot C\ge0$ for
all compact curves $C\subset X$. It is clear that a line bundle with
semipositive curvature is nef. The converse had been conjectured by
Fujita~[Fu83]. Unfortunately this is not true as we shall see in \S1,
example 1.7.

A vector bundle $E$ is called {\sl numerically effective} (nef) if the
line bundle $\cO_E(1)$ on $\pb(E)$, the projective bundle of
hyperplanes in the fibres of $E$, is nef. Again it is clear that a
vector bundle $E$ which admits a metric with semipositive curvature
(in the sense of Griffiths) is nef. A compact K\"ahler manifold $X$
having semipositive holomorphic bisectional curvature has by
definition a tangent bundle $T_X$ with semipositive curvature. Again
the converse does not hold. From an algebraic geometric point of view
it is natural to consider the class of projective manifolds $X$ whose
tangent bundle is nef. This has been done by Campana and
Peternell~[CP91] and --in case of dimension 3-- by Zheng~[Zh90]. In
particular, a complete classification is obtained for dimension at most
three.

The purpose of this paper is to investigate compact K\"ahler manifolds
with nef tangent bundles.  One difficulty in carrying over the
algebraic definition of nefness to the K\"ahler case is the possible
lack of curves. This is overcome by the following

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

This means that the curvature of $L$ can have an arbitrarily small
negative part. Clearly a nef line bundle $L$ satisfies $L\cdot C\ge0$
for all curves $C\subset X$, but the converse is not true. For
projective algebraic $X$ both notions coincide. Our main result is a
structure theorem on the Albanese map.\eject

\than Main Theorem|Let $X$ be a compact K\"ahler manifold with nef
tangent bundle $T_X$. Let $\wx$ be a finite \'etale cover of $X$ of
maximum irregularity $q=q(\wx)=h^1(\wx,\cO_{\wx})$. Then
\item{\rm(i)} $\pi_1(\wx)\simeq\zb^{2q}$.
\item{\rm(ii)} The Albanese map $\alpha:\wx\rightarrow A(\wx)$ is a
smooth fibration over a q-dimensional torus with nef relative tangent
bundle.
\item{\rm(iii)} The fibres $F$ of $\alpha$ are Fano manifolds with nef
tangent bundles.\fintha

As a consequence one gets the 

\than Corollary|As an abstract group, the fundamental group $\pi_1(X)$
is an extension of a finite group by $\zb^{2q}$.\fintha

It is conjectured in [CP91] that Fano manifolds (i.e. manifolds such
that $-K$ is ample) with nef tangent bundle are rational homogeneous.
Once this is proved our main theorem classifies the compact K\"ahler
manifolds with nef tangent bundle up to \'etale cover. In \S1 we prove
basic properties of nef vector bundles. An important technical point
--~used over and over~-- is the following:

\than Lemma|Let $E$ be a nef vector bundle on a compact complex
connected manifold $X$ and let $\sigma\in H^0(X,E^\star)$ be a non trivial
section. Then $\sigma$ has no zeros at all.\fintha

The main result of \S1 characterizes vector bundles $E$ which are
numerically flat, i.e. such that $E$ and $E^\star$ are nef.

\than Theorem|Let $E$ be numerically flat on the compact K\"ahler
manifold~$X$. Then $E$ admits a filtration by vector bundles whose
graded pieces are hermitian flat, i.e. given by unitary representations
of $\pi_1(X)$.\fintha

The proof uses two deep facts: the Uhlenbeck-Yau theorem on the
existence of Hermite-Einstein metrics on stable vector bundles and
Kobayashi's flatness criterion derived from L\"ubke's inequality on
Chern classes. In \S2 it is shown that the Fulton-Lazarsfeld
inequalities~[FL83] for Chern classes of ample vector bundles still
hold for nef vector bundles on compact K\"ahler manifolds.

\than Theorem|Let $X$ be a compact K\"ahler manifold and let $E$ be a
nef vector bundle on $X$. Then $E$ is numerically semipositive, i.e.
$P(c(E))\ge0$ for all positive polynomials $P$.\fintha

As a consequence we get the following

\than Corollary|If $E$ is nef and $c_1(E)^n=0$, $n=\dim\,X$, then all
Chern polynomials $P(c(E))$ of degree $2n$ vanish.\fintha

This will be used in the proof of our main theorem. \S3
contains the proof of our main theorem cited above. One of the key
points is the following

\than Proposition|Let $X$ be a compact K\"ahler n-fold with $T_X$ nef. Then
\item{\rm(i)} If $c_1(X)^n>0$, then $X$ is a Fano manifold.
\item{\rm(ii)} If $c_1(X)^n=0$, then $\chiox=0$ and there exists a non
zero holomorphic $p$-form, $p$ suitable odd and a finite \'etale cover
$\wx\rightarrow X$ such that $q(\wx)>0$.\smallskip
\fintha

The difficult part is (ii). Since $c_1(X)^n=0$ we get by the above
corollary that $\chiox=0$. From this we infer the existence of a non
zero $p$-form $u$ for some suitable odd number $p$. By contraction with
$u$ we get a map $S:\Lambda^{p-1}T_X\rightarrow\Omega_X^1$. The image $E$
has constant rank and is therefore a subbundle of $\Omega_X^1$. Hence $E$
is numerically flat. The theorem in \S1 implies the existence of a
hermitian flat subbundle $E_1\subset E$. If $E_1$ would be trivial
after some finite \'etale cover the proposition would be proved. We use
a result of Tits that every subgroup of a linear group contains either
a non abelian free subgroup or a solvable subgroup of finite index. The
first case cannot occur since we show in~[DPS92] that $\pi_1(X)$ has
sub-exponential growth. The second case leads to the desired
conclusion.

In \S4 we give algebraic proofs of the following two results, stated
resp. proved in 3.6 and 3.10~(i).

\than Theorem|Any Moishezon manifold with nef tangent bundle is projective.
\fintha

\than Theorem|Let $X$ be a compact K\"ahler n-fold with $T_X$ nef and
$c_1(X)^n>0$. Then $X$ is a Fano manifold.\fintha

In \S5 we show that a Mori contraction $\varphi:X\rightarrow Y$ of a
projective manifold $X$ with $T_X$ nef is smooth; moreover $Y$ is
smooth too. As a consequence we get an algebraic proof of the crucial
proposition 3.10. In \S6 we classify all non algebraic surfaces with
nef tangent bundles.

\than Theorem|The smooth non algebraic compact complex surfaces with
nef tangent bundles are precisely the following:
\item{\rm(i)} Non algebraic tori$\,;$
\item{\rm(ii)} Kodaira surfaces$\,;$
\item{\rm(iii)} Hopf surfaces.
\fintha

In the final section we give a classification of all non algebraic
K\"ahler 3-folds.

\than Theorem|Let $X$ be a non algebraic 3-dimensional compact K\"ahler
manifold. Then $T_X$ is nef if and only if $X$ is up to finite \'etale
cover either a torus or of the form $\pb(E)$ where $E$ is a numerically
flat rank 2-bundle over a 2-dimensional torus.\fintha

Let us mention that the classification of projective surfaces
and 3-folds with nef tangent bundle is contained in~[CP91] and~[Zh90].

\medskip\noindent{\bf Acknowledgement}: Our collaboration has been
made possible by {\sl Procope} and the {\sl DFG Schwerpunktprogramm
``Komplexe Mannigfaltigkeiten''}.

\vskip25pt
\titre 1. Basic properties of nef vector bundles|
\titre 1.A. Nef line bundles|

Let $X$ be a compact complex manifold and let $L$ be a holomorphic line
bundle over $X$. When $X$ is projective or Moishezon, $L$ is said to be
numerically effective (nef for short) if $L\cdot C=\int_Cc_1(L)\ge0$
for every curve $C$ in $X$. This definition can no longer be used for a
general compact manifold $X$ because $X$ need not possess any curve. An
alternative definition is suggested by the following lemma, which is an
easy consequence of Seshadri's ampleness criterion (namely, a line
bundle $A$ is ample if and only if there exists $\ep>0$ such that $A\cdot
C\ge\ep\cdot m(C)$ for every curve~$C$, where $m(C)$ is the
maximum multiplicity of singular points in $C$).

\than Lemma 1.1|Suppose that $X$ is projective algebraic and let $A$ be an
ample line bundle  on $X$. Then $L$ is nef if and only if $L^k\otimes A$ is
ample for every integer $k\ge0$.\fintha

Now it is well known that a line bundle $A$ is ample if and only if it 
carries a smooth hermitian metric $h$ such that the associated Chern 
curvature form
$$\Theta_h(A)={i\over2\pi}D^2_{A,h}$$
is positive definite everywhere. Suppose that $h$ is such a metric on
$A$. If $L$ is nef, lemma 1.1 shows that there is a smooth hermitian
metric $h_k$ on $L$ such that 
$$\Theta\left(L^k\otimes A\right)=k\Theta_{h_k}(L)+\Theta_h(A)>0,$$
hence
$$\Theta_{h_k}(L)\ge-{1\over k}\Theta_h(A)$$
can have an arbitrary small negative part. We take this property as the
general definition.

\than Definition 1.2|Let $X$ be an arbitrary compact complex manifold,
equipped with a hermitian metric $\omega$. A line bundle $L$ over $X$
is said to be nef if for every $\ep>0$ there exists a smooth
hermitian metric $h_\ep$ on $L$ such that
$$\Theta_{h_\ep}(L)\ge-\ep\omega.$$
\fintha

If $C$ is a curve in $X$, we infer from this 
$$L\cdot C=
\int_C\Theta_{h_\ep}(L)\ge-\ep\int_C\omega$$
for every $\ep>0$, so $L\cdot C\ge0$. However this property
does not imply nefness in general, as shown by:

\rem Example 1.3|Let
$X=\left(\cb^2\setminus\lbrace0\rbrace\right)/\Gamma$ be the primary
Hopf surface associated to the infinite cyclic group
$\Gamma=\lbrace\lambda^p;p\in\zb,$$\lambda>1\rbrace$. Then $X\simeq
S^3\times S^1$ has zero second Betti number, so every line bundle $L$
is numerically trivial (i.e. $c_1(L)=0$), in particular $L\cdot C=0$
for every curve $C$. However, if $\pi:X\ra\pb^1$ is the natural
projection, the pull-back $L=\pi^\star\cO(-1)$ cannot be nef, otherwise
$\cO(-1)$ itself would be nef by property 1.8 (ii) below.\finpr

One may wonder whether $\ep$ can be made equal to zero in
Definition 1.2. This is true if the hermitian metric $h$ is allowed to
be singular.

\than Proposition 1.4|If $L$ is nef, there exists a singular hermitian
metric $h$ on $L$ $($trivialized locally by weight functions $\varphi\in
L^1_{loc}$ with $h\simeq e^{-\varphi})$ such that 
$$\Theta_h(L)={i\over2\pi}\partial\quer{\partial}\varphi\ge0$$
in the sense of currents, i.e. such that the weight functions $\varphi$
are plurisubharmonic.\fintha

\proof Let $\omega$ be a hermitian metric on~$X$ and let $h_0$ be a
fixed smooth metric on~$L$. For every $\ep\in{}]0,1[$ there is a hermitian
metric $h_\varepsilon=h_0e^{-\psi_\ep}$ such that $\Theta_{h_\varepsilon}(L)
\ge -\ep\omega$. We can always multiply $h_\ep$ by a constant in such a way
that $\max_X\psi_\varepsilon=0$. Then
$${i\over2\pi}\partial\quer{\partial}\psi_\ep=
\Theta_{h_\ep}(L)-\Theta_{h_0}(L)\ge-\omega+\Theta_{h_0}(L).$$
In particular, there is a covering $(U_j)$ of $X$ by coordinate
patches and a constant $C>0$ such that $\psi_\varepsilon(z)+C|z^{(j)}|^2$
is plurisubharmonic in each~$U_j$. The well known properties of
subharmonic functions (namely, the mean value inequality and the
Green-Riesz formula) show that the family $(\psi_\ep)$ is bounded in $L^1$
norm on each patch~$U_j$, and therefore relatively compact in $L^1(X)$.
Hence there exists a subsequence $(\psi_{\ep_\nu})$ converging to a
limit $\psi$ in~$L^1(X)$. The possibly singular hermitian metric
$h=h_0e^{-\psi}$ has curvature
$\Theta_h(L)=\lim_{\nu\to+\infty}\Theta_{h_{\ep_\nu}}(L)$ in the weak
topology of currents. Therefore $\Theta_h(L)\ge 0$.\finpr

\than Corollary 1.5|A line bundle $L$ is nef as well as its dual
$L^\star=L^{-1}$ if and only if $L$ is hermitian flat, that is, $L$ is 
given by a representation $\pi_1(X)\ra S^1$ in the unit circle.\fintha

\proof If $L$ is hermitian flat, then of course both $L$ and $L^\star$
are nef. Conversely, suppose that $L$ and $L^\star$ are nef. By
Proposition~1.4, there are (possibly singular) hermitian metrics
$h$ on $L$ and $h'$ on $L^\star$ with semipositive curvature
in the sense of currents. Then $h\otimes h'$ is a metric of
semipositive curvature on the trivial line bundle $L\otimes L^\star$,
given by a weight $h\times h'=e^{-\psi}$ where $\psi$ is a global
plurisubharmonic function on~$X$. Since $X$ is compact, we infer that
$\psi$ is a constant, hence $\Theta_h(L)+\Theta_{h'}(L^\star)=0$
and therefore $\Theta_h(L)=\Theta_{h'}(L)=0$, each term being semipositive.
This implies that $h$, $h'$ are smooth and flat.\finpr

\rem Remark 1.6|If $L$ admits a singular metric $h$ with
$\Theta_h(L)\ge0$, it does not necessarily follow that $L$ is nef. In
fact, let $X$ be the blow-up of some manifold at one point and let
$L=\cO(E)$ be the line bundle associated to the exceptional divisor
$E$. Then $L$ is not nef because $L\cdot C<0$ for every curve $C\subset
E$. However the inclusion morphism $\cO\subset\cO(E)$ gives a singular
metric $h$ on $\cO(E)$ with poles along $E$, such that
$$\Theta_h(\cO(E))=[E]$$ is the current of integration on $E$. In
particular $\Theta_h(L)\ge0$.\finpr

\medskip It is also important to observe that a smooth hermitian metric
$h$ with $\Theta_h(L)\ge0$ need not exist when $L$ is nef, thus
$\ep$ cannot be taken equal to $0$ in Definition 1.2,
disproving a conjecture of Fujita ([Fu83], 4.14):

\rem Example 1.7|Let $\Gamma=\cb/(\zb+\zb\tau)$, $\Im\,\tau>0$, be
an elliptic curve and let $E$ be the rank 2 vector bundle over $\Gamma$
defined by $$E=\cb\times\cb^2/(\zb+\zb\tau)$$ where the action of
$\zb+\zb\tau$ is given by the two automorphisms
$$\eqalign{
  g_1(x,z_1,z_2) &= (x+1,z_1,z_2),\cr
  g_\tau(x,z_1,z_2) &= (x+\tau,z_1+z_2,z_2),\cr}$$
and where the projection $E\ra\Gamma$ is induced by the first
projection $(x,z_1,z_2)\mapsto x$. Then
$\cb\times\cb\times\lbrace0\rbrace/(\zb+\zb\tau)$ is clearly a trivial
line subbundle $\cO\hookrightarrow E$, and the quotient
$E/\cO\simeq\Gamma\times\lbrace0\rbrace\times\cb$ is also trivial. Let
$L$ be the line bundle $L=\cO_E(1)$ over the ruled surface $X=\pb(E)$.
The exact sequence $$0\ra\cO\ra E\ra\cO\ra0$$ shows by Prop. 1.14 (ii)
below that $L$ is nef over $X$. Now, we are going to determine all
hermitian metrics (including singular ones) such that the curvature
$\Theta(L)$ is semipositive in the sense of currents. As the total
space of $L^{-1}$ is equal to $E^\star$ blown up along the zero section,
the function
$$\varphi(\zeta)=\log\parallel\zeta\parallel^2_{h^{-1}},\quad\zeta\in L^{-1}$$
associated to any hermitian metric $h$ on $L$ can also be seen as a
function on $E^\star$ satisfying the log-homogeneity condition
$$\varphi(\lambda\zeta)=\log\mid\lambda\mid+\varphi(\zeta)$$
for every $\lambda\in\cb$. We have
$${i\over2\pi}\ppq\varphi(\zeta)=\pi^\star_{L^{-1}}\Theta_h(L),
\quad\pi_{L^{-1}}:L^{-1}\ra X,$$
thus $\Theta_h(L)\ge0$ if and only if $\varphi$ is plurisubharmonic on
$E^\star$. The total space of $E^\star$ is the quotient 
$$E^\star=\cb\times\cb^2/(\zb+\zb\tau)$$
by the dual action 
$$\eqalign{
  g^\star_1(x,w_1,w_2) &= (x+1,w_1,w_2),\cr
  g^\star_\tau(x,w_1,w_2) &= (x+\tau,w_1,w_1+w_2). \cr}$$
The function $\varphi$ gives rise to a plurisubharmonic function $\wp$
on $\cb\times\cb^2$ which is invariant by $g_1^\star$ and $g_\tau^\star$
and log-homogeneous with respect to $(w_1,w_2)$. We claim that the only
such functions are 
$$\wp(x,w_1,w_2)=\log\bet{w_1}+a,\quad a\in\rb.\leqno(+)$$
Hence the associated metric $h^{-1}$ on $L^{-1}$ must have zeros along
the curve $C=\lbrace w_1=0\rbrace\simeq\lbrace z_2=0\rbrace$ in
$\pb(E^\star)\simeq\pb(E)$. By the Lelong-Poincar\'e equation, we then find
$$\Theta_h(L)=[C]$$ where $[C]$ denotes the current of integration
over~$C$.

To check $(+)$, we first observe that any periodic
subharmonic function $u$ on $\cb$ with logarithmic growth at infinity
must be constant: suppose $u(z)\le A\log_+\bet{z}+B$ and $u(z+1)=u(z)$,
then $v(z)=\sup_{t\in[0,1]}u(z+t)$ satisfies a similar upper bound and
depends only on $\Im\,z$; since $v$ is a convex function of $\Im\,z$ it
must be a constant, so $u$ is bounded above and constant, too. As 
$$\wp(x+p+q\tau,w_1,w_2)=\wp(x,w_1,w_2+qw_1)$$
for all $p,q\in\zb$, the log-homogeneity of $\wp$ in $(w_1,w_2)$
implies that $\wp$ has logarithmic growth also in $x$. But
$x\mapsto\wp(x,w_1,w_2)$ is 1-periodic, so $\wp$ must be
independent of $x$. Now, the invariance by $g_\tau^\star$ shows that
$w_2\mapsto\wp(w_1,w_2)$ is periodic for $w_1\not=0$. Therefore $\wp$
is also independent of $w_2$. The log-homogeneity implies 
$$\wp(w_1)=\log\bet{w_1}+\wp(1),$$
as desired.

We have thus shown that $\cO_E(1)$ does not admit any smooth hermitian
metric with semipositive curvature. In particular, $E$ is not
semipositive in the sense of Griffiths (see 1.10 (i) for the
definition). This weaker statement could have been proved directly as
follows: if $\Theta_h(E)\ge0$, then
$\Tr\,\Theta_h(E)=\Theta_h(\det\,E)\ge0$ and must be zero, for
$\det\,E\simeq\cO$; therefore the full curvature tensor $\Theta_h(E)$
would itself vanish and so $E$ would be hermitian flat; then the exact
sequence $0\ra\cO\ra E\ra\cO\ra0$ would split holomorphically (by the
orthogonal splitting), which is easily seen not to be the case.\finpr

\than Proposition 1.8|Let $X$, $Y$ be compact complex manifolds, let 
$L$ be a line bundle on~$X$ and let $f:Y\ra X$ be an analytic map.
\item{\rm(i)} If $L$ is nef, then $f^\star L$ is nef.
\item{\rm(ii)} Suppose that $f$ is surjective and has equidimensional
fibres. If $f^\star L$ is nef, then $L$ is nef.\smallskip
\fintha

These results are of course well known in the algebraic case (see
e.g.~[Fu83]).
\vskip 0.2 truecm

\proof Let $\omega$, $\omega'$ be hermitian metrics on $X$, $Y$
respectively. There exists a constant $a>0$ such that
$$f^\star\omega\le a\omega'.$$
\noindent{\bf(i)} If $h_\ep$ is as in Definition 1.2, then
$$\Theta_{f^\star h_\ep}(f^\star L)=f^\star\Theta_{h_\ep}(L)
\ge-\ep f^\star\omega\ge-\ep a\omega'.$$
This shows that $f^\star L$ is nef.

\noindent{\bf(ii)}  Fix an arbitrary hermitian metric $h$ on $L$ and set
$\alpha=\Theta_h(L)$. If $f^\star L$ is nef, there is for every
$\ep>0$ a metric $h'_\ep=e^{-\psi_\ep}f^\star h$ on
$f^\star L$, where $\psi_\ep$ is some smooth real function on~$Y$,
such that
$$\Theta_{h'_\ep}(f^\star L)=f^\star\alpha+{i\over2\pi}\ppq
\psi_\ep\ge-\ep\omega'.$$
The idea, in very rough terms, is to define a metric $h_\ep$ on $L$ by
$$|\xi|_{h_\ep(x)}=\inf_{y\in f^{-1}(x)}|\xi|_{h'_\ep(y)},~~~~
\xi\in L_x=(f^\star L)_y,~~~y\in f^{-1}(x),$$
that is, $h_\ep=e^{-\varphi_\ep}h$ with $\varphi_\ep(x)=\sup_{y\in f^{-1}(x)}
\psi_\ep(y)$. However, this simple-minded idea would not give the desired
lower bound for $\Theta_{h_\ep}(L)$ in a neighborhood of the critical values
of $f$ in~$X$. Therefore, a slightly modified construction is needed.

Let $p$ be the dimension of the fibres of $f$. For any point $y\in Y$,
there are local coordinates $(w)=(w_1,\ldots,w_m)$ centered at $y$ such that
the map $z\mapsto(f(z),w_1(z),\ldots,w_p(z))$ is proper and finite from a
neighborhood of $y$ onto a neighborhood of $f(y)\times\lbrace0\rbrace$
in~$X\times\cb^p$. Thus we can find local coordinates centered at $f(y)$ on
$X$ and a neighborhood $U$ of $y$ such that 
$$\bet{F(z)-F(y)}^2+\sum\limits_{1\le j\le p}\bet{w_j(z)}^2>0$$
on $\partial U$, where $F=(F_1,\ldots,F_n)$ denote the local coordinate
components of $f$. As $Y$ is compact, we can find finitely many points
$y_k\in Y$ and local coordinates $(w^{(k)})=(w^{(k)}_1,\ldots,
w^{(k)}_m)$ on neighborhoods $\quer{U}_k$ of~$y_k$ (with $w^{(k)}(y_k)=0$),
resp. local coordinates on $X$ centered at~$f(y_k)$, such that the
corresponding components $F^{(k)}=(F^{(k)}_1,\ldots,F^{(k)}_n)$ of $f$ satisfy
$$2\delta_k:=\bet{F^{(k)}(z)-F^{(k)}(y_k)}^2+\sum\limits_{1\le j\le p}
\bet{w^{(k)}_j(z)}^2>0~~~~\hbox{\rm on}~~\partial U_k,$$
and such that the collection of open sets $(V_k)$ defined by
$$V_k:=\left\lbrace z\in U_k\,;\,\bet{F^{(k)}(z)-F^{(k)}(y_k)}^2+
\sum\limits_{1\le j\le p}\bet{w^{(k)}_j(z)}^2<\delta_k\right\rbrace$$
is a covering of $Y$. For $z\in\overline{U}_k$ and $x\in X$ we define
$$\eqalign{
\lambda^{(k)}_\ep(z)&:=\ep^3\bet{w^{(k)}(z)}^2-\ep^2\left(
\bet{F^{(k)}(z)-F^{(k)}(y_k)}^2+\sum\limits_{1\le j\le p}
\bet{w^{(k)}_j(z)}^2-\delta_k\right),\cr
\varphi_\ep(x)&:=\sup_{y\in f^{-1}(x),\quer{U}_k\ni y}
\psi_{\ep^4}(y)+\lambda^{(k)}_\ep(y),\cr}$$
where $\ep$ is chosen${}\le\ep_0$ so small that
$\lambda^{(k)}_\ep<0$ on $\partial U_k$. Since
$\lambda^{(k)}_\ep\ge0$ on $V_k$, it follows that the supremum in
$\varphi_\ep(x)$ is never attained for an index $j$ such
that $\partial U_j\ni y$: indeed we then have
$\lambda^{(j)}_\ep(y)<0$, while $\lambda^{(k)}_\ep(y)\ge0$
for some $k$ such that $V_k\ni y$. Therefore $\varphi_\ep$ is
continous. Moreover, for $\ep\le\ep_1$ small
$$\eqalign{
{i\over2\pi}\ppq\left(\ep^3\bet{w^{(k)}}^2\right)
&\ge\ep^4\omega', \cr
{i\over2\pi}\ppq\left(\ep^2\bet{F^{(k)}-F^{(k)}(y_k)}^2\right)
&\le{\ep\over2}f^\star\omega\cr}$$
on each $U_k$. Hence by definition of $\psi_\ep$ we get
$$f^\star\alpha+{i\over2\pi}\ppq\left(\psi_{\ep^4}+
\lambda^{(k)}_\ep\right)\ge-{\ep\over2}
f^\star\omega-\ep^2{i\over2\pi}\ppq\sum\limits_{1\le j\le p}
\bet{w^{(k)}_j}^2.$$
We will check below that this implies
$$\alpha+{i\over2\pi}\ppq\varphi_\ep\ge-{\ep\over2}\omega.
\leqno(\star)$$
By Richberg's regularization theorem~[Ri68], we can find a smooth
function $\wp_\ep$ such that
$\alpha+{i\over2\pi}\ppq\wp_\ep\ge-\ep\omega$. Then the
metric $h_\ep=e^{-\wp_\ep}h$ on $L$ satisfies
$$\Theta_{h_\ep}(L)=\alpha+{i\over2\pi}\ppq
\wp_\ep\ge-\ep\omega,\quad\hbox{\rm QED.}$$
To check $(\star)$, fix a point $x_0\in X$ and take $y_0\in Y$ and
$U_k\ni y_0$ such that $\varphi_\ep(x_0)=
\psi_{\ep^4}(y_0)+\lambda^{(k)}_\ep(y_0)$.
Take an arbitrary quadratic function $q(x)$ near $x_0$ such that
${i\over2\pi}\ppq q>\alpha+{\ep\over2}\omega$ at $x_0$. Then
$${i\over2\pi}\ppq\Psi\ge0\quad\hbox{where}\quad\Psi=
\psi_{\ep^4}+\lambda^{(k)}_\ep+q\circ f
+\ep^2\sum\limits_{1\le j\le p}\bet{w^{(k)}_j}^2$$
i.e. the function $\Psi$ is plurisubharmonic near $y_0$. However, $f$
defines a finite map from the submanifold 
$$S=\left\lbrace z\in U_k\,;\,
w_{1,k}(z)=w_{1,k}(y_0),\ldots,w_{p,k}(z)=w_{p,k}(y_0)\right\rbrace$$
onto a neighborhood of $x_0$ (by construction of $(w^{(k)})$). Hence
the function
$$\Phi(x):=\sup_{y\in f^{-1}(x)\cap S}\Psi(y)$$
is well defined near $x_0$ and satisfies
$$\Phi(x)\le\varphi_\ep(x)+q(x)+\ep^2\sum
\limits_{1\le j\le p}\bet{w^{(k)}_j(y_0)}^2$$
with equality at $x=x_0$. Since $\Phi$ is plurisubharmonic, we
conclude that $\varphi_\ep+q$ satisfies the mean value
inequality at $x_0$. This is also true at all nearby points where
${i\over2\pi}\ppq q>\alpha+{\ep\over2}\omega$, so
$\varphi_\ep+q$ is plurisubharmonic near $x_0$. Taking
${i\over2\pi}\ppq q$ as close as possible to
$\alpha+{\ep\over2}\omega$ at $x_0$, we conclude that $(\star)$
holds.\finpr 

\medskip Of course we expect that property 1.8 (ii) holds for arbitrary
surjective maps $f:Y\ra X$, but there are serious technical
difficulties to overcome.

\vskip15pt
\titre 1.B. Nef vector bundles|

Let $E$ be a holomorphic vector bundle of rank $r$ over a compact
complex manifold. We denote by $\pb(E)$ the projectivized bundle of
hyperplanes of $E$ and by $\cO_E(1)$ the associated canonical line
bundle.

\than Definition 1.9|We say that $E$ is nef over $X$ if $\cO_E(1)$ is
nef over $\pb(E)$.\fintha

Proposition 1.8 can then be immediately extended to the vector bundle
case by taking $L=\cOE, X=\pb(E),Y=\pb(f^\star E)$:

\than Proposition 1.10|Let $X$, $Y$ be compact complex manifolds, let
$f:Y\ra X$ be an analytic map and let $E$ be a vector bundle over $X$.
Then $E$ nef implies $f^\star E$ nef, and the converse is true if $f$
is surjective and has equidimensional fibres. \fintha

It will be useful to have a differential geometric criterion for
nefness directly over $X$. First recall a few basic properties of
hermitian differential geometry (see e.g. Griffiths~[Gr69]).

\than Proposition 1.11|If $(E,h)$ is a hermitian vector bundle, the
Chern curvature tensor 
$$\Theta_h(E)={i\over2\pi}D_{E,h}^2=i\sum\limits_{\buildrel
{\scriptstyle{1\le j,k\le n}}\over {\scriptstyle{1\le\lambda,\mu\le r}}}
a_{jk\lambda\mu}dz_j\wedge d\quer{z}_k\otimes e^\star_\lambda\otimes e_\mu$$
is a hermitian (1,1)-form with values in $\Hom(E,E)$; here
$(z_1,\ldots,z_n)$ are local coordinates on $X$ and
$(e_\lambda)_{1\le\lambda\le r}$ is a local orthonormal frame of $E$. 
\item{\rm(i)} We say that $(E,h)$ is semipositive in Griffiths' sense
and write $\Theta_h(E)\ge0$ if $\Theta_h(E)(\xi\otimes t)=\sum
a_{jk\lambda\mu}\xi_j\quer{\xi}_k v_\lambda\quer{v}_\mu\ge0$ for every
$\xi\in T_x X$, $v\in E_x$, $x\in X$. We write $\Theta_h(E)>0$ in case
there is strict inequality for $\xi\not=0, v\not=0$. 
\item{\rm(ii)} If $(F,h')$ is another hermitian vector bundle, then
$$\Theta_{h\otimes h'}(E\otimes F)=\Theta_h(E)\otimes
\Id_F+\Id_E\otimes\Theta_{h'}(F).$$ 
\item{\rm(iii)} If $0\ra F\ra E\ra Q\ra0$ is an exact sequence and if
the metrics of $F$, $Q$ are induced by the metric of $E$, the orthogonal
splitting $E\simeq F\oplus Q$ yields
$$D_E=\pmatrix{D_F   & -\beta^\star\cr
               \beta & D_Q\cr},\quad
\Theta(E)=\pmatrix{\Theta(F)-\beta^\star\wedge\beta & -D^{1,0}\beta^\star\cr
                   D^{0,1}\beta       & \Theta(Q)-\beta\wedge\beta^\star\cr}$$
where $\beta$ is the (1,0) second fundamental form with values in $\Hom(F,Q)$. 
\item{\rm(iv)} As follows easily from (iii), we have
$\Theta(Q)=\Theta(E)_{\mid Q}+\beta\wedge\beta^\star\ge\Theta(E)_{\mid Q}$
in Griffiths' sense, where $\Theta(E)_{\mid Q}$ is a short-hand
notation for the block of $\Theta(E)$ that lies in $\Hom(Q,Q)$.
\smallskip\fintha

\than Theorem 1.12|Let $\om$ be a fixed hermitian metric on $X$. A
vector bundle $E$ on $X$ is nef if and only if the symmetric powers $S^mE$ 
satisfy the following property:

\noindent There is a sequence of hermitian metrics $h_m$ on $S^mE$ such
that for every $\ep>0$ and $m\ge m_0(\ep)$
$$\Thhm(S^mE)\ge-m\ep\om\otimes\Id_{S^mE}$$ in the sense of Griffiths.
\fintha

\proof If $\pi:\pb(E)\ra X$ is the projection, there is a surjective
morphism $\pi^\star S^mE\ra\cO_E(m)$. Suppose first that $h_m$ exists
and satisfies the above inequality. Let $\wdt{h}_m$ be the quotient
metric on $\cO_E(m)$. By 1.11 (iv) we get
$$\Theta_{\wdt{h}_m}\left(\cO_E(m)\right)\ge
\pi^\star\Thhm\left(S^mE\right)_{\mid\cO_E(m)}
\ge-m\ep\pi^\star\om\otimes\Id_{\cO_E(m)},$$
i.e.
$$\Theta_{\wdt{h}_m^{1/m}}\left(\cO_E(1)\right)\ge-\ep\pi^\star\om.$$
This implies that $\cO_E(1)$ is nef. Conversely, suppose  that
$\cO_E(1)$ is nef and let $\om'$ be a hermitian metric on $\pb(E)$.
Then for every $\ep>0$ there is a hermitian metric $\wdt{h}_\ep$ on
$\cO_E(1)$ such that 
$$\Theta_{\wdt{h}_\ep}\left(\cO_E(1)\right)\ge-\ep\om'.$$
Let $H$ be a fixed hermitian metric on $E$ and let $\wdt{H}$ be the
induced metric on $\cO_E(1)$. Then $\Theta_{\wdt{H}}(\cO_E(1))$ is
positive along the fibres of $\pi$, so there exists a constant $C>0$
such that
$$\Theta_{\wdt{H}}\left(\cO_E(1)\right)+C\pi^\star\om\ge C^{-1}\om'$$
on $\pb(E)$. We consider the barycenter metric
$\hhe=\hse^{1-C\ep}\wdt{H}^{C\ep}$ on  $\cOE$. Then
$$\eqalign{
\Theta_\hhe\lp\cOE\rp
&=\lp1-C\ep\rp\Theta_\hse\lp\cOE\rp+C\ep\Theta_\wtH\lp\cOE\rp\cr
&\ge-\lp1-C\ep\rp\ep\om'+C\ep\lp C^{-1}\om'-C\pi^\star\om\rp\cr
&\ge C\ep^2\om'-C^2\ep\pi^\star\om\cr
&>-C^2\ep\pi^\star\om.\cr}$$
By theorem 4.1 of~[De91] applied with $v=-C^2\ep\om$, we can therefore
find a metric $h_{m,\ep}$ on $S^mE$ such that for $m\ge\mu_0(\ep)$ large
$$\Theta_{h_{m,\ep}}\lp S^mE\rp\ge-m\lp C^2+1\rp\ep\om\otimes\Id_{S^mE}.$$
This implies theorem 1.12 by taking $h_m=h_{m,\ep_m}$ with a sequence
$\ep_m$ converging to 0 so slowly that $\mu_0(\ep_m)\le m$.\finpr

\than Corollary 1.13|Let $\Gamma^aE$ be the irreducible tensor
representation of $Gl(E)$ of highest weight $a=(a_1,\ldots,a_r)\in\zb^r$,
with $a_1\ge\ldots\ge a_r\ge0$. Then there are hermitian metrics $h_a$ on
$\Gamma^aE$ such that for every $\ep>0$
$$\Theta_{h_a}\lp\Gamma^aE\rp\ge-\bet{a}\ep\om\otimes\Id_{\Gamma^aE}$$
for $\bet{a}:=\sum a_j\ge A(\ep)$.\fintha

\proof (Compare with [Ha66]). The representation $\GaE$ is a direct
summand in $S^{a_1}E\otimes\ldots\otimes S^{a_r}E$. Let $h_m$ be a
metric on $S^mE$ such that
$$\Thhm\lp S^mE\rp\ge-m\ep_m\om\otimes\Id_{S^mE}$$
with $\lim\ep_m=0$ and let $\wdt{h}_a$ be the quotient metric of
$h_{a_1}\otimes\ldots\otimes h_{a_r}$ on $\GaE$. Then by 1.11 (ii) and
(iv) we get
$$\Thha\lp\GaE\rp\ge-\sum a_j\ep_{a_j}\om\otimes\Id_\GaE$$
and it is clear that the ratio $\sum a_j\ep_{a_j}/\bet{a}$ tends to 0
when $\bet{a}=\sum a_j$ tends to $+\infty$.\finpr

\vskip 0.2 truecm We can now prove the basic properties of nef vector
bundles. In the case of ample vector bundles over algebraic varieties,
these properties are well known by~[Ha66]; the nef case then follows
essentially by passing to the limit if the base $X$ is algebraic
(see~[CP91] for details).

\than Proposition 1.14|Let $E$, $F$ be holomorphic vector bundles.
\item{\rm(i)} If $E$ and $F$ are nef, then $E\otimes F$ is nef.
\item{\rm(ii)} If $E$ is nef, any tensor power $\GaE$ with
$a_1\ge\ldots\ge a_r\ge0$ is nef.
\item{\rm(iii)} If some symmetric power $\SmE$ is nef ($m>0$), then
$E$ is nef.\fintha

\proof\hfill\break
\noindent{\bf(i)} There is a direct sum decomposition               
$$S^m(E\otimes F)=\bigoplus_{\buildrel {\scriptstyle{\mid a\mid=m}}
\over{\scriptstyle{\mid{b}\mid=m}}}n_{a,b}\GaE\otimes\Gamma^bF$$
into irreducible $Gl(E)\times Gl(F)$ modules of degree $(m,m)$ (the
exact formula is not needed, except that the weights $a,b$ are non
negative). We take the metric $\wdt{h}_{a,E}\otimes\wdt{h}_{b,F}$ on
each summand. Corollary 1.13 and property 1.11 (ii) imply
$$\Theta\lp S^m(E\otimes F)\rp\ge-2m\ep\om\otimes\Id_{S^m(E\otimes F)}$$
for $m\ge A(\ep)$ (note that each summand contributes in a different
block of the curvature matrix and that $\bet{a},\bet{b}\ge A(\ep)$).

\noindent{\bf(ii)} The nefness of $\GaE$ is obtained in the same way, by
observing that $S^m(\GaE)$ is a direct sum of tensor powers $\Gamma^bE$
with $b\ge0$ and $\bet{b}=m\bet{a}$.

\noindent{\bf(iii)} The pull-back $\pi^\star\SmE$ of $\SmE$ to $\pb(E)$ is
nef by Prop. 1.10. Its quotient $\cO_E(m)$ is therefore nef by Prop.
1.15 (i) below. Hence $\cOE=\cO_E(m)^{1/m}$ is nef.\finpr 

\than Proposition 1.15|Let $0\ra F\ra E\ra Q\ra 0$ be an exact sequence
of holomorphic vector bundles. Then 
\item{\rm(i)} $E$ is nef ${}\Rightarrow{}$ $Q$ is nef.
\item{\rm(ii)} $F$, $Q$ nef ${}\Rightarrow{}$ $E$ is nef.
\item{\rm(ii)} $E$ nef, $(\det Q)^{-1}$ nef ${}\Rightarrow{}$ $F$ nef.
\fintha

\proof\hfill\break
\noindent{\bf(i)} Apply theorem 1.12 and take the quotient metric of $\SmE$
on $S^mQ$. Then
$$\Theta\lp S^mQ\rp\ge\Theta\lp\SmE\rp_{\mid S^mQ}
\ge-m\ep\om\otimes\Id_{S^mQ}$$
for $m\ge m_0(\ep)$.

\noindent{\bf(ii)} Fix once for all a $C^\infty$ splitting $E=F\oplus Q$.
Then we have a $C^\infty$ splitting
$$\Phi:\SmE\buildo\simeq\over\Ra\bigoplus_{a+b=m}S^aF\otimes S^bQ.$$
By assumption, there are metrics $h'_a$ on $S^aF$ and $h''_b$ on $S^bQ$
such that
$$\Theta_{h'_a}\lp S^aF\rp\ge-a\ep_a\om\otimes\Id_{S^aF},\quad
  \Theta_{h''_b}\lp S^bQ\rp\ge-b\eta_b\om\otimes\Id_{S^bQ}$$
with $\lim\limits_{a\ra+\infty}\ep_a=\lim\limits_{b\ra+\infty}\eta_b=0$.
We define hermitian metrics $h_{m,\rho}$ on $\SmE$ by
$$h_{m,\rho}=\Phi^\star\bigoplus_{a+b=m}\rho^{-2b}h'_a\otimes h''_b\;,$$
where $\rho>0$ is a small rescaling factor. Changing slightly our
viewpoint, we replace the initial splitting $\Phi$ by the splitting
$\Phi_\rho$ obtained by multiplying the $S^aF\otimes S^bQ$ component
by~$\rho^{-b}$. Then
$$h_{m,\rho}=\Phi_\rho^\star\bigoplus_{a+b=m}h'_a\otimes h''_b\;.$$
With respect to $\Phi_\rho$, the Chern connection $D_{h_{m,\rho}}$ on
$\SmE$ takes the form
$$D_{h_{m,\rho}}=\pmatrix{
D_{h'_m\otimes h''_0} & \cdots & -\rho^{k-j}\beta^\star_{k,j}\cr
             \vdots & D_{h'_{m-j}\otimes h''_j} & \vdots\cr
\rho^{j-k}\beta_{j,k}&\vphantom{\Big(}\cdots & D_{h'_0\otimes h''_m}\cr}$$ 
where $\beta_{j,k}$ is a (1,0)-form with values in
$$\Hom(S^{m-k}F\otimes S^kQ,S^{m-j}F\otimes S^jQ),\quad j>k$$
and is independent of~$\rho$. To check this, recall the formula
$\beta_{j,k}^\star=-\pi_k\circ\quer{\partial}\pi_j^\star$ where $\pi_j$
is the orthogonal projection of $\SmE$ onto $S^{m-j}F\otimes S^jQ$ given
by~$\Phi_1\,$, and \hbox{$\pi_j^\star:S^{m-j}F\otimes S^jQ\ra\SmE$} is the
adjoint (and right inverse) of~$\pi_j$. By definition of~$\Phi_\rho$,
the projection $\pi_k$ is replaced by $\rho^{-k}\pi_k$, thus $\pi_j^\star$
is replaced by $\rho^j\pi_j^\star$ and $\beta_{j,k}^\star$
by~$\rho^{j-k}\beta_{j,k}^\star$, as desired. By computing the matrix
of $D^2_{h_{m,\rho}}$, we~get
$$\Theta_{h_{m,\rho}}\lp\SmE\rp=\bigoplus_{0\le j\le m}
\pi_j^\star\circ\Theta_{h'_{m-j}\otimes h''_j}\circ\pi_j+O(\rho),$$
therefore  
$$\Theta_{h_{m,\rho}}\lp\SmE\rp\ge-\lp\max_{0\le j\le m}
\lp(m-j)\ep_{m-j}+j\eta_j\rp+O(\rho)\rp\om\otimes\Id_\SmE$$
where $O(\rho)$ comes from the contribution of the second fundamental
form $\rho^{j-k}\beta_{j,k}$, $j>k$. For $\rho=\rho_m$ small this
contribution is negligible. On the other hand, it is easy to see that
$$\lim_{m\ra+\infty}{1\over m}\max_{0\le j\le m}
\lp(m-j)\ep_{m-j}+j\eta_j\rp=0,$$
thus $E$ is nef.

\noindent{\bf(iii)} We have $F=\Lambda^{s-1}F^\star\otimes\det F$ where
$s$ is the rank of $F$. By dualizing and taking the $s-1$ exterior
product in the exact sequence, we get a surjection
$$\Lambda^{s-1}E^\star\ra\Lambda^{s-1}F^\star=F\otimes(\det F)^{-1}.$$
We multiply by $\det F=\det E\otimes(\det Q)^{-1}$ to get a surjection
$$\Lambda^{r-(s-1)}E\otimes(\det Q)^{-1}\ra F$$
where $r=\rank(E)$. Now the left hand side is nef by Prop. 1.14 and the
assumption that $(\det Q)^{-1}$ is nef, so its quotient $F$ is nef.\finpr
 
\vskip0.2truecm The next proposition concerns the behaviour of
holomorphic sections in the dual of a nef vector bundle.

\than Proposition 1.16|Let $E$ be a nef vector bundle over a connected
compact $n$-fold $X$ and let $\sigma\in H^0(X,E^\star)$ be a non zero
section. Then $\sigma$ does not vanish anywhere.\fintha

\proof By Gauduchon~[Ga77], we know that any conformal class of
hermitian metrics on $X$ contains a special metric with
$\partial\quer{\partial}\left(\omega^{n-1}\right)=0$.
Let $h_m$ be hermitian metrics as in theorem 1.12 and let
$h^\star_m$ be the dual metric on $\SmE^\star$. Then the closed
(1,1)-current
$$T_m={i\over2\pi}\ppq{1\over m}\log\parallel\sigma^m\parallel^2_{h^\star_m}$$
satisfies 
$$T_m\ge-{1\over m}{\langle\Theta_{h^\star_m}\lp
\SmE^\star\rp\sigma^m,\sigma^m\rangle_{h^\star_m}
\over\parallel\sigma^m\parallel^2_{h^\star_m}}.$$
For $m\ge m_0(\ep)$ we have
$$-\Theta_{h^\star_m}\lp\SmE^\star\rp=
{^t}\Theta_{h_m}\lp\SmE\rp\ge-m\ep\om\otimes\Id_{\SmE^\star}.$$
Hence $T_m\ge-\ep\om$ for $m\ge m_0(\ep)$. Now, if $\om$ is a Gauduchon
metric
$$\int_X\lp T_m+\ep\om\rp\wedge\om^{n-1}=\ep\int_X\om^n$$
because $T_m$ is $\ppq$-exact. This implies that $T_m+\ep\om$ and $T_m$
converge weakly to zero as $m\ra+\infty$. If $\sigma(x)=0$ at some
point $x\in X$, the Lelong number $\nu(T_m,x)$ would be at least 1 (and
equal to the vanishing order of $\sigma$ at $x$). Thus by~[S74], the
Lelong number of any weak limit $T=\lim T_m$ would satisfy
$\nu(T,x)\ge1$, a contradiction.\finpr

\vskip 0.2 truecm We end this section with the following important
analogue of Corollary 1.5 in the vector bundle case.

\than Definition 1.17|We say that a holomorphic vector bundle $E$ is
numerically flat if both $E$ and $E^\star$ are nef (equivalently if $E$ and
$(\det E)^{-1}$ are nef). \fintha

\than Theorem 1.18|Suppose that $X$ is K\"ahler. Then a holomorphic
vector bundle $E$ over $X$ is numerically flat if and only if $E$ admits a 
filtration
$$\lbrace0\rbrace=E_0\subset E_1\subset\ldots\subset E_p=E$$
by vector subbundles such that the quotients $E_k/E_{k-1}$ are
hermitian flat, i.e. given by unitary representations $\pi_1(X)\ra
U(r_k)$.\fintha

It should be observed that $E$ itself need not be hermitian flat as
example 1.7 shows.

\than Corollary 1.19|If $X$ is K\"ahler and $E$ is numerically flat, then
all Chern classes $c_k(E)$ vanish.\fintha

\proof Note that the ''if'' part of the statement is clear by Prop.
1.15 (ii). The proof of the ''only if'' part is based on two deep
facts: first the Uhlenbeck-Yau theorem asserting the existence of
Hermite-Einstein metrics on stable vector bundles, secondly Kobayashi's
flatness criterion derived from L\"ubke's inequality on Chern classes.

Since $\det E$ is numerically flat, Cor.~1.5 implies that
$c_1(E)=c_1(\det E)=0$. Fix~a K\"ahler metric $\om$ on $X$.

{\bf Step 1.} Let $\cF$ be a subsheaf of $\cO(E)$ of rank $p$ and let
$\det\cF=\lp\Lambda^p\cF\rp^{\star\star}$ be its determinant: $\det\cF$
is a reflexive rank 1 sheaf, so it is locally free and we have an
injection of sheaves $$\det\cF\hookrightarrow\Lambda^pE.$$ This morphism
can be seen as a (non necessarily injective!) morphism of bundles. We
know by 1.14 (ii) that $\Lambda^pE^\star$ is nef, so there is a sequence
$h_m$ of hermitian metrics on $S^m(\Lambda^pE^\star)$ such that 
$$\Thhm\lp S^m\lp\Lambda^pE^\star\rp\rp\ge-m\ep\om\otimes\Id$$
for $m\ge m_0(\ep)$. We equip $\det\cF$ with the possibly singular hermitian
metric induced as a quotient of $h_m$ by the non necessarily surjective
morphism $$S^m\lp\Lambda^pE^*\rp\ra\lp\det\cF\rp^{-m}.$$ Then 1.1 (iv)
implies in the sense of currents
$$\Theta_{\wdt{h}_m}\lp(\det\cF)^{-1}\rp\ge{1\over m}\Theta_m\lp
S^m(\Lambda^pE^\star)\rp_{\mid(\det\cF)^{-m}}\ge-\ep\om$$ 
for $m\ge m_0(\ep)$ (one cannot conclude from this that
$(\det\cF)^{-1}$ is nef because $\wdt{h}_m$ is singular). Therefore
$c_1(\cF)=c_1(\det\cF)$ satisfies the cohomology inequality
$$c_1(\cF)\cdot\lbrace\om\rbrace^{n-1}=
\int_X-\Theta_{\wdt{h}_m}\lp(\det\cF)^{-1}\rp\wedge\om^{n-1}\le0.$$
If $c_1(\cF)\cdot\lbrace\om\rbrace^{n-1}=0$, the same arguments as in
the proof of Proposition 1.16 show that
$\lim\Theta_{\wdt{h}_m}((\det\cF)^{-1})=0$ in the weak topology, in
particular $c_1(\cF)=0$  and $\det\cF$ is hermitian flat. In that case,
Prop. 1.16 implies that $\det\cF\ra\Lambda^pE$ has no zeros: consider
the section given by $\cO\ra\Lambda^pE\otimes(\det\cF)^{-1}$ and observe
that the dual of $\Lambda^pE\otimes(\det\cF)^{-1}$ is nef.

{\bf Step 2.} Now take a subsheaf $\cF\subset\cO(E)$ of minimal rank
$p>0$ with $c_1(\cF)\cdot\lbrace\om\rbrace^{n-1}=0$. After replacing
$\cF$ by $\cF^{\star\star}$ we can assume that $\cF$ is reflexive. The
discussion made in Step 1 shows that $\df$ is a line subbundle of
$\Lambda^pE$. The elementary lemma 1.20 below implies that $\cF$ itself
is locally free and is a subbundle of $E$. By definition, $\cF$ cannot
have any subsheaf of degree$\ge0$ and of positive rank${}<p$. Therefore
$\cF$ is $\om$-stable and by Uhlenbeck-Yau~[UY86] $\cF$ possesses a
Hermite-Einstein metric. Since $\cF^\star$ is a quotient of $E^\star$,
1.15 (i) shows that $\cF^\star$ is nef and therefore $\cF$ is numerically
flat because $\df$ is flat. The inequalities for Chern classes which will
be proved in Cor. 2.6 (independently of Cor. 1.19$\,$!) imply
$c_2(\cF)\cdot\lbrace\om\rbrace^{n-2}=0$, because $c_1(\cF)=0$. The
Kobayashi-L\"ubke flatness criterion ([Ko87], Th. 4.11) finally shows
that $\cF$ is hermitian flat. We set $E_1=\cF$ and conclude by
induction on the rank, since $E/E_1$ is again numerically flat by
Prop.~1.15.\finpr

\than Lemma 1.20|Let $E$ be a vector bundle over a complex manifold $M$
and let $\cF$ be a reflexive subsheaf of $\cO(E)$ such that the induced
bundle morphism $\df\ra\Lambda^pE$ is injective,where $p=\rank\cF$.
Then $\cF$ is locally free and is a subbundle of $E$.\fintha

\proof We are indebted to F.-O. Schreyer for the simple and elegant
argument given below. 

Since the question is purely local, we can assume that $M$ is a ball in
$\cb^n$ and that $E=\cO^r$ is a trivial vector bundle. As $\cF$ is
reflexive, $\cF$ is locally free on the complement $M\setminus Z$ of
some analytic subset $Z$ of codimension$\ge3$. As $\df\ra\Lambda^pE$ is
injective, $\cF_{\mid M\setminus Z}$ is actually a subbundle of
$E_{\mid M\setminus Z}$, so we get a holomorphic map into the
Grassmannian $G(p,r)$ of $p$-planes of $\cb^n$:
$$M\setminus Z\ra G(p,r),\quad x\ra\cF_x.$$
Now the composition with the Pl\"ucker embedding 
$$G(p,r)\ra P(\Lambda^p\cb^r),\quad S\mapsto\Lambda^pS$$
is precisely the map $M\setminus Z\ra P(\Lambda^pE)$, $x\mapsto\df_x$.
This map extends across $Z$ by the assumption that $\df\ra\Lambda^pE$
is everywhere injective. Hence the map $x\mapsto\cF_x$ extends as a
holomorphic map $\Phi:M\ra G(p,r)$. By construction $\cF_{\mid
M\setminus Z}$ is contained in the pull-back $\Phi^\star S$ of the
tautological rank $p$ bundle on $G(p,r)$. Therefore
$\cF\subset\Phi^\star S$ with equality on $M\setminus Z$, and
$\Phi^\star S$ is a vector subbundle of $E$. The sheaves
$\Hom(\cF,\Phi^\star S)$ and $\Hom(\Phi^\star S,\cF)$ are themselves
reflexive, so the identity homomorphism extends across $Z$ into an
isomorphism. Hence $\cF=\Phi^\star S$ is a subbundle of $E$.\finpr

\rem Remark 1.21|It would be interesting to know whether Theorem~1.18
and Cor.~1.19 are still true in the non K\"ahler case. In fact
Li-Yau~[LY87] proved the existence of Hermite-Einstein metrics also in
that case, assuming that $\om$ is a Gauduchon metric. The difficulty is
with the Fulton-Lazarsfeld inequalities for Chern classes, which we can
only prove for K\"ahler manifolds (the hard Lefschetz theorem is used,
see~\S2).\finpr


\vskip25pt
\titre 2. Inequalities for Chern classes|

The goal of this section is to show that the Fulton-Lazarsfeld
inequalities~[FL83] for Chern classes of ample vector bundles still
hold for nef vector bundles over compact K\"ahler manifolds. The proof
follows basically the same lines, but the arguments involving tensor
products with ample line bundles must be replaced by arguments using
the K\"ahler metric. This gain in generality will fortunately be
accompanied by a substantial simplification of the technique (covering
tricks are eliminated). The first step is an extension of the
Bloch-Gieseker inequalities~[BG71].

\than Proposition 2.1|Let $X$ be a compact complex $n$-fold possessing a
K\"ahler metric $\om$ and let $E$ be a nef vector bundle of rank $r$
over $X$. Then for every $\lambda\ge0$ we have
$$\sum\limits_{k=0}^n{r\vee n-k\choose n-k}
\lambda^{n-k}\int_Xc_k(E)\wedge\om^{n-k}\ge0$$
where $r\vee n=\max(r,n)$. In particular $\int_Xc_n(E)\ge0$ and more
generally
$$\int_Xc_k(E)\wedge\om^{n-k}\ge0$$
if the integral vanishes for higher values of $k$.\fintha

\proof We first observe that for every $\lambda>0$
$$\xi_\lambda=c_1(\cOE)+\lambda\pi^\star\lbrace\om\rbrace$$
is a K\"ahler class on $\pb(E)$. Indeed, the construction of $\hhe$ in
the proof of Th. 1.12 shows that
$$\Theta_\hhe\lp\cOE\rp+\lambda\pi^\star\om>0$$
for $\ep>0$ small. We can of course suppose that $r\ge n$, otherwise we
replace $E$ by $E\oplus\cO^{n-r}$. Then $\xi_0=c_1(\cOE)$ satisfies the
relation 
$$\xi_0^r-\pi^\star c_1(E)\xi_0^{r-1}+\ldots+(-1)^n\pi^\star
c_n(E)\xi_0^{r-n}=0.$$
It follows that $\xi_\lambda=\xi_0+\lambda\pi^\star\lbrace\om\rbrace$
satisfies
$$\xi_\lambda^r-\pi^\star
a_1(\lambda)\xi_\lambda^{r-1}+\ldots+(-1)^n\pi^\star
a_n(\lambda)\xi_\lambda^{r-n}=0,$$
where 
$$a_p(\lambda)=\sum\limits_{k=0}^p{r-k\choose p-k}
\lambda^{p-k}c_k(E)\cdot\lbrace\om\rbrace^{p-k}$$
($a_p(\lambda)$ is just the $p$-th elementary symmetric function in
$\rho_j+\lambda\lbrace\om\rbrace$, where $\rho_1,\ldots,\rho_r$ are 
the Chern roots). If the inequality of Prop. 2.1 does not hold, there
is $\lambda\ge0$ such that $a_n(\lambda)<0$. Since
$a_n(\lambda)\sim{r\choose n}\lbrace\om\rbrace^n\lambda^n$ tends to
$+\infty$ when $\lambda\ra+\infty$, there would exist $\lambda>0$ such
that $a_n(\lambda)=0$. Suppose that $\lambda$ is chosen in this way and
consider the cohomology class
$$\alpha=\xi_\lambda^{n-1}-\pi^\star
a_1(\lambda)\xi_\lambda^{n-2}+\ldots+(-1)^{n-1}\pi^\star
a_{n-1}(\lambda)\in H^{2n-2}(\pb(E),\rb).$$
Then $\alpha\not=0$, for the projection formula implies
$$\pi_\star\lp\alpha\cdot\xi_\lambda^{r-n}\rp=
\pi_\star\lp\xi_\lambda^{r-1}\rp=1\in H^0(X,\rb).$$
On the other hand $\alpha\cdot\xi_\lambda^{r+1-n}=0$ by our choice of
$\lambda$. This contradicts the hard Lefschetz theorem.\finpr

\than Corollary 2.2|If $Y$ is a $d$-dimensional analytic subset of $X$
with $d\ge r$, then \hbox{$\int_Yc_r(E)\wedge\om^{d-r}\ge0$}.\fintha

\proof Let $\wx\ra X$ be an embedded resolution of singularities of $Y$
in $X$ (Hironaka~[Hi64], Bierstone-Milman~[BM90]), and let $f:\wy\ra X$
be the induced map to $X$. Then $f^\star E$ is nef over $\wy$ and $\wy$
is K\"ahler, so
$$\int_\wy c_r(f^\star E)\wedge\ww^{d-r}\ge0$$
for every K\"ahler metric $\ww$ on $\wy$ (clearly the integrals
corresponding to degrees greater than $r=\rank E$ vanish, so Prop. 2.1
can be applied). Replacing $\ww$ by  $\ww_\ep=f^\star\om+\ep\ww$ with
$\ep\ra0$, we get 
$$\int_Y c_r(E)\wedge\om^{d-r}=
\int_\wy c_r(f^\star E)\wedge f^\star\om^{d-r}\ge0.\eqno\finpr$$

The second step is a semipositivity statement for the intersection
with cone classes. Let $\quer{E}=P(E\oplus\cO)$ be the compactification
of $E$ by the hyperplane at infinity and let $Z_E$ be the zero section
of $E$. If $C$ is a cone in $E$, that is, a $\cb^\star$-invariant
analytic subset in the total space of $E$, we denote by
$\quer{C}=P(C\oplus\cO)$ its closure in $\quer{E}$. There are well
defined cohomology classes
$$\lbrace Z_E\rbrace\in H^{2r}(\eq,\rb),\quad\lbrace\cq\rbrace\in
H^{2p}(\eq,\rb)$$
with $p=\codim C$, which can be seen for instance as the cohomology
classes of the associated closed positive currents $[Z_E]$, $[\cq]$. We
are interested in computing the intersection class
$$\lbrace Z_E\rbrace\cdot\lbrace\cq\rbrace$$
in $H^{2(r+p)}(\eq,\rb)$ (note that $\dim\eq=r+n$).  Following~[FL83],
we introduce the tautological quotient vector bundle
$Q=\pi^\star(E\oplus\cO)/\cO_\eq(-1)$ of rank $r$ on $\eq$. Then we
have $c_r(Q)=\lbrace Z_E\rbrace$ since the canonical section of $Q$
(=image of $0\oplus1$ in $Q$) admits $Z_E$ as its transversal zero
locus in $\eq$. In this way, by using the fact that $\eq$ is K\"ahler,
Cor.~2.2 implies

\than Proposition 2.3|If $E$ is nef, then for every cone $C$ in $E$ of
dimension $d=r+n-p\ge r$, we have 
$$\lbrace Z_E\rbrace\cdot\lbrace\cq\rbrace\cdot\lbrace\pi^\star
\om^{d-r}\rbrace=\int_\cq c_r(Q)\wedge\pi^\star\om^{d-r}\ge0.$$
\fintha

The third and final step is an application of the determinantal formula
of Kempf-Laksov~[KL74]. First recall a few notations about Schur
polynomials and Schubert calculus. Denote by $\Lambda(k,r)$ the set of
all partitions of $k$ by non negative integers${}\le r$. Thus an element
$a\in\Lambda(k,r)$ is a sequence
$$r\ge a_1\ge a_2\ge\ldots\ge a_r\ge0\quad\hbox{with}\,\,\sum a_j=k.$$
Each $a\in\Lambda(k,r)$ gives rise to a Schur polynomial
$P_a\in\zb[c_1,\ldots,c_r]$ of weighted degree $2k$ (with $\deg
c_i=2i$), defined as the $r\times r$ determinant
$$P_a(c)=\det(c_{a_i-i+j})_{1\le i,j\le r}$$
where by convention $c_0=1$ and $c_i=0$ if $i\notin[0,r]$. For every
vector bundle $E$, the Kempf-Laksov formula expresses $P_a(c(E))$ as
the class of a cone intersection.

\than Formula 2.4|Let $E$ be a complex vector bundle of rank $r$ on a
compact differentiable manifold $M$. Let $V$ be a complex vector space
of dimension $k+r$ and let $V_M$ be the trivial vector bundle $M\times
V$. Fix a flag of subspaces $\lbrace0\rbrace\subset
V_1\subset\ldots\subset V_k\subset V$ with $\dim V_i=r+i-a_i$. Consider
the cone $\Omega_a(E)\subset\Hom(V_M,E)$ whose fibre over $x\in M$
consists of all $u\in\Hom(V,E_x)$ such that
$$\dim(\ker u\cap V_i)\ge i.$$
Then $\Omega_a(E)$ is a locally trivial cone bundle over $M$ of
codimension $k=\sum a_i$ in $\Hom(V_M,E)$ and we have 
$$P_a(c(E))=\pi_\star\lp\lbrace
Z_{\Hom(V_M,E)}\rbrace\cdot\lbrace\quer{\Omega_a(E)}\rbrace\rp$$
where $\pi_\star$ denotes the integration along the fibres of 
$\pi:\quer{\Hom(V_M,E)}\ra M$.\fintha

\proof Observe first that the current $[\Oaq]$ is well defined,
although $\Oaq$ is singular and $M$ is not analytic; indeed $\Oaq$ is
locally the product of an open set in the base with an analytic set in
the fibre of $\Hom(V_M,E)$. Formula 2.4 is proved in~[KL74] in the
context of schemes and non singular projective varieties. Since we
allow ourselves to deal with $C^\infty$ objects, the proof becomes in
fact much easier. First observe that the formula is functorial with
respect to base change. Now, every $C^\infty$ vector bundle is
generated by a finite number $N$ of sections, hence there is a
commutative diagram
$$\matrix{
E & \longrightarrow & Q\cr
\downarrow    & & \downarrow\cr
M & \longrightarrow & G(N-r,N)\cr}$$
where $Q$ is the universal quotient vector bundle of rank $r$ over the
Grassmannian. Hence it is sufficient to check the formula for $E=Q$. In
that case the formula is simply a sophisticated formulation of
Giambelli's formula for Schubert cycles in $G(N-r,N)$ (see e.g.~[GH78],
p.206 and 415).\finpr

\medskip A combination of formula 2.4 with Prop. 2.3 gives 
$$\int_X P_a(c(E))\wedge\om^{n-k}=
\lbrace Z_{\Hom(V_M,E)}\rbrace\cdot\lbrace\quer{\Omega_a(E)}\rbrace\cdot
\lbrace\pi^\star\om^{n-k}\rbrace\ge0$$
when $E$ is nef. More generally, the argument of Cor. 2.2 implies:

\than Theorem 2.5|Let $E$ be a nef vector bundle over a compact complex
manifold $X$ equipped with a K\"ahler metric $\om$. Then for any Schur
polynomial $P_a$ of degree $2k$ and any analytic subset $Y\subset X$ of
dimension $d\ge k$ we have 
$$\int_Y P_a(c(E))\wedge\om^{d-k}\ge0.\eqno\finpr$$\fintha

Even in the case of projective algebraic manifolds, this result is new
because $\om^{d-k}$ need not be representable by an algebraic cycle.
Interesting examples of Schur polynomials are 
$$\eqalign{
P_{k00\ldots}(c)&=\left|\matrix{ 
c_k & c_{k+1} & c_{k+2} & \ldots & \ldots\cr
0   &    1    &    0    & \ldots & 0     \cr  
\vdots &      &  \ddots &        & \vdots\cr  
0   &         & \ldots &         & 1     \cr}\right|=c_k,\cr
&\phantom{-}\cr
P_{11\ldots10\ldots0}(c)&=\left|\matrix{ 
c_1 & c_2     & c_3     & \ldots & c_k   \cr     
1   & c_1     & c_2     &        & \vdots\cr  
0   &  1      & c_1     &        & \vdots\cr  
\vdots&       &   1     &        & \vdots\cr  
\vdots&       &         &  c_1   & c_2   \cr  
0   & \ldots  & 0       &  1     & c_1   \cr}\right|
=\hbox{Segre class}\,s_k,\cr
&\phantom{-}\cr
P_{k-1,10\ldots0}(c)&=\left|\matrix{ 
c_{k-1} & c_k\cr
     1  & c_1\cr}\right|=c_1c_{k-1}-c_k.\cr}$$
The Schur polynomials $P_a$ with $a\in\Lambda(k,r)$ form a basis of the
$\qb$-vector space of polynomials of weighted degree $2k$. A Polynomial
$$P=\sum\lambda_aP_a$$
is said to be numerically positive if all coefficients $\lambda_a$ are
$\ge0$. It is well known in the theory of Schur polynomials that a
product of numerically positive polynomials is again numerically
positive (see~[FL83] or~[Md79] p.68 for a purely combinatorial proof
based on the Littlewood-Richardson rule). In particular 
$$c_1^k-c_k=\sum\limits_{j=2}^kc_1^{k-j}(c_1c_{j-1}-c_j)\ge0,$$
i.e. $c_k\le c_1^k$. Similarly, any Chern monomial $c_I=c_1^{i_1}\ldots
c_r^{i_r}$ of degree $2k$ satisfies $0\le c_I\le c_1^k$. By Th.~2.5 we
thus get:

\than Corollary 2.6|If $E$ is nef, any Chern monomial $c_I$ of degree
$2k$ satisfies
$$0\le\int_X c_I(E)\wedge\om^{n-k}\le\int_X
c_1(E)^k\wedge\om^{n-k}.$$\fintha

\than Corollary 2.7|If $E$ is nef and $c_1(E)^n=0$, all Chern
polynomials $P(c(E))$ of degree $2n$ vanish.\fintha

\vskip25pt
\titre \hbox{$\eqalign{
\hbox{\bf 3.~}&\hbox{\bf Compact manifolds with nef tangent bundles.}\cr
&\hbox{\bf Structure of the Albanese map (K\"ahler case)}\cr}$}|
\titre 3.A. Some examples|

The first examples are produced by

\than Proposition 3.1| Every homogeneous compact complex manifold has a
nef tangent bundle.\fintha

Indeed, if $X$ is homogeneous, the Killing vector fields generate $T_X$,
so $T_X$ is a quotient of a trivial vector bundle. In particular, we get
the following 

\rem Examples 3.2 {\rm(homogeneous case)}|
\item{\rm(i)} Rational homogeneous manifolds: $\pb_n$, flag manifolds,
quadrics $Q_n\,$, and more generally quotients $X=G/P$ of a simply
connected complex Lie group $G$ by a parabolic subgroup $P$\hfill\break
(all projective algebraic with $K_X^{-1}$ ample).
\item{\rm(ii)} Tori $\cb^n/\Lambda$\hfill\break
(K\"ahler, possibly non algebraic).
\item{\rm(iii)} Hopf manifolds $(\cb^n\setminus\lbrace0\rbrace)/H$ where
$H$ is a discrete group of homotheties\hfill\break
(non K\"ahler for $n\ge2$).
\item{\rm(iv)} Iwasawa manifolds $G/\Lambda$ where $G$ is the group of
unipotent upper triangular $p\times p$ matrices and $\Lambda$ the
subgroup of matrices with entries in the ring of integers of some
imaginary quadratic field, e.g. $\zb[i]$\hfill\break
(non K\"ahler if $p\ge3$, although $T_X$ is trivial).\finpr
 
\medskip We must remark at this point that not all manifolds $X$ with
nef tangent bundles are homogeneous, the automorphism group may even be
discrete:

\rem Example 3.3|Let $\Gamma=\cb/(\zb+\zb\tau)$, $\Im\,\tau>0$, be
an elliptic curve. Consider the quotient space
$X=(\Gamma\times\Gamma\times\Gamma)/G$ where
$G=\lbrace1,g_1,g_2,g_1g_2\rbrace\simeq\zb_2\times\zb_2$ is given by
$$\eqalign{
g_1(z_1,z_2,z_3)&=(z_1+\half,-z_2,-z_3),\cr
g_2(z_1,z_2,z_3)&=(-z_1,z_2+\half,-z_3+\half),\cr
g_1g_2(z_1,z_2,z_3)&=(-z_1+\half,-z_2+\half,z_3+\half).\cr}$$
Then $G$ acts freely, so $X$ is smooth. It is clear also that $T_X$ is
nef (in fact $T_X$ is unitary flat). Since the pull-back of $T_X$ to
$\Gamma\times\Gamma\times\Gamma$ is trivial, we easily conclude that
$T_X$ has no sections, thanks to the change of signs in
$g_1,g_2,g_1g_2$. Therefore the automorphism group $\Aut(X)$ is
discrete. The same argument shows that  $H^0(X,\Omega_X^1)=0$.

Similar examples can be found in higher dimensions: for instance, one
can replace one or several of the factors $\Gamma$ in
$\Gamma\times\Gamma\times\Gamma$ by an arbitrary torus, the translation
by $\half$ being replaced by a translation with an element in the
half-lattice. In this way we get examples of K\"ahler non algebraic
manifolds with $T_X$ nef and  $H^0(X,\Omega_X^1)=0$ for all $n=\dim
X\ge4$, possibly of algebraic dimension 0  when  $n\ge6$ (by Prop. 7.3
below, such non algebraic examples do not exist in dimension${}\le3$).\finpr

\medskip The above example suggests that we should try to classify
manifolds with nef tangent bundles only up to finite \'etale covers. In
fact Prop. 1.10 immediately implies:

\than Proposition 3.4|Let $X$ be a compact complex manifold and let
$\wx$ be a finite \'etale cover of $X$. Then $T_X$ is nef if and only if 
$T_\wx$ is nef.\fintha

\rem Example 3.5|Let $X$ be the ruled surface $\pb(E)$ over the
elliptic curve $\Gamma=\cb/(\zb+\zb\tau)$ defined in 1.7. Then the
relative tangent bundle of $\pb(E)\ra\Gamma$ (=relative anticanonical
line bundle) is $\pi^\star(\det E^\star)\otimes\cO_E(2)\simeq\cO_E(2)$
and $T_\Gamma$ is trivial, so $T_X$ is nef. Moreover $X$ is almost
homogeneous, with automorphisms induced by 
$$(x,z_1,z_2)\mapsto(x+a,z_1+b,z_2),\quad(a,b)\in\cb^2$$
and has a single closed orbit equal to the curve $\lbrace z_2=0\rbrace$.
Here, no finite \'etale cover of $X$ can be homogeneous, otherwise
$K_X^{-1}=\cO_E(2)$ would be semipositive. Observe that no power of
$K_X^{-1}$ is generated by sections, although $K_X^{-1}$ is nef.\finpr

\vskip15pt
\titre 3.B. General properties|

Let $X$ be a compact complex n-dimensional manifold. We collect here a
few remarkable facts which hold under the assumption that $T_X$ is nef.

\than Proposition 3.6 {\rm[De91]}|Suppose that $T_X$ is nef. If $X$ belongs
to the Fujiki class $\cC$ (class of manifolds bimeromorphic to K\"ahler
manifolds), then $X$ is K\"ahler. In particular, if $X$ is Moishezon,
then $X$ is projective algebraic.\fintha

An algebraic proof of 3.6 for the Moishezon case will be given in \S4.
The next proposition was also proved in~[CP91] in the algebraic case.

\than Proposition 3.7|If $T_X$ is nef, then the line bundle $\cO(D)$ is
nef for every effective divisor $D$ in $X$. \fintha

\proof Propositions 3.6 and 3.7 are obtained in~[De91] as consequences
of a general regularization theorem for closed positive currents. Since
3.7 is quite elementary we indicate here briefly the argument.

Let $\sigma\in H^0(X,\cO(D))$ be a section with divisor $D$.
Then for $k$ larger than the maximum vanishing order of $\sigma$ on $X$, 
the $k$-jet section $J^k\sigma\in H^0(X,J^k\cOD)$ has no zeros. Therefore, 
there is an injection
$\cO\ra J^k\cOD$ and a dual surjection 
$$\lp J^k\cOD\rp^\star\otimes\cOD\ra\cOD.$$
Now, we observe that ~$J^k\cOD$~ has a filtration whose graded bundle
is\break $\bigoplus_{0\le p\le k}S^pT_X^\star\otimes\cOD$, so
$(J^k\cOD)^\star\otimes\cOD$ has a dual filtration with graded bundle
$\bigoplus_{0\le p\le k}S^pT_X$. By 1.14 (ii) and 1.15 (ii), we conclude
that $(J^k\cOD)^\star\otimes\cOD$ is nef, so its quotient $\cOD$ is nef
by 1.15~(i).\finpr

\medskip A combination of 3.6 and 3.7 leads to the following result.

\than Proposition 3.8|Let $X$ be a K\"ahler n-fold such that $T_X$ is
nef. If $X$ admits a nef line bundle $L$ such that $c_1(L)^n\not=0$,
then $X$ is projective algebraic and $L$ is ample. \fintha

\proof Let $\om$ be a K\"ahler metric on $X$. Then for every $\ep>0$
there is a hermitian metric $h_\ep$ on $L$ such that
$\Thhe(L)\ge-\ep\om$. We apply the holomorphic Morse inequalities
proved in~[De85]: let $X(h_\ep,q)$ be the open set of points $x\in X$
where $\Thhe(L)$ has $q$ negative eigenvalues and $n-q$ positive ones,
and let $X(h_\ep,\le q)$ be the union of $X(h_\ep,j)$ for $j\le q$. On
$X(h_\ep,q)$ we have 
$$0\le{1\over n!}(-1)^q\Thhe(L)^n
\le{1\over q!(n-q)!}\ep^q\om^q\wedge\lp\Thhe(L)+\ep\om\rp^{n-q}.$$
Indeed, ${1\over n!}\Thhe(L)^n/\om^n$ is the product
$\lambda_1\ldots\lambda_n$ of the eigenvalues of $\Thhe(L)$, which
satisfy 
$$-\ep\le\lambda_1\le\ldots\le\lambda_q<0<\lambda_{q+1}\le\ldots\le\lambda_n
\hbox{\ on\ } X(h_\ep,q).$$
Our inequality then follows from the obvious bound
$$\bet{\lambda_1\ldots\lambda_n}\le\ep^q\lp\lambda_{q+1}+
\ep\rp\ldots\lp\lambda_n+\ep\rp.$$
By integration over $X$, we infer from this that 
$$\left|\,\int_{X(h_\ep,\ge2)}\Thhe(L)^n\right|\le
\sum\limits_{q=2}^n{n!\over q!(n-q)!}\ep^q\lbrace\om\rbrace^q\cdot
\lp c_1(L)+\ep\lbrace\om\rbrace\rp^{n-q}=O(\ep^2).$$
Hence
$$\int_{X(h_\ep,\le1)}\Thhe(L)^n = c_1(L)^n-
\int_{X(h_\ep,\ge2)}\Thhe(L)^n>0 $$
for $\ep>0$ small enough. By~[De85], it follows that $X$ is Moishezon
and $\kappa(L)=n$, and so $X$ is projective algebraic thanks to
Proposition~3.6. It remains to show that $L$ is ample.
Let $A$ be a very ample smooth divisor. The exact sequence
$$0\ra H^0(X,\cO(kL-A)) \ra H^0(X,\cO(kL))\ra H^0(A,\cO(kL)_{\mid A})$$
shows that $\cO(kL-A)$ has sections for $k$ large: in fact since $L$ is
nef and big we have $h^0(X,\cO(kL))\ge ck^n$,  while
$h^0(A,\cO(kL)_{\mid A})\le ck^{n-1}$. Hence there is an effective
divisor $D$ such that $\cO(kL)=\cO(A+D)$. Proposition 3.7 implies that
$D$ is nef, therefore $A+D$ and $L$ are ample.\finpr

\vskip15pt
\titre 3.C. Structure of the Albanese map (K\"ahler case)|

We now investigate the structure of the Albanese map when $X$ is
K\"ahler. Let  $q=h^1(X,\cox)=h^0(X,\Omega_X^1)$ be the irregularity of
$X$ and let $u_1,\ldots,u_q$ be a basis of holomorphic 1-forms. Since
$X$ is K\"ahler, the forms $u_j$ are always closed. The Albanese map is
classically defined as follows: consider the map
$$\wx\ra\cb^q,\quad x\mapsto\lp\,\int_{x_0}^xu_j\rp_{1\le j\le q}$$
where $\wx$ is the universal covering of $X$ and $x_0\in\wx$ a given
point. Then the orbit of $x_0$ under $\pi_1(X)$ is mapped onto a
lattice $\Lambda\subset\cb^q$ (by Hodge theory). Therefore the map
$\wx\ra\cb^q$ factorizes into a map 
$$\alpha:X\ra A(X):=\cb^q/\Lambda,$$
which is precisely the Albanese map. By construction
$d\alpha=(u_1,\ldots,u_q)$ at every point. The following fact was
already remarked in~[CP91].

\than Proposition 3.9|Let $X$ be a K\"ahler manifold with $T_X$ nef. Then
the Albanese map $\alpha$ is a submersion onto the Albanese torus
$A(X)$ and the relative tangent bundle to $\alpha$ is nef. The fibres
are connected and have nef tangent bundles. \fintha

\proof If we had rank $(d\alpha(x))<q$ at some point $x\in X$, there
would exist a non zero linear combination
$u=\lambda_1u_1+\ldots+\lambda_qu_q$ with $u(x)=0$. Since $T_X$ is nef,
$u\in H^0(X,T_X^\star)$ would contradict Prop. 1.16. Therefore $\alpha$
is a submersion. The relative tangent bundle sequence
$$0\Ra T_{X/A(X)}\Ra T_X{\buildrel {d\alpha}\over\longrightarrow}
\alpha^\star T_{A(X)}\Ra0$$
in which $T_{A(X)}$ is trivial shows by 1.15 (iii) that
$T_{X/A(X)}$ is nef. This implies in particular that the fibres have nef
tangent bundles. If the fibres were not connected, their connected
components would give rise to a factorization $\alpha:X\ra Y\ra A(X)$
where $X\ra Y$ has connected fibres and $Y\ra A(X)$ is a finite \'etale
cover. Then $Y$ is a complex torus, so the universal property of the
Albanese map gives a factorization $X\ra A(X)\ra Y$ where $A(X)\ra Y$
is again \'etale. But then the fibres of $X\ra Y$ would not be
connected, contradiction.\finpr

\medskip The crucial step in the proof of our structure theorem is
contained in the following proposition (we denote here by
$c_k(X)=c_k(T_X)$ the Chern classes of~$X$).

\than Proposition 3.10|Let $X$ be a K\"ahler manifold with $T_X$ nef. One
of the two following situations occurs:
\item{\rm(i)} $c_1(X)^n>0$.\hfill\break
Then $X$ is a Fano manifold (that is, a projective algebraic manifold
with $K_X^{-1}$ ample) and $\chi(X,\cO_X)=1$.
\item{\rm(ii)} $c_1(X)^n=0$.\hfill\break
Then $\chi(X,\cO_X)=0$ and $X$ admits a non trivial holomorphic
$p$-form of odd degree. Moreover, there is a finite \'etale cover $\wx$
of $X$ such that 
$$q(\wx):=h^0(X,\Omega_\wx^1)>0.$$\fintha

\proof Since $c_1(X)=c_1(\kme)$ and $\kme=\Lambda^nT_X$ is nef, we
certainly have $c_1(X)^n\ge0$.

\vskip 0.2 truecm
\noindent{\bf(i)} If $c_1(X)^n>0$, Prop. 3.8 shows that $X$ is
projective algebraic and that $\kme$ is ample. Then the Kodaira
vanishing theorem implies
$$H^q\xox=H^q(X,K_X\otimes\kme)=0\quad\hbox{for}\,q\ge1,$$
therefore $\chi\xox=h^0\xox=1$.

\vskip 0.2 truecm
\noindent{\bf(ii)} If $c_1(X)^n=c_1(T_X)^n=0$, Corollary 2.7 implies
that all Chern polynomials  $P(c_1(X),$ $\ldots,c_n(X))$ of degree $2n$
vanish. Hence the Hirzebruch-Riemann-Roch formula
$$\chi\xox=\int_X \ch(\cO_X)\cdot \Todd (T_X)$$
shows that $\chi\xox=\sum(-1)^ph^p\xox=0$. Since $h^0\xox=1$, it
follows necessarily that $h^0(X,\Omega_X^p)=h^p\xox\ge1$ for some odd
degree $p$.

Let $u\in H^0(X,\Omega_X^p)$ be a non zero $p$-form. We consider the
contraction morphism
$$\Lambda^{p-1}T_X{\buildrel S\over\longrightarrow}
\Omega_X^1,\quad S(v)=v\ort u.$$
We claim that $S$ has constant rank (as a bundle morphism). In fact, if
$r$ is the generic rank of $S$, then
$$\Lambda^rS:\Lambda^r\lp\Lambda^{p-1}T_X\rp\ra\Omega_X^r$$
can be seen as a non zero holomorphic section of 
$$\lp\Lambda^r(\Lambda^{p-1}T_X)\otimes\Lambda^rT_X\rp^\star,$$
and $\Lambda^r(\Lambda^{p-1}T_X)\otimes\Lambda^rT_X$ is nef by Prop.~1.14.
Therefore $\Lambda^rS$ has no zeros and so the rank of $S$ is
equal to $r$ at every point (note that $r>0$ since $u\not=0$). Now
$E=\Im\,S$ is a rank $r$ subbundle of $\Omega_X^1$ which is a quotient
of $\Lambda^{p-1}T_X$. By Prop. 1.14 (ii) and 1.15 (i) we conclude that
$E$ is nef; since $E^\star$ is a quotient of $(\Omega_X^1)^\star=T_X$,
we also infer that $E^\star$ is nef. 

Finally, Th. 1.18 implies that $E$ admits a hermitian flat subbundle
$E_1$ of positive rank $r_1\le r$, given by some unitary representation
$\rho:\pi_1(X)\ra U(r_1)$. We would like $E_1$ to have a
finite monodromy group $G:=\rho(\pi_1(X))$, so that the pull-back of
$E_1$ by some finite \'etale cover $\pi:\wx\ra X$ gives rise to a
trivial subbundle $\pi^\star E_1\subset\pi^\star\Omega_X^1=\Omega_\wx^1$.
A well-known theorem of Tits~[Ti72] asserts that every subgroup $G$ of
a linear group either contains a non abelian free subgroup or a
solvable subgroup of finite index. The first case of Tits' alternative
is not possible here: in fact, Th.~1.1 of~[DPS92] shows, since $\kme$
is nef, that $\pi_1(X)$ has subexponential growth and cannot contain a
non abelian free subgroup. Therefore $G$ contains a solvable subgroup
$\Gamma$ of finite index. Let
$$\Gamma=\Gamma_0\supset\Gamma_1\supset\ldots\supset\Gamma_N=\lbrace1\rbrace$$
be the series of derived subgroups $\Gamma_{i+1}=[\Gamma_i,\Gamma_i]$.
If $G$ is finite, we have already seen that there is a trivial finite
\'etale pull-back $\pi^\star E_1\subset\Omega_\wx^1$ and so
$H^0(\wx,\Omega_\wx^1)\not=0$. Otherwise $G$ and $\Gamma$ are infinite.
Let $i$ be the smallest index such that $\Gamma_i/\Gamma_{i+1}$ is infinite.
Then $G/\Gamma$, $\Gamma/\Gamma_1,\ldots,\Gamma_{i-1}/\Gamma_i$ are finite,
and so is $G/\Gamma_i$. Therefore \hbox{$\Pi_i=\rho^{-1}(\Gamma_i)$}
is of finite index in $\pi_1(X)$ and gives rise to a finite \'etale cover
$\wx\ra X$ with $\pi_1(\wx)=\Pi_i$. The representation $\rho$ induces
a surjective homomorphism
$$H_1(\wx,\zb)=\Pi_i/[\Pi_i,\Pi_i]\longrightarrow
\Gamma_i/[\Gamma_i,\Gamma_i]=\Gamma_i/\Gamma_{i+1}.$$
Hence $H_1(\wx,\zb)$ is infinite, $\wx$ has non zero first Betti number
$b_1$ and 
$$h^0(\wx,\Omega_\wx^1)=\half b_1>0,$$
as desired.\finpr

\rem Remark 3.11|Example 3.3 shows that $H^0(\xomx^1)$ is not necessarily
different from $0$ in the second case (ii) of Prop. 3.10. In this
example, it is easily verified that
$$h^0\xox=1,\quad h^0(\xomx^1)=h^0(\xomx^2)=0,\quad h^0(\xomx^3)=1.
\eqno\finpr$$
\smallskip

The proof of the structure theorem proceeds by induction on the
dimension. In the induction step, we need to compute the irregularity
of the fibres of the Albanese map. For convenience, we introduce the
notations
$$q(X)=h^1\xox,\quad\wdt{q}(X)=\sup q(\wx)$$
where the supremum is taken over all finite \'etale covers $\wx\ra X$.
The following topological result generalizes and makes more precise
Prop. 2.8 of~[CP91].

\than Proposition 3.12|Let $X,Y$ be compact K\"ahler manifolds and let
$g:X\ra Y$ be a smooth fibration with connected fibres. If $F$ denotes
any fibre of $g$, then
\item{\rm(i)} $q(X)\le q(Y)+q(F)$,
\item{\rm(ii)} $\qq(X)\le\qq(Y)+\qq(F)$.
\item{\rm(ii)} Suppose that $Y$ is a complex torus and that the fibres
$F$ have the following properties: $\pi_1(F)$ contains an abelian
subgroup of finite index, and for every finite \'etale cover $\Fq$ of
$F$ the Albanese map $\Fq\ra A(\Fq)$ has constant rank. Then
$$\qq(X)=\qq(Y)+\qq(F).$$
\fintha

\proof {\bf(i)} The Leray spectral sequence of the constant sheaf $\rx$
over $X$ satisfies
$$\eqalign{
E_2^{s,t}&=H^s(Y,R^tg_\star\rx),\cr
E_r^{s,t} &\Rightarrow H^{s+t}(X,\rb).\cr}$$
Since the fibres of $g$ are connected and since $g$ is differentiably a
locally trivial fibre bundle, we have $R^0g_\star\rx=\rb_Y$, while
$R^1g_\star\rx$ is a locally constant real vector bundle of rank $2q$,
$q=q(F)$, containing $R^1g_\star\zb_X$ as a lattice; in particular the
monodromy group is contained in $SL_{2q}(\zb)$. Therefore
$$\dim H^0(Y,R^1g_\star\rx)\le 2q(F),$$
the equality occuring if and only if the locally constant system
$R^1g_\star\zb_X$ is trivial, i.e. $\pi_1(X)$ acts trivially on
$H^1(F,\zb)$ by monodromy. We know by Blanchard~[Bl56] that the Leray
spectral sequence always degenerates in $E_2$ for K\"ahler fibrations.
Hence
$$\eqalign{
2q(X) &=\dim H^1(X,\rb)\cr
      &=\dim H^1(Y,\rb)+\dim H^0(Y,R^1g_\star\rx)\cr
      &\le 2q(Y)+2q(F),\cr}$$
with equality if and only if $\pi_1(X)\ra\Aut(H^1(F,\zb))$ is trivial.

\vskip 0.2 truecm
\noindent{\bf (ii)} The homotopy exact sequence of the fibration yields
$$\pi_2(Y)\ra\pi_1(F)\ra\pi_1(X)\ra\pi_1(Y)\ra1.$$
Any finite \'etale cover $\wx$ of $X$ corresponds to a subgroup
$\pi_1(\wx)\subset\pi_1(X)$ of finite index. If we take the image of
$\pi_1(\wx)$ in $\pi_1(Y)$ and its inverse image in $\pi_1(F)$, we
obtain subgroups of finite indices, corresponding to \'etale covers
$\wy$ and $\wf$. Moreover $\pi_2(Y)=\pi_2(\wy)$ is mapped onto the
kernel of $\pi_1(\wf)\ra\pi_1(\wx)$, so we get a smooth fibration
$\wx\ra\wy$ with fibre $\wf$. By (i) this implies
$$q(\wx)\le q(\wy)+q(\wf)\le\wq(Y)+\wq(F).$$
Inequality (ii) follows by taking the supremum over all $\wx$.

\vskip 0.2 truecm
\noindent{\bf(iii)} Since $\pi_1(F)$ is finitely generated, the
additional hypotheses in (iii) imply that $\pi_1(F)$ contains a free
finitely  generated abelian subgroup $\Lambda$ of finite index. Take a
finite \'etale cover $\wf$ of $F$ with $\pi_1(\wf)=\Lambda$, so that 
$q(\wf)={1\over 2}\rank\Lambda=\wdt{q}(F)$. It is easy to see that
$\Lambda$ has only finitely many conjugates by $\Aut(\pi_1(F))$: if
$\varphi\in\Aut(\pi_1(F))$ and $N=[\pi_1(F):\Lambda]$, then
$\Lambda\cap\varphi(\Lambda)$ has index at most
$[\pi_1(F):\varphi(\Lambda)]=N$ in $\Lambda$, so
$\Lambda\cap\varphi(\Lambda)\supset N\Lambda$. Therefore
$\Lambda'=\bigcap_{\varphi\in\Aut(\pi_1(F))}\varphi(\Lambda)$ is such
that $N\Lambda\subset\Lambda'\subset\Lambda$, has finite index in
$\pi_1(F)$ and is left invariant by the monodromy
$\pi_1(Y)\ra\Aut(\pi_1(F))$. This gives rise to a semidirect product
$G'=\Lambda'\tri\pi_1(Y)$ and to a commutative diagram
$$\matrix{
0  & \Ra &      \Lambda' & \Ra &         G' & \Ra & \pi_1(Y)    & \Ra & 0\cr
   &     & \longdownhook &     & \longdownhook &     & \downarrow= &     &\cr  
0  & \Ra & \pi_1(F)      & \Ra & \pi_1(X)   & \Ra & \pi_1(Y)    & \Ra & 0\cr}
$$
where the injections have finite index (note that $\pi_2(Y)=0$). The
upper row corresponds to a K\"ahler fibration $X'\ra Y$ whose fibre
$F'$ covers $\wf$. We check below in Steps (iv-vii) that the monodromy
action
$$\pi_1(Y)\ra\Aut(H^1(F',\zb))\simeq\Aut({\Lambda'}^\star)$$
has a finite image, and therefore there is a subgroup
$\wh\subset\pi_1(Y)$ of finite index which has trivial action. We set
$\wg=\Lambda'\tri\wh\subset G'$ and obtain in this way a fibration
$\wx\ra\wy$ with fibre $\wf=F'$, such that the monodromy is trivial.
This implies 
$$\wq(X)\ge q(\wx)=q(\wy)+q(\wf)=\wq(Y)+\wq(F)$$
as desired.

\vskip 0.2 truecm
\noindent{\bf(iv)} It is sufficient to show, under the hypotheses of
(iii) and after changing $(X,Y,F)$ into $(X',Y,F')$, that 
$$\pi_1(Y)\ra\Aut(H^1(F,\zb))$$
has a finite image, i.e. that the monodromy group of $R^1g_\star\rx$ is
finite. For this, we define a euclidean structure on 
\def\rgr{{R^1g_\star\rb_X}} $\rgr$ as follows: if $u,v$ are local
sections of $\rgr$ near a point $y\in Y$ corresponding to classes
$u,v\in H^1(F_y,\rb)$, Hodge theory shows that one can write
$u=Re(u')$, $v=Re(v')$ for some uniquely defined holomorphic 1-forms
$u',v'$ on $F_y$. Then we set 
$$\langle u,v\rangle_y=Re\int_{F_y}i u'\wedge\quer{v}'\wedge\om^{p-1},
\quad p=\dim F_y,$$
where $\om$ is a K\"ahler metric on $X$. If the euclidean structure can
be proved to be locally flat, then the monodromy group is contained in
$SL_{2q}(\zb)\cap O_{2q}(\langle~,~\rangle)$ which is a finite group.

\vskip 0.2 truecm
\noindent{\bf(v)}  Consider the relative Albanese factorization
$$X\ra A(X/Y)\ra Y$$
of the fibration: $A(X/Y)$ is a smooth torus fibration
over $Y$ whose fibres are the Albanese tori of the fibres $F_y$; the
existence of such a relative Albanese map is proved in~[Fj83] and [Cm85]
in a much more general context, but is very easy to check here since
everything is smooth. We prove later that $A(X/Y)\ra Y$ is a holomorphically
locally trivial torus bundle over $Y$. Then if $U$ is a small
neighborhood of a point $y\in Y$ and $g^{-1}(U)$ its inverse image in
$X$, there are well defined holomorphic maps
$$g^{-1}(U)\ra A(g^{-1}(U)/U)\simeq A(F_y)\times U{\buildrel
{pr_2}\over\longrightarrow} A(F_y)$$
over $U$. This shows that every holomorphic 1-form on $F_y$ can be
extended to a d-closed holomorphic 1-form on $g^{-1}(U)$. Therefore in
the above definition of $\langle u,v\rangle_y$ we may suppose that
$u',v'$ are d-closed holomorphic 1-forms on $g^{-1}(U)$. Stokes'
formula then shows that  $\langle u,v\rangle_y$  does not depend on
$y\in U$, as desired.

\vskip 0.2 truecm
\noindent{\bf(vi)} We first check that the total space $\axy$ is
K\"ahler. In fact, for every fibre $F$ the image $\alpha(F)$ in the
Albanese torus $A(F)$ is by assumption a submanifold $S$ of $A(F)$. The
tangent spaces $T_xS$ at various points $x\in S$ generate the vector
space $T_{A(F)}$, otherwise $S$ would be contained in a subtorus of $A(F)$
(this is of course impossible by the universal property of the Albanese
map). Therefore there is an integer $N$ such that the map
$$\eqalign{
F^N&\Ra A(F)\cr
(x_1,\ldots,x_N)&\longmapsto
\alpha(X_1)+\sum\limits_{j=2}^N\lp\alpha(x_j)-\alpha(x_1)\rp\cr}$$
is a submersion. Doing this fibre by fibre and using the compactness of
$Y$, we find a $N$-th power fibered product of $X$ over $Y$ and a
submersion 
$$X\times_YX\times_Y\ldots\times_YX\ra\axy.$$
The fibered product is a closed submanifold of $X^N$, thus K\"ahler,
and therefore its submersive image $\axy$ is K\"ahler by pushing
forward an appropriate power $\om^p$ of a K\"ahler metric. 
 
\vskip 0.2 truecm
\noindent{\bf(vii)} By (vi), the local triviality of $\axy\ra Y$ is a
consequence of the fact that any smooth K\"ahler fibration $T\ra\cb$ by
complex tori is trivial (join any two points of $Y$ by a line and take
the pull-back of $\axy\ra Y$ to the line). In fact any such fibration
$T\ra\cb$ is given by a map from $\cb$ to the Siegel domain $D$
parametrizing the moduli space of complex tori which are polarized by a
K\"ahler form $\om$ with given periods. It is known by general results
of Griffiths and Schmid (see~[GS69], Cor.9.4) that there are no non
constant holomorphic maps $\cb\ra D$. In our case the argument is so
simple that we can explain briefly the details. Let $\cb\ni z\mapsto
P(z)$ be the $n\times2n$ holomorphic matrix representing a lattice
basis corresponding to the torus $T_z$. If $H(z)$ is the positive
hermitian matrix representing the K\"ahler class $\om_{\mid T_z}$, the
periods of $\om$ are given by
$$\Im\,\lp{^t}PH\quer{P}\rp=Q$$
where $Q$ is a real symplectic $2n\times2n$ matrix with constant
coefficients. Let $U$ be an invertible real $2n\times2n$ matrix such
that ${^t}UQU$ is the standard symplectic matrix and set $PU=(AB)$
where $A,B$ are holomorphic $n\times n$ matrices. We may assume
(possibly after changing $U$) that $A(z)$ is generically invertible,
and thus that $A(z)=I$ by changing the basis of $\cb^n$ via
$A(z)^{-1}$. We then get a meromorphic matrix $B$ on $\cb$ such that
$$\Im\,{I\choose{^t}B}H(IB)={^t}UQU=\pmatrix{
 0 & I\cr
-I & 0\cr},$$
i.e. ${^t}B=B$ and $\Im\,B=H^{-1}$ positive definite. The second
condition clearly implies that $B$ must be constant by Liouville's
theorem, therefore the lattice matrix $P$ is constant.\finpr

\rem Remark 3.13|A careful examination of the proof shows that the equality
$\wq(X)=\wq(Y)+\wq(F)$ holds under the following much more general
hypotheses:

\noindent{\bf(iii')} $X,Y$ are K\"ahler, the boundary
map $\pi_2(Y)\ra\pi_1(F)$ is zero, $\pi_1(F)$ contains an abelian (or
polycyclic) subgroup of finite index and $Y$ contains a subvariety $S$
with $\pi_1(S)\simeq\pi_1(Y)$, such that any two generic points in the
universal covering $\widehat{S}$ can be joined through a chain of
holomorphic images $\cb\ra\widehat{S}$. 

The proof of Prop.~3.12 actually shows that the monodromy group of
$\pi_1(Y)$ in $\Aut(H^1(F,\zb))$ is finite and that the relative
Albanese fibration $\axy\ra Y$ is locally trivial over $S$. 

The constant rank assumption made in (iii) is not necessary because we
always get in (vi) a surjective map $X\times_Y\ldots\times_YX\ra\axy$,
in particular $\axy$ always admits a K\"ahler current with poles. This
shows that the polarization used in (vii) exists for almost all fibres
$T_z$ of the relative Albanese space $\axy$, after pulling back
everything to $\cb$ by $\cb\ra\widehat{S}\ra S\ra Y$.\finpr

\medskip We are now ready to prove our main theorem on the structure of
the Albanese map.

\than Theorem 3.14|Let $X$ be a compact K\"ahler manifold with nef
tangent bundle $T_X$. Let $\wx$ be a finite \'etale cover of maximum
irregularity $q=q(\wx)=\wq(X)$. Then 
\item{\rm(i)} $\pi_1(\wx)\simeq\zb^{2q}$.
\item{\rm(ii)} The Albanese map $\alpha:\wx\ra A(\wx)$ is a smooth
fibration over a q-dimensional torus with nef relative tangent bundle.
\item{\rm(iii)} The fibres $F$ of $\alpha$ are Fano manifolds with nef
tangent bundles (i.e. $K_F^{-1}$ is ample and $T_F$ is nef).\smallskip
\fintha

\than Corollary 3.15|As an abstract group, the fundamental group
$\pi_1(X)$ is an extension of a finite group by $\zb^{2q}$.\fintha

\proof We prove 3.14 and 3.15 simultaneously by induction on $n=\dim X$.
Suppose that the result is known in dimension${}<n$. Then $\wx\ra
A(\wx)$ satisfies 3.14~(ii) by Prop. 3.9. In particular the fibres $F$ have
nef tangent bundle, and the induction hypothesis for 3.15 implies that
$\pi_1(F)$ contains an abelian subgroup of finite index. We can now
apply Prop. 3.12 (iii)  to conclude that $\wq(F)=0$. Hence the
alternative 3.10 (ii) cannot occur for $F$ and so $F$ is a Fano
manifold. Property 3.14~(iii) is thus proved. Finally it is well
known~[Ko61] that Fano manifolds are simply connected. The homotopy
exact sequence of the Albanese fibration shows that
$\pi_1(\wx)\simeq\pi_1(A(\wx))\simeq\zb^{2q}$. There is a composition
of finite \'etale covers $\wt\wx\to\wx\to X$ such that $\wt\wx$ is
a Galois cover of~$X$ (take $\pi_1(\wt\wx)$ to be the normal subgroup
of finite index in $\pi_1(X)$ obtained by taking the intersection
of all conjugates of~$\pi_1(\wx)$). Again $\pi_1(\wt\wx)\simeq
\zb^{2q}$ as a subgroup of finite index in $\pi_1(X)\simeq\zb^{2q}$,
and 3.15 follows.\finpr

\rem Remark 3.16|In order to complete the classification of K\"ahler
manifolds with nef tangent bundles, it would remain essentially to
understand the case when $X$ is a Fano manifold. This case is by far
the most difficult. It is conjectured in~[CP91] that every Fano
manifold with nef tangent bundle is rational homogeneous.

\vskip15pt
\titre 3.D. Numerical flatness of the Albanese map|

In this subsection, we want to analyze more precisely how the fibres of
the Albanese map can vary over the base torus. The Albanese map appears
in fact to be extremely rigid. We will show, at least when $X$ is
projective, that the fibres can be embedded ``in a numerically flat way''
in a projectivized numerically flat vector bundle over the Albanese
torus. We start with the special case of projective bundles.

\rem Example 3.17|Let $E$ be a holomorphic vector bundle of rank $r$
over a complex torus $Y$ and let $X=\pb(E)$. Then the natural
projection $\alpha:X\ra Y$ coincides with the Albanese map. The
relative and Euler exact sequences for the tangent bundles give
$$\eqalign{
0\Ra T_{X/Y}&{}\Ra T_X\Ra\alpha^\star T_Y\Ra 0,\cr
0\Ra\cO_X\Ra\alpha^\star E^\star&{}\otimes\cOE\Ra T_{X/Y}\Ra0.\cr}$$
By Prop.1.15 it follows immediately from this that $X=\pb(E)$ has a nef
tangent bundle as soon as $E$ is numerically flat. This is true more
generally if $E$ is projectively flat in the sense that $E\otimes(\det
E)^{-1/r}$ is numerically flat (i.e. $G:=S^rE\otimes(\det E)^{-1}$ is
numerically flat). To see this, observe that
$$\eqalign{
S^r\lp\alpha^\star E^\star\otimes\cOE\rp
&=\alpha^\star S^rE^\star\otimes\cO_E(r)\cr
&=\alpha^\star G^\star\otimes\cO_E(r)\otimes\alpha^\star(\det E)^{-1}\cr}$$
and that $\cO_E(r)\otimes\alpha^\star(\det E)^{-1}$ is a quotient of
$\alpha^\star G$; hence $\alpha^\star E^\star\otimes\cOE$ is nef by 1.14
(iii). The converse is also true and was proved in~[CP91] in the
algebraic case:

\than Proposition 3.18|Let $E$ be a holomorphic vector bundle of rank
$r$ over a torus. Then $T_{\pb(E)}$ is nef if and only if 
$E\otimes(\det E)^{-1/r}$ is numerically flat.\fintha

\proof If $T_{\pb(E)}$ is nef, the above exact sequences imply that
$\alpha^\star E^\star\otimes\cOE$ is nef (note that $T_Y$ is trivial).
In particular
$$\det\lp\alpha^\star E^\star\otimes\cOE\rp^{1\over r}
=\lp\alpha^\star(\det E)^\star\otimes\cO_E(r)\rp^{1\over r}
=\cO_{\edete}(1)$$
is nef over $\pb(E)$. Theorem 1.12 then shows that $\edete$ is nef
(this formal tensor product is not  really a vector bundle over $Y$ but
the proof of 1.12 together with 1.14 (ii) implies that
$S^rE\otimes(\det E)^{-1}$ is nef). Since $S^rE\otimes(\det E)^{-1}$ 
has trivial determinant, we conclude that  $S^rE\otimes(\det E)^{-1}$
is numerically flat.\finpr
\vskip7pt

We now suppose for the rest of this subsection that $X$ is projective.
It is then easy to show that the numerical dimension $\nu(K_X^{-1})$
coincides with the fibre dimension of the Albanese map of $\wt X$
(if~$L$ is a nef line bundle, $\nu(L)$ is by definition the
largest integer $m$ such that $c_1(L)^m\ne0$). 

\than Proposition 3.19|Let $X$ be a projective manifold with nef
tangent bundle. Then
$$\nu(K_X^{-1})=\dim X-\wt q(X).$$
\fintha

\proof Replacing $X$ with a suitable finite \'etale covering, we can
suppose that \hbox{$q(X)=\wt q(X)=q$}. Set $n=\dim X$ and let $\alpha:X\ra
A(X)$ be the Albanese map. Since $\alpha$ is surjective and the fibres
are Fano manifolds, we have $c_1(X)^{n-q}\ne 0$, hence $m:=\nu(-K_X)\ge n-q$.
Assume that $m>n-q$. This implies $c_1(X)^{n-q+1}\ne0$. Hence the
Kawamata-Viehweg vanishing theorem [Ka82, Vi82] applied to $K_X^{-1}$ gives
$$H^i(X,\cO_X)=0~~~\hbox{for\ }i\ge q\,.$$
Since $q=\dim A(X)$ we have $H^0(X,\Omega_X^q)\not=0$, contradicting
$H^q(X,\cO_X)=0$.\finpr
\vskip7pt

We now suppose that $X$ is a projective manifold with $T_X$ nef and such that
\hbox{$q(X)=\wt q(X)$}. Then $K_X^{-1}$ is relatively ample along the fibres
of $\alpha:X\Ra A(X)$ since the fibres are Fano varieties. The Kodaira
vanishing theorem implies that $R^q\alpha_\star(K_X^{-m})=0$ for all
$m,q\ge 1$. Therefore $\alpha_\star(K_X^{-m})$ is a locally free
sheaf, which we can identify with a vector bundle $E_m$ over~$A(X)$.
For $m\ge m_0$ large, each fibre $F$ has a very ample pluricanonical
line bundle $K_F^{-m}$ and we thus get a pluricanonical embedding of $F$
into $\pb\big(H^0(F,K_F^{-m})\big)$. By compactness of the base, the
integer $m_0$ can be chosen independent of~$F$. In this way we obtain
a relative embedding over $A(X)\,$:
$$\eqalign{
X&\buildo{\displaystyle j}\over\Longhook\pb(E_m)\cr
\alpha&\searrow\hbox{\qquad$~$}\swarrow\pi\cr
&\hbox{\quad$~$}A(X)\cr}$$
for all $m\ge m_0$. Set
$$V_{m,p}=\pi_\star\big(\cI_X\otimes\cO_{\pb(E_m)}(p)\big)\subset
\pi_\star\big(\cO_{\pb(E_m)}(p)\big)=\cO(S^pE_m).$$
For $p\ge p_0(m)$ large enough, the relative sections of
$\cI_X\otimes\cO_{\pb(E_m)}(p)$ define $X$ in the fibres of $\pb(E_m)$
as a transversal intersection of hypersurfaces of degree~$p$, and we have
$R^q\pi_\star\big(\cI_X\otimes\cO_{\pb(E_m)}(p)\big)=0$ for all $q\ge 1$.
In particular $V_{m,p}$ is a locally free subsheaf of~$\cO(S^pE_m)$.

\than Theorem 3.20|With the above notations and hypotheses, the locally
free sheaves $E_m=\alpha_\star(K_X^{-m})$ and $V_{m,p}=
\pi_\star\big(\cI_X\otimes\cO_{\pb(E_m)}(p)\big)$
are numerically flat.
\fintha

We first need some general results about the nefness of direct images of
nef line bundles by smooth morphisms.

\than Lemma 3.21|Let $\varphi:X\Ra Y$ be a smooth morphism of projective
manifolds and let $L$ be a nef and relatively ample line bundle on~$X$.
\smallskip
\item{\rm(i)} Let $G$ be a very ample line bundle over $Y$ and let $m=\dim Y$.
Then
$$\varphi_\star(K_{X/Y}\otimes L)\otimes K_Y\otimes G^{m+1}$$
is generated by sections.
\smallskip
\item{\rm(ii)} Suppose that $Y$ has the following property: there exists
a holomorphic self-map $\theta:Y\to Y$ and an ample line bundle $H$ over
$Y$ such that $\theta^\star H\otimes H^{-2}$ is nef. Then
$\varphi_\star(K_{X/Y}\otimes L)$ is nef over~$Y$.
\smallskip\fintha

\proof First observe that the sheaf $\varphi_\star(K_{X/Y}\otimes L)$ is
locally free, because we have $R^q\varphi_\star(K_{X/Y}\otimes L)=0$ for
$q\ge 1$ by the Kodaira vanishing theorem.

\noindent{\bf(i)} The assertion means that every section $\sigma$ of
$$K_{X/Y}\otimes L\otimes\varphi^\star(K_Y\otimes G^{m+1})=
K_X\otimes L\otimes\varphi^\star(G^{m+1})$$
along a fibre $F=\varphi^{-1}(x_0)$ can be extended to all
of~$X$. Let $\mu:\wt X\ra X$ (resp. $\nu:\wt Y\ra Y$) be the blow-up of
$X$ with center $F$ (resp. of $Y$ at $x_0$) and let $D\subset\wt X$
(resp. $E\subset\wt Y$) be the exceptional divisor. Then there is a map
$\wt\varphi:\wt X\ra\wt Y$ extending $\varphi$ to the blow-ups, i.e. such
that $\nu\circ\wt\varphi=\varphi\circ\mu$, and we have
$D=\wt\varphi^\star E$, $K_{\wt X}=\mu^\star K_X\otimes\cO((m-1)D)$. Now,
$$B:=\mu^\star\big(L\otimes\varphi^\star(G^{m+1})\big)\otimes\cO(-mD)$$
is easily seen to be ample on~$\wt X\,$; indeed $\mu^\star L$ is nef and
$\wt\varphi$-ample, while $\mu^\star\varphi^\star(G^{m+1})\otimes\cO(-mD)$
is the pull-back by $\wt\varphi$ of $\nu^\star(G^{m+1})\otimes\cO(-mE)$
which is ample on~$\wt Y$ (note that $\nu^\star G^p\otimes\cO(-E)$
is ample for $p$ large and generated by sections for $p=1$ thanks to the
very ampleness of~$G$, and make an interpolation). By the Kodaira vanishing
theorem and the Leray spectral sequence we get
$$\eqalign{
       0=H^1(\wt X,K_{\wt X}\otimes B)
       &=H^1\big(\wt X,\mu^\star(K_X\otimes L\otimes\varphi^\star(G^{m+1}))
            \otimes\cO(-D)\big)\cr
       &=H^1\big(X,K_X\otimes L\otimes\varphi^\star(G^{m+1})\otimes\cI_F\big).
\cr}$$
This implies the desired extension property.

\noindent{\bf(ii)} By replacing if necessary $H$ by some power, we may
assume that
$$H\otimes(K_Y\otimes G^{m+1})^{-1}$$
is nef. Then (i) implies that $\varphi_\star(K_{X/Y}\otimes L)\otimes
H$ is nef. Observe that the line bundles $G$ and $H$ can be chosen
independent of $X$, $L$ and~$\varphi$. Let $X_p$ be the fibered product
$X\,{}_\varphi{\times}_{\theta^p}\,Y$, so that we have a commutative diagram
$$\eqalign{
        X_p&\buildo{\displaystyle\Theta_p}\over\Ra X\cr
        \varphi_p{\big\downarrow}~&\qquad{\big\downarrow}\varphi\cr
        Y&\buildo{\displaystyle\theta^p}\over\Ra Y\cr}$$
Set $L_p=\Theta_p^\star L$. An obvious base change yields
$$(\theta^p)^\star\big(\varphi_\star(K_{X/Y}\otimes L)\big)=
(\varphi_p)_\star\big(K_{X_p/Y}\otimes L_p\big).$$
By (i) applied to $\varphi_p:X_p\to Y$, we infer that $(\theta^p)^\star
\big(\varphi_\star(K_{X/Y}\otimes L)\big)\otimes H$ is nef. The assumption
on $\theta$ implies $(\theta^p)^\star H\ge H^{2^p}$, hence
$$(\theta^p)^\star\big(\varphi_\star(K_{X/Y}\otimes L)\otimes H^{-2^p}\big)$$
is nef, and therefore so is $\big(\varphi_\star(K_{X/Y}\otimes L)\big)
\otimes H^{-2^p}$ by Proposition 1.10. Since this result holds for
arbitrarily large~$p$, we conclude that $\varphi_\star(K_{X/Y}\otimes L)$
is nef (one can avoid fractional powers by taking symmetric powers
$S^{2^p}({\scriptstyle\bullet})\,$; note also that the equidimensionality
of the fibres of the base change is not needed in the projective case;
we will anyway only use the equidimensional case later).\finpr\vskip7pt

\noindent{\sl Proof of Theorem 3.20}. The existence of a self-map $\theta:
Y\to Y$ as in Lemma~3.21~(ii) is clear when $Y$ is a torus (take $\theta$
to be the isogeny $\theta(x)=2x$) or when $Y$ is a projective space $\pb^N$
(take $\theta[z_0,\ldots,z_N]=[z_0^2,\ldots,z_N^2]$). Since $K_X^{-1}=
K_{X/A(X)}^{-1}$ is nef and relatively ample, Lemma~3.21$\,$(ii) shows that
$$E_m=\alpha_\star(K_X^{-m})=\alpha_\star(K_{X/A(X)}^{-m})$$
is a nef vector bundle. Now, since $R^q\alpha_\star(K_X^{-m})=0$ for
$q\ge 1$, the Riemann-Roch-Grothendieck formula yields (see~[BS58]):
$${\rm Ch}(E_m)=\alpha_\star\big({\rm Ch}(K_X^{-1})\cdot{\rm Todd}
(T_{X/A(X)}\big).$$
In particular, we see that $c_1(E_m)$ is the direct image (integration
along the fibres) of a polynomial of degree $\dim F+1$ in the Chern
classes of~$X$. All these polynomials must vanish by Proposition~3.19
and Corollary~2.6. Therefore $c_1(E_m)=0\,$; the nefness of $E_m$ then
shows that $E_m$ is numerically flat. Now, take the tensor product of the
exact sequence
$$0\Ra\cI_X\Ra\cO_{\pb(E_m)}\Ra j_\star\cO_X\Ra 0$$
by $\cO_{\pb(E_m)}(p)$ and compute the direct image by~$\pi$. We have by
construction $j^\star\cO_{\pb(E_m)}(1)=K_X^{-m}$ and so
$$\pi_\star\big(j_\star\cO_X\otimes\cO_{\pb(E_m)}(p)\big)=
\alpha_\star(K_X^{-mp})=E_{mp}.$$
This gives rise to an exact sequence
$$0\Ra V_{m,p}\Ra S^pE_m\Ra E_{mp}\Ra 0,$$
and we conclude that $V_{m,p}$ is numerically flat by
Proposition~1.15.\finpr

\rem Remark 3.22|We expect Proposition 3.19 and Theorem~3.20
to be true also in the K\"ahler case. This would require a K\"ahler
version of the Kawamata-Viehweg vanishing theorem, at least in our
special situation. The rest of the proof would be almost unchanged,
except that we would have to replace very ample line bundles
in the proof of Lemma~3.21 by high multiples of a K\"ahler form on~$Y$
(and invoke the results of [De91] on the almost generation of nef
vector bundles over arbitrary compact complex manifolds).

\vskip25pt
\titre 4. Moishezon manifolds with nef tangent bundles|

In this section we show in a purely algebraic way 
that Moishezon manifolds with nef tangent bundles
are projective. This is just the algebraic part of 3.6.
Moreover we give another proof that any compact K\"ahler n-fold $X$ with $T_X$
nef and $K^n_X\not=0$ is automatically projective and even Fano.

\than Theorem 4.1| Let $X$ be a Moishezon manifold with $T_X$ nef. Then $X$
is projective.\fintha

\proof Let $\pi:\xd\ra X$ be a sequence of blow-up's such that $\xd$ is a
projective manifold. Let $\ld$ be very ample on $\xd$ and put
$\cL=\pi_\star(\ld)^{\star\star}$. 
Since $H^0(X,\cL)\not=0$ and since every effective
divisor on $X$ is nef by~[CP91] or by 3.7, $\cL$ is nef.
Since $\kappa(X,\cL)=\dim X$, $\cL$ is also big. As $K_X^{-1}$ is nef, we
deduce nefness of $\cL\otimes K_X$, hence by the base point free theorem, 
$\cL^m$ is generated by sections for some $m\gg0$ (the base point free 
theorem also holds for Moishezon manifolds by exactly the same proof, see
e.g. [KMM85]). We obtain a morphism $\phi:X\ra Y$ to a normal projective
variety $Y$ and it is sufficient to prove that $\phi$ is an isomorphism.
Assume the contrary and let $E$ be the degeneracy set ($\cL$ being big,
$\phi$ is a modification). By substituting $Y$ by local hypersurface
sections, we may assume that $E$ is mapped to a point $y_0$, i.e. $E$ is
exceptional. Thus we substitute $X$ by a lower-dimensional manifold
containing $E$ whose tangent bundle is still nef (near $E$). 
Now we equip $E$ with a complex structure whose conormal sheaf
$N^\star_{E/X}$ is (weakly) positive, i.e.
$\cO_{\pb(N^\star_{E/X})}(1)$ is ample ([An82], [Pe82]). We find some $m\in\nb$
(the ''minimal vanishing order'') and a canonical generically non zero map
$$\alpha:N^\star_{E/X\vert\red E}\ra S^m\Omega^1_{X\vert\red E},$$ 
by associating to a function the homogeneous part of degree $m$ in the power
series expansion. Now we take a general curve $C\subset\red E$, which may
be assumed to be smooth (otherwise normalize). Let $S=N^\star_{E/X\vert C}
/$torsion. Then $\alpha_C:S\ra S^m\Omega^1_{X\vert C}$ is
non zero. Since $\pb(S)\subset\pb(N^\star_{E/X\vert C})$ and since 
$$\cO_{\pb(S)}(1)\simeq{\cO_{\pb(N^\star_{E/X\vert C})}(1)}_{\vert\pb(S)},$$
$S$ is positive, i.e. ample.  But this contradicts the nefness of $T_X$ 
(for this consider $\det S\ra\Lambda^rS^m{\Omega^1}_{X\vert C}$,
$r=\rk(S)$).\finpr

\medskip We will use 4.1 to prove

\than Theorem 4.2| Let $X$ be a compact K\"ahler manifold of dimension
$n$. Assume that $T_X$ is nef and that $K_X^n\not=0$. Then $X$ is Fano,
in particular projective.\fintha
 
\proof Since $\kxe$ is nef and $(\kxe)^n>0$ by our assumption, it follows
from~[De85] that $X$ must be Moishezon, compare the proof of 3.8.
By 4.1, $X$ is projective. 
By the base point free theorem $K_X^{-m}$ is generated by sections for some
$m\gg0$. Let $\varphi:X\ra Y$ be the map defined by $\vert K_X^{-m}\vert$. 
$\varphi$
is birational, and if $\kxe$ is not ample, $\varphi$ is not finite. Since
$\kxe$ is $\varphi$-nef, we can apply [Ka91, appendix] to produce a
rational curve $C$ contracted by $\varphi$. $T_X$ being nef, $C$ deforms to
fill up $X$,which gives a contradiction since $K_X\cdot C'=0$ if and only 
if $C'$ is contracted by $\varphi$. Hence $\kxe$ is ample, i.e. $X$ 
is~Fano.\finpr

\vskip25pt
\titre 5. Two structure Theorems|

In this section we prove two general theorems on the structure of
projective manifolds with nef tangent bundles. In particular we prove a
criterion when $X$ is Fano. 

\than Proposition 5.1|Let $X$ be a complex manifold, $Y$ a reduced
locally irreducible complex space and $\varphi:X\ra Y$ a holomorphic
surjective map. Assume $\dim Y<\dim X$ and $T_X$ to be $\varphi$-nef, i.e.
$T_X$ is nef on every curve contracted by $\varphi$.
Assume for a fixed $y_0\in Y$ that every component of $\varphi^{-1}(y_0)$ of
dimension $\dim\varphi^{-1}(y_0)$ is compact. Let
$F\subset\varphi^{-1}(y_0)$ be such a component of maximal dimension (with
the canonical structure). Then:
\item{\rm(i)} $\red F$ is smooth.
\item{\rm(ii)} $N^\star_{\red F/X}$ is topologically trivial up to
tensoring with a torsion line bundle.
\item{\rm(iii)} $N^\star_{\red F/X}$ is numerically flat.
\fintha

\proof We let $F'=\varphi^{-1}(y_0)$ be equipped with the canonical fibre
structure. Let $F_0=\red F$. We prove the proposition by induction on
$n=\dim X$. For $n=1$ we have nothing to prove. So assume 5.1 to be proved
for $\dim X<n$.

\vskip 0.2 truecm
\noindent 1. First case: $\dim F=\dim F'\le n-2$.

Take $f_{y_0}\in\cO_{Y,y_0}$ general with $f(y_0)=0$. Then $\lbrace
f=0\rbrace$ will be a locally irreducible reduced hypersurface in $Y$
(possibly shrink $Y$). Let $Y_1=\lbrace f=0\rbrace$ and
$X_1=\lbrace\varphi^\star(f)=0\rbrace$, with reduced structure. Let
$\wdt{X}_1\subset X$ be the analytic preimage of $Y_1$, i.e. defined 
ideal-theoretically by $\varphi^\star(f)$. $F$ is (set-theoretically) still
contained in $X_1$ since $\dim F\le n-2$. Then the conormal sheaf satisfies
$$N^\star_{\wdt{X}_1/X\vert F_0}\simeq\cO_{F_0}.$$
Since $\wdt{X}_1=kX_1$ as divisors on $X$, we conclude
$$N^{\star k}_{X_1/X\vert F_0}\simeq\cO_{F_0}.$$
So $N^\star_{X_1/X\vert F_0}$ is a torsion line bundle. 
Now consider the exact sequence
$$0\ra N^\star_{X_1/X\vert F_0}
{\buildrel\alpha\over\ra}\Omega^1_{X\vert F_0}\ra\Omega^1_{X_1\vert F_0}\ra0.$$
Since $T_X$ is nef, $\alpha$ must have constant rank, otherwise
$S^k\Omega^1_{X\vert F_0}$ would have a section with zeros which is
impossible by [CP91,~2.12] or by 1.16. Hence $\Omega^1_{X_1\vert F_0}$
is locally free, consequently $X_1$ must be smooth near $F_0$. By
shrinking we may assume $X_1$ to be smooth everywhere. By induction we
conclude smoothness of~$F_0$. Moreover the exact sequence
$$0\ra N^\star_{X_1/X\vert F_0}\ra N^\star_{F_0/X}\ra
N^\star_{F_0/X_1}\ra0$$
together with induction proves that $N^\star_{F_0/X}$ is topologically
trivial up to tensoring with a torsion line bundle. 
Since $N_{F_0/X}$ is nef as a quotient of $T_{X\vert F_0}$ and $\det N_{F_0/X}$
is topological trivial we conclude that $N_{F_0/X}$ is numerically flat.

\vskip 0.2 truecm
\noindent 2. Second case: $\dim F= n-1$.

In this case we even don't need induction. Since $F'$ carries the full
fibre structure $N^\star_{F'/X}$ is generated by global sections.
Choose $k$ maximal, such that the $k$-th infinitesimal neighborhood
$F_{0,k}$ is contained in $F'$. Let
$$\beta:N^\star_{F'/X\vert F_0}\ra N^\star_{F_{0,k}/X\vert F_0}=
N^{\star k}_{F_0/X}$$
be the canonical map. Then we conclude the existence of $s\in
H^0(N^\star_{F'/X})$, such that $\beta(s)\not=0$.
Now consider the exact sequence
$$0\ra N^\star_{F_0/X}\ra\Omega^1_{X\vert F_0}\ra\Omega_{F_0}^1\ra0.$$
Since $T_{X\vert F_0}$ is nef, $\beta(s)$ cannot have zeros (pass to $S^k$).
So $N^{\star k}_{F_0/X}\simeq\cO_{F_0}$ and moreover $F_0$ is
smooth.\finpr

\than Theorem 5.2|Let $X$ be a projective manifold with $T_X$ nef, let
$\varphi:X\ra Y$ be the contraction of an extremal ray {\rm([Mo82],
[KMM85])}. Then $Y$ and $\varphi$ are smooth.\fintha
 
\proof Of course we may assume $\dim Y>0$, moreover by~[CP91] we know that
$\varphi$ cannot be a modification, hence $\dim Y<\dim X$.
We let $F'=\varphi^{-1}(y)$ be a fibre of $\varphi$ equipped with the full
fibre structure. It is sufficient to prove that $F'$ is smooth with $\dim
F'=\dim X-\dim Y$. This follows easily from [Fi76, p.100] since by our
assumption $\Omega^1_{X/Y}$ is locally free of rank $=\dim X-\dim Y$.
So let $F\subset F'$ be a component of maximal dimension in $F'$, with
induced structure. By 5.1, $\red F$ is smooth and $N^\star_{\red F/X}$ is
topologically trivial up to tensoring with a torsion line bundle. So 
$\det N^\star\equiv0$ (numerically).
By the adjunction formula and the fact that $-K_X$ is $\varphi$-ample
($\varphi$ being a Mori contraction), $K_{\red F}$ is negative, i.e.
$F_0=\red F$ is Fano.
By 5.1 $N^\star_{F_0/X}$ is nef as well as its dual.
By 1.18, $N^\star_{F_0/X}$ has a filtration by hermitian flat quotient vector
bundles. $F_0$ being simply connected, all these are trivial, hence
$N^\star_{F_0/X}$ is trivial. Now 
$N_{F_0/X}$ is generated by sections and 
$$H^1(N_{F_0/X})=0,$$
so the deformations of~$F_0$ fill up $X$~[Gr66].
Since $\dim\varphi(F_0)=0$, all deformations $F_{0,t}$ fulfill
$$\dim\varphi(F_{0,t})=0$$
as well (consider $(F_{0,t}\cdot\varphi^\star(L^k))$, $L$ ample on $Y$). Hence
$\dim F_0=\dim X-\dim Y$ and moreover $F=F_0$ as complex spaces. So $F$ is
smooth.\finpr

\rem Remark 5.3|Instead of assuming $\varphi$ to be a contraction of an
extremal ray in 5.2, it is sufficient to assume:
\item{\rm(i)} $X$ to be a compact manifold,
\item{\rm(ii)} $Y$ to be a normal compact complex space which is a subspace
of a K\"ahler manifold.
\item{\rm(iii)} $K_X$ to be negative on every fibre of $\varphi$,
$\varphi_\star(\cO_X)=\cO_Y$.\finpr

\medskip We now prove 3.10 in the algebraic case.

\than Theorem 5.4|Let $X$ be a projective manifold with $T_X$ nef.
Assume $c_1(X)^n=0$ with $n=\dim X$. Then $\wdt{q}(X)>0$.\fintha
 
\proof We proceed by induction on $n$. For $n=1$ we have nothing to
show. Now let $X$ be of dimension $n$. If $K_X\equiv0$ we are done
since $X$ is covered by an abelian variety. So assume $K_X$ not to be
nef. Then there is a contraction of an extremal ray, say $\varphi:X\ra
Y$. $Y$ cannot be a point, because then $X$ would be Fano. So by 5.1
$Y$ and $\varphi$ are smooth and $0<\dim Y<n$. Thus $T_Y$ is nef again.
Let $m=\dim Y$. If $c_1(Y)^m\not=0$ then $Y$ is Fano by 4.2, hence
$\chi(\cO_Y)=1$, hence  $\chi(\cO_X)=1$ contradicting $c_1(X)^n=0$
(which implies $\chi(\cO_X)=0$ by 2.7 and Riemann-Roch). So
$c_1(Y)^m=0$. Then $\wdt{q}(Y)>0$ by induction, hence $\wdt{q}(X)>0$.\finpr

\than Corollary 5.5|Let $X$ be a projective manifold with $T_X$ nef. Then
$X$ is Fano if and only if $\wdt{q}(X)=0$, respectively if and only if 
$c_1(X)^n\not=0$ $(n=\dim X)$.\fintha

\vskip25pt
\titre 6. Surfaces with nef tangent bundles|

In this section we give a classification of all (non necessarily K\"ahler)
surfaces whose tangent bundles are nef. In the algebraic case this has been
carried out in~[CP91]:

\than Theorem 6.1|Let $X$ be a smooth projective surface and assume $T_X$
to be nef. Then $X$ is minimal and is exactly one of the surfaces in the 
following list:
\item{\rm(1)} $X$ is abelian$\,;$
\item{\rm(2)} $X$ is hyperelliptic$\,;$
\item{\rm(3)} $X=\P2\,;$
\item{\rm(4)} $X=\P1 \times \P1\,;$
\item{\rm(5)} $X=\pb(E)$, where $E$ is a rank$\,$2-vector bundle on an
              elliptic curve $C$ with either
\itemitem{\rm(a)} $E=\cO\oplus L$, $L\in\Pic^0(C)$, or
\itemitem{\rm(b)} $E$ is given by a non split extension
$0\ra\cO\ra E\ra L\ra0$ with $L=\cO$ or $\deg L=1$.\smallskip
\fintha

In the non algebraic case the classification is as follows:

\than Theorem 6.2|Let $X$ be a smooth non algebraic compact surface with
$T_X$ nef. Then $X$ is minimal and is exactly one of the surfaces in the
following list:
\item{\rm(1)} $X$ is a torus$\,;$
\item{\rm(2)} $X$ is a Kodaira surface$\,;$
\item{\rm(3)} $X$ is a Hopf surface.
\fintha

For the convenience of the reader we recall the Kodaira classification of
non algebraic surfaces, cf.~[BPV84] or Kodaira's original papers. 
Let $X$ be a minimal non algebraic surface. Then  the algebraic dimension
$a(X)$ is either one or zero.

\vskip 0.2 truecm
\noindent If $a(X)=1$, then either
\item{\rm(a)} $\kappa(X)=1$ and $X$ is elliptic or
\item{\rm(b)} $\kappa(X)=0$ and $X$ is a torus or a K3-surface or a Kodaira 
           surface or
\item{\rm(c)} $\kappa(X)=-\infty$ and $X$ is a Hopf surface.

\vskip 0.2 truecm
\noindent If $a(X)=0$, then either
\item{\rm(a)} $\kappa(X)=0$ and $X$ is a torus or a K3-surface or
\item{\rm(b)} $\kappa(X)=-\infty$ and $X$ is said to be of type VII.

\vskip 0.2 truecm Recall that a Hopf surface is a compact surface whose
universal cover is $\cb^2\setminus\lbrace0\rbrace$, a K3-surface is a
surface with $K_X\simeq\cO$ and $b_1(X)=0$ and a Kodaira surface is a
surface with $\kappa(X)=0$ and $b_1(X)$ odd.

The proof of 6.2 splits into several steps according to this classification.

\than Proposition 6.3|Let $X$ be a compact surface with $T_X$ nef and
$a(X)=1$. Then $X$ is exactly one of the surfaces in the following list:
\item{\rm(1)} $X$ is a torus$\,;$
\item{\rm(2)} $X$ is a Kodaira surface$\,;$
\item{\rm(3)} $X$ is a Hopf surface.
\fintha

\proof By [CP91,~2.1] $X$ is minimal. Let $f:X\ra C$ be the algebraic
reduction. The map $f$ is an elliptic fibration over a smooth curve
$C$. The genus $g(C)$ is bounded by $1$ since otherwise $X$ would admit
a holomorphic 1-form with zeros contradicting [CP91,~2.1]. Using the
classification of singular fibres of $f$ (cf.~[BPV84]) and [CP91,~2.1]
the only possible singular fibres are multiples of smooth elliptic
curves.

First observe that $\kappa(X)\le0$ : clearly $\kappa(X)\le1$ and if
$\kappa(X)=1$, $X$ would admit an elliptic fibration over an elliptic
curve $C$ (observe $c_1^2=c_2=0$). Since $X\ra C$ must have a singular
fibre (a multiple elliptic curve) we obtain again a holomorphic 1-form
with zeros.

\vskip 0.2 truecm
\noindent{\bf($\alpha$)} $\kappa(X)=0$.

Since $0=c_1(X)^2\ge c_2(X)$ and $T_X$ is nef we get $c_2(X)=0$.
Therefore $X$ cannot be a K3-surface and by surface classification $X$ is
a torus or a Kodaira surface. Hence it remains to be shown that Kodaira
surfaces have nef tangent bundles.  
A primary Kodaira surfaces $X$ is an elliptic fibre bundle over an
elliptic curve $C$ with $b_1(X)=3$. By the exact sequence
$$0\ra T_{X/C}\ra T_X\ra f^\star T_C\ra0$$
and the equality $K_X=\cO_X$, we deduce that $T_X$ is nef by 
1.14 (observe $\omega_{X/C}\simeq\cO_X$ since $\omega_X\simeq\cO_X$).
A non primary (often called secondary) Kodaira surface admits a primary
Kodaira surface as a finite \'etale cover and hence has nef tangent bundle.

\vskip 0.2 truecm
\noindent{\bf($\beta$)} $\kappa(X)=-\infty$.
Then $X$ is a Hopf surface by classification. Therefore $X$ is an elliptic
fibre bundle over $\P1$ (via algebraic reduction).By the exact sequence 
$$0\ra T_{X/\P1}\ra T_X\ra f^\star T_{\P1}\ra0$$
it is enough to show that $T_{X/\P1}$ is trivial. For this purpose
consider the line bundle $f_\star(\omega_{X/\P1})$. Since $f$ is smooth
we have $\deg f_\star(\omega_{X/\P1})=0$ (cf.~[BPV84]) and hence
$f_\star(\omega_{X/\P1}) \simeq\cO_{\P1}$. This gives a section of
$\omega_{X/\P1}$ which has no zeros (restrict to the fibres of $f$).

\than Proposition 6.4|Let $X$ be a compact surface with $a(X)=0$. Then
$T_X$ is nef precisely if $X$ is a Hopf surface or a torus.
\fintha

\proof Assume first that $T_X$ is nef. By the same argument as in 6.3 $X$
cannot be a K3-surface. Hence by classification $X$ is a torus or of type
VII. Assume now that $X$ is of type VII. Since $T_X$ is nef,        
$c_1(X)^2=0$. Moreover $\chi(\cO_X)=0$ and therefore we get by
Riemann-Roch that $c_2(X)=0$. This implies $b_2(X)=0$. By~[Bo83] and~[LYZ90]
$X$ is either a Hopf surface or an Inoue surface.

Every Hopf surface can be covered by a primary Hopf surface, i.e. a
Hopf surface such that \hbox{$\pi_1(X)\simeq\zb$}. By [Kd66b],
a primary Hopf surface is isomorphic to the quotient of
$\cb^2\setminus\lbrace0\rbrace$ by an infinite cyclic group of
automorphisms generated either by a linear automorphism
$$g(z_1,z_2)=(\alpha_1z_1,\alpha_2z_2),~~~~
0<\vert\alpha_1\vert\le\vert\alpha_2\vert<1,$$
or by a polynomial automorphism of the type
$$g(z_1,z_2)=(\lambda^pz_1+z_2^p,\lambda z_2),~~~~
0<\vert\lambda \vert<1,~~p\in\nb\ssm\{0\}.$$
In the first case the directions $\partial/\partial z_1$,
$\partial/\partial z_2$ are invariant and give rise to a splitting 
$T_X=L_1\oplus L_2$ into two line bundles. Set
$c=\log\vert\alpha_2\vert/ \log\vert\alpha_1\vert$ and fix a positive
integer $k$. Then $L_1$, $L_2$ can be equipped with hermitian metrics
associated to the plurisubharmonic weights:
$$\left(\bet{z_1}^{2kc}+\bet{z_2}^{2k}\right)^{-{1\over2kc}}
       \bet{dz_1}^2,\quad\hbox{\rm resp.}\quad
  \left(\bet{z_1}^{2kc}+\bet{z_2}^{2k}\right)^{-{1\over2k}}
       \bet{dz_2}^2.$$
These metrics are of class $C^2$ for $k$ large and have semipositive
curvature. Hence $T_X$ is Griffiths semipositive (with a metric of
class $C^\infty$ only when $c\in\qb$ and $kc\in\nb$).

In the second case, only the first direction $\partial/\partial z_1$ is
invariant and we get instead an exact sequence 
$$0\ra L_1\ra T_X\ra L_2\ra0.$$
Observe that the metric defined on the pull-back of $L_1$, $L_2$ to
$\cb^2\setminus\lbrace0\rbrace$ by
$$\left(\ep^2\bet{z_1}^2+\bet{z_2}^{2p}\right)^{-1}\bet{dz_1}^2,~~~~
\hbox{\rm resp.}~~\left(\ep^2\bet{z_1}^2+\bet{z_2}^{2p}\right)^{-1/p}
\bet{dz_2}^2$$
is almost invariant, since the inequality $2\ep\bet{z_1}\bet{z_2^p}\le
\ep^2\bet{z_1}^2+\bet{z_2}^{2p}$ implies
$$\eqalign{
\left(\bet{\lambda}^{2p}-\ep\bet{\lambda}^p\right)
\left(\ep^2\bet{z_1}^2+\bet{z_2}^{2p}\right)&\le
\ep^2\bet{\lambda^p z_1+z_2^p}^2+\bet{\lambda z_2}^{2p}\cr
&\le\left(\bet{\lambda}^{2p}+\ep\bet{\lambda}^p+\ep^2\right)
\left(\ep^2\bet{z_1}^2+\bet{z_2}^{2p}\right).\cr}$$
We use a partition of unity to glue the weight $w_1(z_1,z_2)=\log
\big(\ep^2\bet{z_1}^2+\bet{z_2}^{2p}\big)$ with its transform $w_2$ by
the automorphism on two overlapping spherical shells, so as to obtain a
well defined metric on $X$. The resulting weight function $w$ can be
made almost plurisubharmonic, with an arbitrary small negative part
$O(\ep)$ in its complex Hessian: apply e.g. Lemma 3.5 of [De91] to 
$$w={1\over k}\log\left(\theta_1^2e^{kw_1}+\theta_2^2e^{kw_2}\right),\quad
k\sim{1\over\ep},$$
after adding bump functions of order of magnitude $O(\ep)$ to $w_1$ and
$w_2$ so that $C_1=C_2=0$. Therefore $L_1$, $L_2$ and $T_X$ are nef
vector bundles by 1.15(ii).

The Inoue surfaces $S_M$ are constructed as follows (see~[In74]). Let 
$M$ be a matrix in $SL_3(\zb)$ admitting one real eigenvalue $\alpha>1$ and
two complex conjugate eigenvalues $\beta\not=\quer{\beta}$. Let $(a_1,a_2,a_3)$
be a real eigenvector of $M$ corresponding to $\alpha$ and let
$(b_1,b_2,b_3)$ be an eigenvector of $M$ corresponding to $\beta$. Then
$X=S_M$ is the quotient of $\cH\times\cb$ (where $\cH$ is the upper
half plane in $\cb$) by the group of affine automorphisms generated by
$$\eqalign{
  g_0(w,z) &= (\alpha w,\beta z), \cr
  g_i(w,z) &= (w+a_i,z+b_i),\quad i=1,2,3.\cr}$$
It is clear that the directions $\partial/\partial w$ and
$\partial/\partial z$ are left invariant by the automorphisms. Hence
they give rise to a splitting $T_X=L_w\oplus L_z$. The line bundles
$L_w$, $L_ z$ can be equipped respectively with the invariant hermitian
metrics
$$\left(\Im\,w\right)^{-2}\bet{dw}^2, \quad  
  \left(\Im\,w\right)\bet{dz}^2$$
(note that $\alpha\bet{\beta}^2=\det M=1$). Their curvature forms are
$$\eqalign{
\Theta(L_w)&= {i\over2\pi} \partial\quer{\partial}\log(\Im\,w)^2 =
    -{i\over4\pi} {dw\wedge d\quer{w} \over (\Im\,w)^2 } \le 0,\cr
\Theta(L_ z)&= {i\over2\pi} \partial\quer{\partial}\log(\Im\,w)^{-1} =
    {i\over8\pi} {dw\wedge d\quer{w} \over (\Im\,w)^2 } \ge 0.\cr}$$
Since $L_w$ is seminegative and not flat (use e.g. a Gauduchon
metric to see this), $T_X$ cannot be nef.

The Inoue surfaces $X=S^+_{N,p,q,r;t}$ are defined as the quotient of
$\cH\times\cb$ by the group of affine automorphisms generated by
$$\eqalign{
  g_0(w,z) &= (\alpha w,z+t),\cr
  g_i(w,z) &= (w+a_i,z+b_iw+c_i),\quad i=1,2, \cr
  g_3(w,z) &= (w,z+{b_1a_2-b_2a_1\over r}),\cr}$$
where $(a_1,a_2)$ and $(b_1,b_2)$ are the eigenvectors of some matrix
$N\in SL_2(\zb)$ admitting real eigenvalues $\alpha>1$, $1\over\alpha$.
Moreover $t\in\cb$ and $p,q,r\,(r\not=0)$ are integers, while $(c_1,c_2)$
depends on $(a_i,b_i),p,q,r$ (see~[In74]). The vector field $\partial/\partial
z$ is invariant by the automorphisms, thus we get an exact sequence 
$$0\ra\cox\ra T_X\ra L_w\ra0.$$
The quotient line bundle $L_w$ can be equipped with the hermitian metric
$(\Im\,w)^{-2}\bet{dw}^2$ which has seminegative curvature but is not flat.
Hence $T_X$ cannot be nef. 

Finally, the Inoue surfaces of type $S^-_{N,p,q,r}$
possess a 2:1 \'etale cover of the type $S^+_{N_1,p_1,q_1,r_1;0}$ and it 
follows that their tangent bundles cannot be nef.

Since every Inoue surface is of type $S_M$, $S^+$ or $S^-$, the proof of
Prop.~6.4 is finished.\finpr

\vskip25pt
\titre 7. K\"ahler 3-folds with nef tangent bundles|

Due to the complexity of the classification in the non K\"ahler case we
will restrict ourselves now to K\"ahler manifolds. In the 3-dimensional case
we can give a complete classification. In the algebraic case this has
been carried out in~[CP91].

\than Theorem 7.1|Let $X$ be a projective 3-fold. Then $T_X$ is nef if and
only if $X$ is up to finite \'etale cover one of the manifolds in the
following list:
\item{\rm(1)} $X=\P3\,;$
\item{\rm(2)} $X=Q_3$, the 3-dimensional quadric$\,;$
\item{\rm(3)} $X=\P1\times\P2\,;$
\item{\rm(4)} $X=\P1\times\P1\times\P1\,;$
\item{\rm(5)} $X=\pb(T_{\P2})\,;$
\item{\rm(6)} $X=\pb(E)$, with $E$ a numerically flat rank $3$ bundle
              over an elliptic curve~$C\,;$
\item{\rm(7)} $X=\pb(E){\times}_C\pb(F)$, with $E$, $F$ numerically flat
              rank $2$ bundles over an elliptic curve $C\,;$
\item{\rm(8)} $X=\pb(E)$, with $E$ a numerically flat rank $2$ bundle over
              an abelian surface$\,;$
\item{\rm(9)} $X={}$ abelian variety.
\fintha

In the non algebraic case the classification is as follows.

\than Theorem 7.2|Let $X$ be a non algebraic 3-dimensional compact K\"ahler
manifold. Then $T_X$ is nef if and only if $X$ is up to finite \'etale
cover either a torus or of the form $\pb(E)$, where $E$ is a numerically
flat rank 2-bundle over a 2-dimensional torus.
\fintha

The proof is divided into several steps. First we treat the case that
$q(X)=0$. 

\than Proposition 7.3| Let $X$ be a compact K\"ahler 3-fold with $T_X$ nef
and $q(X)=0$. Then $X$ is projective.
\fintha

\proof By a well-known result of Kodaira it is enough to show that
$H^2(X,\cO_X)=0$. Assume that $H^2(X,\cO_X)\not=0$. This gives non zero
sections of $\Omega_X^2$. These sections have no zeros since $\Lambda^2T_X$ is
nef by~1.12. This provides us with an exact sequence
$$0\ra\cO_X\ra\Omega_X^2\ra V\ra0,$$
where $V$ is a rank 2-bundle on $X$. Tensoring this sequence by $K_X^{-1}$
gives a sequence
$$0\ra K_X^{-1}\ra T_X\ra W\ra0$$
with $\det W\cong\cO_X$. Since $W$ is nef all Chern classes of $W$ vanish.
In particular $c_2(X)=0$. By Riemann-Roch we have
$\chi(\cO_X)={1\over24}c_1(X)c_2(X)=0$. On the other hand
$\chi(\cO_X)=1+h^2(\cO_X)-h^3(\cO_X)$. But $h^3(\cox)=h^0(K_X)\le1$,
$-K_X$ being nef. Therefore we obtain $h^2(\cox)\le0$,
a contradiction.\finpr

\vskip 0.2 truecm
We should remark that 7.3 does not extend to arbitrary dimension. But it
does extend if we assume ${\wt q}(X)=0$, see 3.10, 3.11.
For the case of positive irregularity we need 

\than Lemma 7.4|Let $X$ be a 3-dimensional compact K\"ahler manifold with
$T_X$ nef and $q(X)>0$. Then the Albanese map $\alpha:X\ra A(X)$ is smooth
and surjective and the fibres have nef tangent bundles. If
$\wdt{q}(X)=q(X)$, any fibre $F$ of $\alpha$ satisfies $\wdt{q}(F)=0$.
\fintha

\proof Special case of 3.9 and 3.12.\finpr

\vskip 0.2 truecm
\noindent{\sl Proof of Theorem 7.2.}
Let $T_X$ be nef. By 7.3 we have $q(X)\ge1$.
Let $\alpha:X\ra A$ be the Albanese map. By 7.4 the map $\alpha$ is smooth
and surjective. We may assume that $\dim A\le2$ and $\wdt{q}(X)=q(X)$.
\vskip 0.2 truecm
\noindent First we treat the case $\dim A=1$. Let $F$ be a fibre of 
$\alpha$. By 7.4
we have $\wdt{q}(F)=0$ and $T_F$ is nef. By 6.1 and 6.2 the fibre $F$ is
either $\P2$ or $\P1\times\P1$. Hence $X$ is either a $\P2$-bundle or a
$\P1\times\P1$-bundle over an elliptic curve. Therefore $X$ is algebraic
(use~[CP91], 7.2 in the case of $\P1\times\P1$).

Now let $\dim A=2$. Arguing as in the first case $X$ is a $\P1$-bundle
over~$A$. Then after finite \'etale cover $X$ is of the form $\pb(E)$, $E$
a flat 2-bundle over a 2-dimensional torus. This can be proved as in
([CP91], 8.2) with the following modification: instead of quoting Donaldson we
use the existence of a Hermite-Einstein metric on a stable vector bundle
over a K\"ahler manifold, due to Uhlenbeck and Yau~[UY86].\finpr
\vskip25pt

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\tenpoint\vskip5pt
\noindent{\sl E-mail addresses:}\hfill\break
demailly@ccomm.grenet.fr\hfill\break
thomas.peternell@uni-bayreuth.dbp.de\hfill
michael.schneider@uni-bayreuth.dbp.de
\vskip15pt
\tenpoint\noindent(October 15, 1991, revised on August 19, 1992)


\end

