% Proceedings of the Conference in honor of Narasimhan and Seshadri
% J.P. Demailly, Th. Peternell, M. Schneider
% TeX file

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

\def\than{\tha\noindent}
\def\rem#1|{\remarque\noindent{\petcap#1}|}
\def\proof{\noindent{\sl{Proof.}\ }}
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\def\tri{{\,\triangleright\!\raise1pt\hbox{$\scriptstyle{<}$}\, }}
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\def\ep{{\varepsilon}}
\def\ssm{{\smallsetminus}}
\def\lp{\left(}
\def\rp{\right)}
\def\wtH{{\widetilde{H}}}
\def\cOE{{\cO_E(1)}}
\def\cOD{{\cO(D)}}
\def\hse{{\widetilde{h}_\ep}}
\def\hhe{{\widehat{h}_\ep}}
\def\om{{\omega}}
\def\Id{{\rm Id}}
\def\ppq{{\partial\quer{\partial}}}
\def\con{{\cb\ssm\lbrace0\rbrace}}
\def\Thh{{\Theta_h}}
\def\Thhe{{\Theta_{h_\ep}}}
\def\Thhm{{\Theta_{h_m}}}
\def\Thha{{\Theta_{h_a}}}
\def\quer#1{{\overline{#1}}}
\def\bet#1{{\vert #1\vert}}
\def\wdt#1{{\widetilde{#1}}}
\def\df{{\det\cF}}
\def\dfr{{(\df)^{-1}}}
\def\cC{ {\cal C} }
\def\cE{ {\cal E} }
\def\cF{ {\cal F} }
\def\cG{ {\cal G} }
\def\cH{ {\cal H} }
\def\cO{ {\cal O} }
\def\cS{ {\cal S} }
\def\cG{ {\cal G} }
\def\cL{ {\cal L} }
\def\Ricci{{\rm Ricci}}
\def\Alb{{\rm Alb}}
\def\Vol{{\rm Vol}}
\def\Pic{{\rm Pic}}
\def\alb{{\rm alb}}
\def\deg{{\rm deg}\,}
\def\rank{{\rm rank}}
\def\diam{{\rm diam}\,}
\def\ort{\hbox{~\vrule width4.0pt height0.4pt depth0pt
                \vrule width0.4pt height6.0pt depth0pt~$\,$}}
\let\la=\longrightarrow
\let\ov=\overline
\let\wt=\widetilde
\def\cox{{\cO_X}}
\def\PE{ {\pb(\cE) }}
\def\P#1{{\pb_{#1}}}
\def\ch{\mathop{\rm ch}}    
\def\Todd{\mathop{\rm Todd}}    
\def\mod{\mathop{\rm mod}}    
\def\Aut{\mathop{\rm Aut}}    
\def\Hom{\mathop{\rm Hom}}    
\def\codim{\mathop{\rm codim}}    
\def\Vol{{\mathop{\rm Vol}}}    
\def\rank{\mathop{\rm rank}}    
\def\red{\mathop{\rm red}\,}    
\def\rk{\mathop{\rm rk}}    
\def\Pic{\mathop{\rm Pic}}    
\def\supp{\mathop{\rm supp}}    
\def\ra{\rightarrow}
\def\Ra{{\longrightarrow}}
\def\Longhook{\lhook\joinrel\relbar\joinrel\relbar
                       \joinrel\relbar\joinrel\rightarrow}
\def\sp#1{{\buildrel {#1}\over\wedge }}                       
\def\Pn{$\pb_n$}
\def\wx{{\widetilde X}}
\def\wp{{\widetilde\varphi}}
\def\chiox{{\chi({\cal O}_X)}}
\def\GaE{{\Gamma^aE}}
\def\SmE{{S^mE}}
\def\wg{{\widetilde{G}}}
\def\wh{{\widetilde{H}}}
\def\wy{{\widetilde{Y}}}
\def\ww{{\widetilde{\omega}}}
\def\wf{\wdt{F}}
\def\wq{\wdt{q}}
\def\xd{{\widehat X}}
\def\ld{{\widehat\cL}}
\def\eq{{\quer{E}}}
\def\cq{{\quer{C}}}
\def\Oaq{{\quer{\Omega_a(E)}}}
\def\xox{{(X,\cO_X)}}
\def\xomx{X,\Omega_X}
\def\kme{{K_X^{-1}}}
\def\qq{{\wdt{q}}}
\def\Fq{{\wdt{F}}}
\def\rx{{\rb_X}}
\def\axy{{A(X/Y)}}
\def\edete{{E\otimes(\det E)^{-1/r}}}
\def\kxe{K_X^{-1}}

\centerline{\bftwelve Compact complex manifolds whose tangent bundles}\vskip4pt
\centerline{\bftwelve satisfy numerical effectivity properties}

\vskip15pt
\centerline{\sl by Jean-Pierre Demailly${}^\star$}
\centerline{\sl $($joint work with Thomas Peternell${}^\dagger$ and
Michael Schneider${}^\dagger)$}
\vskip5pt

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

}}
\vskip15pt
\noindent Dedicated to Prof. M.S. Narasimhan and C.S. Seshadri on their
sixtieth birthday
\vskip15pt
\titre 0. Introduction|

A compact Riemann surface always has a hermitian metric with constant
curvature, in particular the curvature sign can be taken to be
constant: the negative sign corresponds to curves of general type
(genus $\ge 2$), while the case of zero curvature corresponds to
elliptic curves (genus~$1$), positive curvature being obtained only
for~$\pb^1$ (genus~$0$). 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 strengthened
Frankel's conjecture and asserted that \Pn\ is the only compact complex
manifold with ample tangent bundle. In his famous paper~[Mo79], Mori solved
Hartshorne's 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 semi-positive
holomorphic bisectional curvature.

From the point of view of algebraic geometry, it is natural to consider
the class of projective manifolds $X$ whose tangent bundle is numerically
effective (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 main purpose of this work is to investigate compact (most often K\"ahler)
manifolds with nef tangent or anticanonical bundles in arbitrary dimension.
We first discuss some basic properties of nef vector bundles
which will be needed in the sequel in the general context of compact
complex manifolds. We refer to [DPS91] and [DPS92] for detailed proofs.
Instead, we put here the emphasis on some unsolved questions.
\vskip15pt

\titre 1. Numerically effective vector bundles|

In algebraic geometry a powerful and flexible notion of semi-positivity
is  {\sl numerical effectivity} ("nefness"). We will explain here how
to extend this notion to arbitrary compact complex manifolds.

\than Definition 1.1|A line bundle $L$ on a projective manifold $X$ is
said to be numerically effective $($nef for short$)$ if $L\cdot C\ge0$
for all compact curves $C\subset X$.\fintha

It is clear that a line bundle with semi-positive curvature is nef. The
converse had been conjectured by Fujita~[Fu83]. Unfortunately this is
not true, a simple counterexample can be obtained as follows:

\remarque Example 1.2|Let $\Gamma$ be an elliptic curve and let $E$ be
a rank $2$ vector bundle over $\Gamma$ which is a non split extension of
$\cO$ by~$\cO\,$; such a bundle $E$ can be described as the locally constant
vector bundle over $\Gamma$ whose monodromy is given by the matrices
$$\pmatrix{1&0\cr 0&1\cr},~~~~\pmatrix{1&1\cr 0&1\cr}$$
associated to a pair of generators of $\pi_1(\Gamma)$.
We take $L=\cO_E(1)$ over the ruled surface~$X=\pb(E)$.
Then $L$ is nef and it can be checked that, up to a positive constant
factor, there is only one (possibly singular) hermitian metric on $L$
with semi-positive curvature; this metric is unfortunately singular and
has logarithmic poles along a curve. Thus $L$ cannot be semi-positive
for any smooth hermitian metric.\finpr

\than Definition 1.3|A vector bundle $E$ is called nef if the line bundle
$\cO_E(1)$ is nef on $\pb(E)$ $(={}$~projectivized bundle of hyperplanes
in the fibres of~$E)$. \fintha

Again it is clear that a vector bundle $E$ which admits a metric with
semi-positive curvature (in the sense of Griffiths) is nef. A compact
K\"ahler manifold $X$ having semi-positive holomorphic bisectional
curvature has by definition a tangent bundle $TX$ with semi-positive
curvature. Again the converse does not hold. 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 1.4|Let $X$ be a compact complex manifold with a fixed
hermitian metric $\omega$. A line bundle $L$ over $X$ is {\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 ($X$ may have
no curves at all, as is the case for instance for generic complex tori).
For projective algebraic $X$ both notions coincide; this is an easy
consequence of Seshadri's ampleness criterion: take $L$ to be a nef line
bundle in the sense of Def.~1 and let $A$ be an ample line bundle; then
$L^{\otimes k}\otimes A$ is ample for every integer $k$ and thus $L$ has
a smooth hermitian metric with curvature form $\Theta(L)\ge-{1\over
k}\Theta(A)$.

Definition~1.3 can still be used to define the notion of nef vector
bundles over arbitrary compact manifolds. If $(E,h)$ is a hermitian
vector bundle, recall that 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 $\Hom(E,E)$. We
say that $(E,h)$ is {\sl semi-positive in Griffiths' sense} [Gr69]
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 $\xi\not=0, v\not=0$. Numerical effectivity
can then be characterized by the following differential geometric criterion
(see [De91]).

\than Criterion 1.5|Let $\om$ be a fixed hermitian metric on $X$. A
vector bundle $E$ on $X$ is nef if and only if there is a sequence of
hermitian metrics $h_m$ on $S^mE$ and a sequence $\varepsilon_m$ of
positive numbers decreasing to~$0$ such that
$$\Thhm(S^mE)\ge-m\ep_m\om\otimes\Id_{S^mE}$$ in the sense of Griffiths.
\fintha

The main functorial properties of nef vector bundles
are summarized in the following proposition.

\than Proposition 1.6|Let $X$ be an arbitrary compact complex manifold and
let $E$ be a holomorphic vector bundle over~$X$.
\item{\rm(i)} If $f:Y\ra X$ is a holomorphic map with equidimensional
fibres, then $E$ is nef if and only if $f^\star E$ nef.
\item{\rm(ii)} 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 $\Gamma^aE$ is nef. In particular,
all symmetric and exterior powers of $E$ are nef.
\item{\rm(iii)} let $F$ be a holomorphic vector bundle over~$X$.
If $E$ and $F$ are nef, then $E\otimes F$ is nef.
\item{\rm(iv)} If some symmetric power $\SmE$ is nef ($m>0$), then
$E$ is nef.
\item{\rm(v)} Let $0\ra F\ra E\ra Q\ra 0$ be an exact sequence
of holomorphic vector bundles over~$X$. Then 
\itemitem{$(\alpha)$} $E$ nef ${}\Rightarrow{}$ $Q$ nef.
\itemitem{$(\beta)$} $F$, $Q$ nef ${}\Rightarrow{}$ $E$ nef.
\itemitem{$(\gamma)$} $E$ nef, $(\det Q)^{-1}$ nef ${}\Rightarrow{}$ $F$ nef.
\fintha

The proof of these properties in the general analytic context can be
easily obtained by curvature computations. The arguments are parallel
to those of the algebraic case and will therefore be omitted (see
[Ha66] and [CP91] for that case). Another useful result which will be
used over and over in the sequel is

\than Proposition 1.7|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 We merely observe that if $h_m$ is a sequence of hermitian metrics
on $S^mE$ as in criterion~5, then
$$T_m={i\over\pi}\partial\overline\partial{1\over m}\log||\sigma^m||_{h_m}$$
has zero $\partial\overline\partial$-cohomology class and
satisfies $T_m\ge-\varepsilon_m\omega$. It follows that $T_m$ converges to
a weak limit $T\ge 0$ with zero cohomology class. Thus
$T=i\partial\overline\partial\varphi$ for some global plurisubharmonic
function $\varphi$ on~$X$. By the maximum principle this implies~$T=0$.
However, if $\sigma$ vanishes at some point~$x$, then all $T_m$ have Lelong
number ${}\ge 1$ at~$x$. Therefore so has $T$, contradiction.\finpr
\vskip7pt

One of our key results is a characterization of vector bundles $E$ which are
numerically flat, i.e. such that both $E$ and $E^\star$ are nef.

\than Theorem 1.8|Suppose that $X$ is K\"ahler. Then a holomorphic
vector bundle $E$ over $X$ is numerically flat iff $E$ 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

\remarque Sketch of proof|It is clear by 1.6~(v) that every vector bundle
which is filtrated with hermitian flat quotients is nef as well as its dual.
Conversely, suppose that $E$ is numerically flat. This assumption implies
$c_1(E)=0$. Fix a K\"ahler metric~$\omega$. If $E$ is $\omega$-stable, then
$E$ is Hermite-Einstein by the Uhlenbeck-Yau theorem [UY86].
Moreover we have $0\le c_2(E)\le c_1(E)^2$ by Theorem~1.9 below, so
$c_2(E)=0$. Kobayashi's flatness criterion derived from L\"ubke's
inequality on Chern classes then implies that $E$ is hermitian flat.
Now suppose that $E$ is unstable and take $\cF\subset\cO(E)$ to be a 
destabilizing subsheaf of minimal rank~$p$. We then have by definition
$c_1(\cF)=c_1(\det\cF)=0$ and the morphism $\det\cF\to\Lambda^pE$ cannot
have any zero (otherwise the zeroes would contribute to give a
semi-negative non zero curvature current on the line bundle $\det\cF$,
contradiction). This implies easily that $\cF$ is locally free, and we infer
that $\cF$ is also numerically flat. Since $\cF$ is stable by definition,
$\cF$ must be hermitian flat. We set $E_1=\cF$, observe that $E'=E/E_1$ is
again numerically flat and proceed by induction on the rank.\finpr
\vskip7pt

Another key point, which has been indeed used in the above proof, is the
fact that the Fulton-Lazarsfeld inequalities~[FL83] for Chern classes
of ample vector bundles still hold for nef vector bundles over compact
K\"ahler manifolds:

\than Theorem~1.9|Let $(X,\omega)$ be a compact K\"ahler manifold and
let $E$ be a nef vector bundle on $X$. Then for all positive
polynomials~$P$ the cohomology class $P(c(E))$ is numerically positive,
that is, $\int_Y P(c(E))\wedge\omega^k\ge 0$ for any $k$ and any
subvariety $Y$ of~$X$.\fintha

By a positive polynomial in the Chern classes, we mean as usual a homogeneous
weighted polynomial $P(c_1,\ldots,c_r)$ with deg$\,c_i=2i$, such that $P$
is a positive integral combination of Schur polynomials:
$$P_a(c)=\det(c_{a_i-i+j})_{1\le i,j\le r},~~~~
r\ge a_1\ge a_2\ge\ldots\ge a_r\ge0$$
(by convention $c_0=1$ and $c_i=0$ if $i\not=[0,r]$, $r={}$rank$\,E$).
The proof of Theorem~1.9 is based essentially on the same arguments as
the original proof of [FL83] for the ample case: the starting point is
the nonnegativity of all Chern classes $c_k(E)$ (Bloch-Gieseker [BG71]);
the general case then follows from a formula of Schubert calculus known
as the Kempf-Laksov formula~[KL74], which expresses any Schur polynomial
$P_a(c(E))$ as a Chern class $c_k(F_a)$ of some related vector
bundle~$F_a$. The only change occurs in the proof of Gieseker's result,
where the Hard Lefschetz theorem is needed for arbitrary K\"ahler
metrics instead of hyperplane sections (fortunately enough, the
technique then gets simplified, covering tricks being eliminated). Since
$c_1c_{k-1}-c_k$ is a Schur polynomial, we get by induction
$$0\le c_k(E)\le c_1(E)^k~~~~\hbox{\rm for all}~k.$$
Therefore all Chern monomials are bounded above by corresponding powers
$c_1(E)^k$ of the same degree, and we infer:

\than Corollary 1.10|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
\vskip10pt

\titre 2. Compact K\"ahler manifolds with nef anticanonical line bundle|
  
Compact K\"ahler manifolds with zero or semi-positive Ricci curvature have been 
investigated by various authors (cf. [Ca57], [Ko61], [Li67,$\,$71,$\,$72],
[Bo74a,$\,$b], [Be83], [Ko81] and [Kr86]). The purpose of this section
is to discuss the following two conjectures.

\than Conjecture 2.1|Let $X$ be a compact K\"ahler manifold with numerically
effective anticanonical bundle $K_X^{-1}$. Then the fundamental group
$\pi_1(X)$ has polynomial growth.\fintha

\than Conjecture 2.2|Let $X$ be a compact K\"ahler manifold with $K_X^{-1}$
numerically effective. Then the Albanese map $\alpha : X \to \Alb(X)$
is a smooth fibration onto the Albanese torus.\fintha

These conjectures are known to be true if $K_X^{-1}$ is semi-positive.
In both cases, the proof is based on differential geometric techniques
(see e.g. [Bi63], [HK78] for Conjecture~2.1 and [Li71] for Conjecture~2.2).
However, the methods of proof are not so easy to carry over to the nef case.
Our main contributions to these conjectures are derived from Theorem~2.3
below.

\than Theorem 2.3|Let $X$ be a compact K\"ahler manifold with $K_X^{-1}$ nef.
Then $\pi_1 (X)$ is a group of subexponential growth.\fintha

The proof actually gives the following additional fact (this was already
known before, see [Bi63]).

\than Corollary 2.4|If moreover $-K_X$ is hermitian semi-positive,
then $\pi_1(X)$ has polynomial growth of degree ${}\le 2 \dim X$,
in particular $h^1 (X,\cO_X) \le \dim X$.\fintha 

As noticed by F.~Campana (private communication), Theorem~2.3 also implies
the following consequences.

\than Corollary 2.5|Let $X$ be a compact K\"ahler manifold with $K_X^{-1}$
nef. Let $\alpha:X\to\Alb(X)$ be the Albanese map and set
$n=\dim X$, $d=\dim\alpha(X)$. If $d=0,1$ or $n$, then $\alpha$ is
surjective. The same is true if $d=n-1$ and if $X$ is projective
algebraic.\fintha

\than Corollary 2.6|Let $X$ be a K\"ahler surface or a projective $3$-fold
with $K_X^{-1}$~nef. Then the Albanese map $\alpha:X\to\Alb(X)$ is
surjective.\fintha

We now explain the main ideas required in the proof of Theorem~2.3.
If $G$ is a finitely generated group with generators $g_1,\ldots,g_p$,
we denote by $N(k)$ the number of elements $\gamma\in G$ which can be
written as words
   $$\gamma = g_{i_1}^{\varepsilon_1}\ldots g_{i_k}^{\varepsilon_k},~~~~
                                  \varepsilon_j = 0,~1~~\hbox{\rm or}~~-1$$
of length ${}\le k$ in terms of the generators. The group $G$ is said
to have {\sl subexponential growth} if for every $\varepsilon > 0$ there is
a constant $C(\varepsilon)$ such that
   $$N(k) \le  C(\varepsilon)e^{\varepsilon k}~~~~\hbox{\rm for}~~k\ge 0.$$
This notion is independent of the choice of generators. In the free
group with  two generators, we have $N(k) = 1+4(1+3+3^2+...+3^{k-1}) =
2\cdot 3^k -1$, thus a group with subexponential growth cannot
contain a non abelian free subgroup.

 The first step consists in the construction of suitable K\"ahler
metrics by means of the Calabi-Yau theorem [Y77]. Let $\omega$ be a
fixed K\"ahler metric on~$X$. Since $K_X^{-1}$ is nef, for every
$\varepsilon >0$ there exists a smooth hermitian metric $h_\varepsilon$
on $K_X^{-1}$ such that
   $$ u_\varepsilon = \Theta_{h_\varepsilon} (K_X^{-1}) \ge
                                                  -\varepsilon\omega.$$
By [Y77] and [Y78] there exists a unique K\"ahler metric
$\omega_\varepsilon$ in the cohomology class $\{\omega\}$ such that
   $$\Ricci(\omega_\varepsilon) = -\varepsilon\omega_\varepsilon +
                            \varepsilon\omega + u_\varepsilon.\leqno(+)$$
In fact $u_\varepsilon$ belongs to the Ricci class $c_1 (K_X^{-1}) =
c_1 (X)$, hence so does the right hand side
$-\varepsilon\omega_\varepsilon + \varepsilon\omega + u_\varepsilon$.
In particular there exists a function $f_\varepsilon$ such that 
   $$u_\varepsilon = \Ricci(\omega) + i\partial\ov\partial f_\varepsilon.$$
If we set $\omega_\varepsilon = \omega +i\partial\ov\partial\varphi$ (where
$\varphi$ depends on $\varepsilon$), equation $(+)$ is equivalent to the 
Monge-Amp\`ere equation
   $$ {(\omega +i\partial\ov\partial\varphi)^n\over\omega^n}=
                        e^{\varepsilon\varphi -f_\varepsilon} \leqno(++)$$
because
   $$\eqalign{
   i\partial\ov\partial \log (\omega +i\partial\ov\partial\varphi)^n/\omega^n
   & = \Ricci(\omega) - \Ricci(\omega_\varepsilon)\cr
   & = \varepsilon(\omega_\varepsilon-\omega) + \Ricci(\omega)-u_\varepsilon\cr
   & = i\partial\ov\partial (\varepsilon\varphi - f_\varepsilon).\cr}$$
It follows from the general results of [Y78] that (++) has a unique
solution~$\varphi$, thanks to the fact the right hand side of (++) is
increasing in~$\varphi$. Since $u_\varepsilon \ge -\varepsilon\omega$,
equation (+) implies in particular that $\Ricci(\omega_\varepsilon) \ge
-\varepsilon\omega$.

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

\than Lemma 2.7|Let
  $$ \Phi : T_a\wt M \to \wt M,~~~~\Phi (\zeta)=\exp_a (\zeta) $$
be the (geodesic) exponential map. Denote by 
  $$ \Phi^*dV_g  = a(t,\zeta)\,dt\, d\sigma (\zeta)$$
the expression of the volume element in spherical coordinates with
$t\in\rb_+$ and $\zeta\in S_a (1)={}$ unit sphere in~$T_a\wt M$. Suppose
that $a(t,\zeta)$ does not vanish for $t\in{}]0,\tau (\zeta)[\,$, with
$\tau (\zeta)=+\infty$ or $a(\tau (\zeta),\zeta)=0$. Then $b(t,\zeta)=
a(t,\zeta)^{1/(m-1)}$ satisfies on $]0,\tau (\zeta)[$ the inequality
   $${\partial^2\over\partial t^2}\,b(t,\zeta) +
     {1\over m-1}\,\Ricci_g (v(t,\zeta),v(t,\zeta))b(t,\zeta) \le 0$$
where
   $$v(t,\zeta) = {d\over dt}\,\exp_a (t\zeta) \in  S_{\Phi(t\zeta)}(1)
                        \subset  T_{\Phi(t\zeta)}\wt M.$$\fintha

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

In our application, the difficulty is that the metric
$g=\omega_\varepsilon$ varies with $\varepsilon$ as well as the
constants $\alpha =\alpha_\varepsilon$, $\beta =\beta_\varepsilon$
in~$(*)$, and $\alpha_\varepsilon \sqrt{(m-1)\varepsilon}$ need not
converge to $0$ as $\varepsilon$ tends to~$0$. We overcome this
difficulty by the following lemma.

\than Lemma 2.8|Let $U_1$, $U_2$ be compact subsets of~$\wt X$. Then for every
$\delta >0$, there are closed subsets $U_{1,\varepsilon,\delta} \subset U_1$
and $U_{2,\varepsilon,\delta} \subset U_2$ with $\Vol_\omega (U_j \ssm
U_{j,\varepsilon,\delta}) < \delta$, such that any two points
$x_1\in U_{1,\varepsilon,\delta}$, $x_2\in U_{2,\varepsilon,\delta}$ can be
joined by a path of length ${}\le C\,\delta^{-1/2}$ with
respect to $\omega_\varepsilon$, where $C$ is a constant independent of
$\varepsilon$ and~$\delta$.\fintha

We will not explain the details. The basic observation is that
   $$\int_X \omega_\varepsilon \wedge \omega^{n-1} = \int_X \omega^n$$
does not depend on $\varepsilon$, therefore $\Vert\omega_\varepsilon
\Vert_{L^1(X)}$ is uniformly bounded. This is enough to imply the
existence of sufficiently many paths of bounded length between random
points taken in $X$ (this is done for example by computing the average
length of piecewise linear paths).\finpr

We let $U$ be a compact set containing the fundamental domain~$E$,
so large that $U^\circ\cap g_j (U^\circ)\ne \emptyset$ for each
generator~$g_j$. We apply Lemma~2.8 with $U_1 =U_2 =U$ and 
$\delta >0$ fixed such that
    $$\delta <{1\over 2} \Vol_\omega (E),~~~~
                 \delta <{1\over 2} \Vol_\omega\big(U\cap g_j (U)\big).$$
We get $U_{\varepsilon,\delta}\subset U$ with $\Vol_\omega(U\ssm
U_{\varepsilon,\delta}) < \delta$ and $\diam_{\omega_\varepsilon}
(U_{\varepsilon,\delta})\le C\delta^{-1/2}$. The ine\-qualities on volumes
imply that $\Vol_\omega(U_{\varepsilon,\delta}\cap E)\ge{1\over 2}
\Vol_\omega (E)$ and $U_{\varepsilon,\delta}\cap g_j (U_{\varepsilon,\delta})
\ne \emptyset$ for every~$j$ (note that all $g_j$ preserve volumes).
It is then clear that
   $$W_{k,\varepsilon,\delta}:=\bigcup_{\gamma\in\pi_1(X),~
               {\rm length}(\gamma)\le k} \gamma (U_{\varepsilon,\delta})$$
satisfies
   $$\eqalign{
      \Vol_\omega(W_{k,\varepsilon,\delta})&{}\ge N(k)\,\Vol_\omega
      (U_{\varepsilon,\delta}\cap E)\ge N(k)\,{1\over 2}\,\Vol_\omega(E)~~~~
      \hbox{\rm and}\cr
      \diam_{\omega_\varepsilon}(W_{k,\varepsilon,\delta})
                &{}\le k\,\diam_{\omega_\varepsilon} U_{\varepsilon,\delta}
                                      \le kC\delta^{-1/2}.\cr}$$
Since $m=\dim_\rb X=2n$, inequality $(**)$ implies
   $$\Vol_{\omega_\varepsilon}(W_{k,\varepsilon,\delta})
              \le \Vol_{\omega_\varepsilon}\big(B(a,kC\delta^{-1/2})\big)
              \le C_4 k^{2n} e^{C_5 \sqrt{\varepsilon}\,k}.$$
Now $X$ is compact, so there is a constant $C(\varepsilon)>0$ such that
$\omega^n\le C(\varepsilon)\omega_\varepsilon^n$. We conclude that
   $$N(k) \le {2\,\Vol_\omega(W_{k,\varepsilon,\delta})\over \Vol_\omega(E)}
          \le C_6 C(\varepsilon)k^{2n}e^{C_5 \sqrt{\varepsilon}\,k}.$$
The proof of Theorem~2.3 is complete.\finpr

\rem Remark 2.9|It is well known and easy to check that equation (++) implies
    $$C(\varepsilon) \le \exp\Big(\max_X f_\varepsilon -
                                         \min_X f_\varepsilon\Big).$$
Therefore it is reasonable to expect that $C(\varepsilon)$ has polynomial
growth in $\varepsilon^{-1}\,$; this would imply that $\pi_1 (X)$ has
polynomial growth by taking $\varepsilon =k^{-2}$. When $K_X^{-1}$
has a semipositive metric, we can even take $\varepsilon =0$ and find a
metric $\omega_0$ with $\Ricci(\omega_0)=u_0\ge 0$. This implies
Corollary~2.4.
\rem Proof of Corollary 2.5|If $d=0$, then by definition $H^0(X,\Omega^1_X)=0$
and $\Alb(X)=\{0\}$.

If $d=n$, the Albanese map has generic rank $n$, so there exist
holomorphic $1$-forms $u_1,\ldots,u_n$ such that $u_1\wedge\ldots\wedge u_n
\not\equiv 0$. However $u_1\wedge\ldots\wedge u_n$ is a section of $K_X$
which has nef dual, so $u_1\wedge\ldots\wedge u_n$ cannot vanish by
Proposition~1.7 and $K_X$ is trivial. Therefore $u_1\wedge\ldots\hat u_k\ldots
\wedge u_n\wedge v$ must be a constant for every holomorphic $1$-form $v$
and $(u_1,\ldots,u_n)$ is a basis of $H^0(X,\Omega^1_X)$.
This implies $\dim A(X)=n$, hence $\alpha$ is surjective.

If $d=1$, the image $C=\alpha(X)$ is a smooth curve. The genus $g$ of
$C$ cannot be ${}\ge 2$, otherwise $\pi_1(X)$ would be mapped
onto a subgroup of finite index in $\pi_1(C)$, and thus
would be of exponential growth, contradicting Theorem~2.3. Therefore
$C$ is an elliptic curve and is a subtorus of~$\Alb(X)$. By the
universal property of the Albanese map, this is possible only if
$C=\Alb(X)$.

The case $d=n-1$ is more subtle and uses Mori theory (this is why we
have to assume $X$ projective algebraic). We refer to [DPS92] for the
details.\finpr
\vskip15pt

\titre 3. Compact complex manifolds with nef tangent bundles|

Several interesting classes of such manifolds are produced by the following
simple observation.

\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 $TX$,
so $TX$ 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 $Q_n$\hfill\break
(all are Fano manifolds, i.e. 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 $\cb^n\ssm\lbrace0\rbrace/H$ 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 for $p\ge3$, although $TX$ 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 $TX$ is
nef (in fact $TX$ is unitary flat). Since the pull-back of $TX$ to
$\Gamma\times\Gamma\times\Gamma$ is trivial, we easily conclude that
$TX$ 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  $H^0(X,\Omega_X^1)=0$.

\rem Example 3.4|Let $X$ be the ruled surface $\pb(E)$ over the
elliptic curve $\Gamma=\cb/(\zb+\zb\tau)$ defined in Example~1.2. 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 $TX$ 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 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 semi-positive. Observe that no power of
$K_X^{-1}$ is generated by sections, although $K_X^{-1}$ is nef.\finpr
\vskip7pt

Our main result is a structure theorem on the Albanese map of compact
K\"ahler manifolds with nef tangent bundles.

\than Main Theorem 3.5|Let $X$ be a compact K\"ahler manifold with nef
tangent bundle $TX$. 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

Recall that a Fano manifold is by definition a compact complex manifold
with ample anticanonical bundle~$K_X^{-1}$. It is well known that Fano
manifolds are always simply connected (Kobayashi [Ko61]).
As a consequence we get

\than Corollary 3.6|With the assumtions of $3.5$, the fundamental
group $\pi_1(X)$ is an extension of a finite group by $\zb^{2q}$.\fintha

In order to complete the classification of compact K\"ahler manifolds
with nef tangent bundles (up to finite \'etale cover), a solution of
the following two conjectures would be needed.

\than Conjecture 3.7 {\rm(Campana-Peternell [CP91])}|Let $X$ be a Fano
manifold. Then $X$ has a nef tangent bundle if and only if $X$ is
rational homogeneous.\fintha

The evidence we have for Conjecture~3.7 is that it is true up to
dimension~3. In dimension 3 there are more than 100 different
types of Fano manifolds, but only five types have a nef tangent bundle,
namely $\pb_3$, $Q_3$ (quadric), $\pb_1\times\pb_2$, $\pb_1\times\pb_1
\times\pb_1$ and the flag manifold $F_{1,2}$ of lines and planes
in~$\cb^3\,$; all five are homogeneous.

A positive solution to Conjecture~3.7 would clarify the structure
of fibers in the Albanese map of Theorem~3.5. To get a complete picture
of the situation, we would still need to know how the fibers are deformed
and glued together to yield a holomorphic family over the Albanese torus.
We note that $K_\wx^{-1}$ is relatively ample, thus for $m$ large
the fibers can be embedded in the projectivized bundle of the direct
image bundle $\alpha_\star(K_\wx^{-m})$.

\than Conjecture 3.8|In the situation of Theorem~3.5, all direct image
bundles $\alpha_\star(K_\wx^{-m})$ are numerically flat over the
Albanese torus.\fintha

It is easy to check that Conjecture~3.8 holds if the fibers of $\alpha$
are projective spaces or quadrics, but the general case seems to be
much more involved. We now explain the main steps in the proof of
Theorem~3.5. One of the key points is the following

\than Proposition 3.9|Let $X$ be a compact K\"ahler $n$-fold with $TX$ 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 $q(\wx)>0$.\vskip1pt
\fintha

\proof We first check that every effective divisor $D$ of $X$ is nef. In
fact, 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 zeroes. 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, $J^k\cOD$ has a filtration whose graded bundle is
$\bigoplus_{0\le p\le k}S^pT^\star X\otimes\cOD$, so
$(J^k\cOD)^\star\otimes\cOD$ has a dual filtration with graded bundle
$\bigoplus_{0\le p\le k}S^pTX$. By 1.6~(ii) and 1.6~(v)$(\beta)$, we conclude
that $(J^k\cOD)^\star\otimes\cOD$ is nef, so its quotient $\cOD$ is nef
by 1.6~(v)$(\alpha)$.

Part (i) is based on the solution of the Grauert-Riemenschneider conjecture
as proved in~[De85]. Namely, $L=K_X^{-1}=\Lambda^nTX$ is nef and satisfies
$c_1(L)^n>0$, so $L$ has Kodaira dimension $n$ (holomorphic Morse
inequalities are needed at that point because $X$ is not supposed a priori
to be algebraic). It follows that $X$ is Moishezon, thus projective
algebraic, and for $m>0$ large we have $L^m=\cO(D+A)$ with divisors $D$, $A$
such that $D$ is effective and $A$ ample. Since $D$ must be in fact nef, it
follows that $L=K_X^{-1}$ is ample, as desired.

The most difficult part is (ii). Since $c_1(X)^n=0$, Corollary~1.10
implies $\chiox=0$. By Hodge symmetry, we get $h^0(X,\Omega^p_X)=
h^p(X,\cO_X)$ and
$$\chiox=\sum_{0\le p\le n}(-1)^ph^0(X,\Omega^p_X)=0.$$
From this and the fact that $h^0(X,\cO_X)=1$, we infer the existence
of a non zero $p$-form $u$ for some suitable odd number~$p$. Let
$$\sigma:\Lambda^{p-1}TX\longrightarrow\Omega_X^1$$
be the bundle morphism obtained by contracting $(p-1)$-vectors with~$u$.
For every $k>0$, the morphism $\Lambda^k\sigma$ can be viewed as a section of
the bundle $\Lambda^k(\Lambda^{p-1}T^\star X)\otimes\Lambda^kT^\star X$
which has nef dual. Hence by Proposition~1.7 we know that
$\Lambda^k\sigma$ is either identically zero or does not vanish. This means
that $\sigma$ has constant rank. Let $E$ be the image of $\sigma$. Then $E$ is
a quotient bundle of $\Lambda^{p-1}TX$, so $E$ is nef, and $E$ is a
subbundle of $\Omega^1_X=T^\star X$, so $E^\star$ is likewise nef.
Theorem~1.8 implies the existence of a hermitian flat
subbundle $E_1\subset E$. If $E_1$ would be trivial after pulling-back to
some finite \'etale cover $\wx$, we would get a trivial subbundle of
$\Omega^1_\wx$, hence $q(\wx)>0$ and the proposition would be proved.
Otherwise $E_1$ is given by some infinite representation of $\pi_1(X)$
into some unitary group. Let $\Gamma$ be the monodromy group (i.e. the
image of $\pi_1(X)$ by the representation). We use a result of Tits [Ti72]
according to which 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 by Theorem~2.?. In the second case, we may assume
$\Gamma$ solvable by taking some finite \'etale cover. We consider
the series of derived groups
$$\Gamma\supset\Gamma_1\supset\ldots\supset\Gamma_N=\{0\}$$
and the largest index $k$ such that $\Gamma_k$ has finite index
in~$\Gamma$. Then the inverse image of $\Gamma_k$ in $\pi_1(X)$ defines
a finite \'etale cover $\wx$ of $X$ with infinite first homology group
(the representation maps this group onto $\Gamma_k/\Gamma_{k+1}$ which
is infinite). Hence $q(\wx)>0$, as desired.\finpr
\vskip7pt
\rem Proof of the Main Theorem|Let $X$ be compact K\"ahler manifold with
nef tangent bundle. Since a non zero holomorphic form $u\in
H^0(X,\Omega^1_X)$ can never vanish by Proposition~1.7, it follows
immediately that the Albanese map $\alpha$ has rank equal to
$q(X)$ at every point, hence $\alpha$ is a submersion and $q(X)\le n$.
Let $\wx$ be a finite \'etale cover with maximum irregularity
$q=q(\wx)$ (note that $\wx$ also has a nef tangent bundle, so
$q(\wx)\le n$). Let $F$ denote the fibers of the Albanese map
$\alpha:\wx\to A(\wx)$. The relative tangent bundle exact sequence 
$$0\ra TF\ra TX{\buildrel {d\alpha}\over\longrightarrow}
\alpha^\star TA(X)\ra0$$
in which $TA(X)$ is trivial shows by 1.6~(v)$(\gamma)$ that $TF$ is
nef. Lemma~3.10~(iii) below implies that all finite \'etale covers $\wf$ of
$F$ satisfy $q(\wf)=O$. Hence the fibers $F$ must be Fano by
Proposition~3.9 and the Main Theorem follows.\finpr

\than Lemma 3.10|Let $X,Y$ be compact K\"ahler manifolds and let
$g:X\ra Y$ be a smooth fibration with connected fibres. We let $q(X)$
be the irregularity of $X$ and $\qq(X)$ be the sup of the irregularity
of all finite \'etale covers. 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(iii)} Suppose that the boundary
map $\pi_2(Y)\ra\pi_1(F)$ is zero, that $\pi_1(F)$ contains an abelian
subgroup of finite index and that $Y$ 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 $\cb\ra\widehat{S}$. Then
$$\qq(X)=\qq(Y)+\qq(F).$$
\fintha

The proof is based on a use of the Leray spectral sequence and a study
of the resulting monodromy on $H^1(F,\cb)$. Triviality of the monodromy
is achieved in case (iii) because all K\"ahler deformations of tori
over $Y$ must be trivial. We refer the reader to [DPS91] for the
details. In our application, $Y$ is taken to be the Albanese torus,
so assumption (iii) is satisfied with $S=Y$ ($\pi_1(F)$ contains an
abelian subgroup of finite index thanks to Corollary~3.6, by using
an induction on dimension).

\vskip15pt

\titre 4. Classification in dimension 2 and 3|

By using the Kodaira classification of surfaces and the structure theorems
of Section~3, it is not difficult to classify all K\"ahler surfaces with
nef tangent bundles; except for tori, the K\"ahler classification is
identical to the projective one. The projective case was already mentioned
in [CP91] and [Zh90].

\than Theorem 4.1|Let $X$ be a smooth K\"ahler surface such that $TX$
is nef. Then $X$ is minimal and is exactly one of the surfaces in the 
following list:
\item{\rm(i)} $X$ is a torus$\,;$
\item{\rm(ii)} $X$ is hyperelliptic$\,;$
\item{\rm(iii)} $X=\P2\,;$
\item{\rm(iv)} $X=\P1 \times \P1\,;$
\item{\rm(v)} $X=\pb(E)$, where $E$ is a rank$\,$2-vector bundle on an
              elliptic curve $C$ with either
\itemitem{$(\alpha)$} $E=\cO\oplus L$, $L\in\Pic^0(C)$, or
\itemitem{$(\beta)$} $E$ is given by a non split extension
$0\ra\cO\ra E\ra L\ra0$ with $L=\cO$ or $\deg L=1$.\smallskip
\fintha

The list of non K\"ahler surfaces in the Kodaira classification is much
smaller. It is then rather easy to check nefness in each case:

\than Theorem 4.2|The smooth non K\"ahler compact complex surfaces with
nef tangent bundles are precisely:
\item{\rm(i)} Kodaira surfaces $($that is, surfaces of Kodaira dimension
$0$ with $b_1(X)$ odd)$\,;$
\item{\rm(ii)} Hopf surfaces $($that is, surfaces whose universal cover
is $\cb^2\ssm\{0\})$.
\fintha

A similar classification can be obtained for $3$-dimensional compact
K\"ahler manifolds.

\than Theorem 7.1|Let $X$ be a K\"ahler $3$-fold. Then $TX$ is nef if and
only if $X$ is up to finite \'etale cover one of manifolds in the
following list:
\item{\rm(i)} $X=\P3\,;$
\item{\rm(ii)} $X=Q_3$, the 3-dimensional quadric$\,;$
\item{\rm(iii)} $X=\P1\times\P2\,;$
\item{\rm(iv)} $X=\P1\times\P1\times\P1\,;$
\item{\rm(v)} $X=F_{1,2}$, the flag manifold of subspaces of $\cb^3\,;$
\item{\rm(vi)} $X=\pb(E)$, with $E$ a numerically flat rank 3-bundle
              on an elliptic curve~$C\,;$
\item{\rm(vii)} $X=\pb(E){\times}_C\pb(F)$, with $E$, $F$ numerically
                flat rank 2-bundles over an elliptic curve~$C\,;$
\item{\rm(viii)} $X=\pb(E)$, with $E$ a numerically flat rank 2-bundle
                 over a $2$-dimensional complex torus$\,;$
\item{\rm(ix)} $X={}$ $3$-dimensional complex torus.
\fintha

The only non algebraic manifolds appear in classes (viii) and (ix)
when the Albanese torus is not algebraic.
Let us mention that the classification of projective 3-folds with nef
tangent bundles was already carried out in~[CP91] and~[Zh90]. In
addition to Theorem~3.5, the main ingredient is the classification of
Fano $3$-folds by Shokurov and Mori-Mukai. An inspection of the list
yields the five first classes (i)-(v).
\vskip20pt

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\vskip30pt
\tenpoint\noindent(June 15, 1992)

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