% Holomorphic line bundles with partially vanishing cohomology
% J.P. Demailly, Th. Peternell, M. Schneider
% Plain-TeX file

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\centerline{\bftwelve HOLOMORPHIC LINE BUNDLES WITH}
\vskip4pt
\centerline{\bftwelve PARTIALLY VANISHING COHOMOLOGY}
\vskip8pt
\centerline{\bf Jean-Pierre Demailly, Thomas Peternell,  Michael Schneider}



\vskip15pt
\titre 0. Introduction and notation|

One of the most fundamental facts of algebraic geometry is the possibility
of characterizing ampleness of line bundles by numerical criteria
(Nakai-Moishezon, Kleiman-Seshadri, $\ldots$), or by cohomology vanishing
theorems. Over the complex numbers, ampleness is moreover equivalent
to the existence of a metric of positive curvature (Kodaira).

The case of line bundles with curvature of mixed signature is also of a
considerable importance. Andreotti and Grauert [AG62] have proved the
following result:

{\sl Given $X$ a compact complex manifold and $L$ a holomorphic line
bundle over $X$ carrying a hermitian metric $h$ whose curvature form
$\Theta_h(L)$ is a $(1,1)$-form with at least $n-q$ positive eigenvalues
at every point, then for every coherent sheaf $\Fc$ over $X$ the
cohomology groups $H^j(X,\Fc\otimes\Oc(mL))$ vanish for $j>q$ and
$m\ge m_0(\Fc)$.}

The purpose of this paper is to investigate line bundles satisfying partial
positivity properties in a systematic way. For this we introduce the following

\than Definition|Let $L$ be a holomorphic line bundle over a projective
manifold~$X$. We let $\sigma_+(L)$ be the smallest integer $q$ with
the following property: there exists an ample divisor $A$ on $X$ and a
constant $C>0$ such that \hbox{$H^j(X,mL-pA)=0$} for all integers $j>q$
and $m,p\ge 0$, $m\ge C(p+1)$.
\fintha

One of the reasons of introducing $A$ in the definition is to recover
the notion of ampleness: indeed, $L$ is ample precisely if $\sigma_+(L)=0$.
On the other hand $\sigma_+(L)=n=\dim X$ if and only if $c_1(L^\star)$
is in the closure of the cone of effective divisors. Moreover
$\sigma_+(L)$ is an upper semicontinuous function of $c_1(L)$ in the
N\'eron-Severi group of $X$ over~$\qb$.

The above mentioned vanishing theorem of Andreotti-Grauert takes in this
context a slightly more precise form.

\than Proposition|If $\Theta_h(L)$ has at least $n-q$ positive eigenvalues at
every point for some integer $q=0,1,\ldots,n$, then $\sigma_+(L)\le q$.
\fintha

In view of Kodaira's characterization of ampleness by positive
curvature, it would be interesting to know the answer to the following
problem.

\than Problem|Let $L$ be a holomorphic line bundle over a
projective algebraic manifold~$X$ and let $q=\sigma_+(L)$. Is there a
smooth hermitian metric $h$ on $L$ such that $\Theta_h(L)$ has at least
$n-q$ positive eigenvalues at each point$\,$?\fintha

We prove that the problem has a positive answer in case $X=\pb(E)$
and $L=\Oc_{\pb(E)}(\pm 1)$, where $E^\star\to Y$ is an ample rank $r$
vector bundle and $Y$ is a curve, or $Y$ is arbitrary and $E^\star$ is
generated by sections. This makes use of results of Umemura [Um73]
and Sommese [So78] relating the notions of ampleness and Griffiths
positivity for vector bundles.

A more algebraic approach leads us to introduce the following definition
of a purely numerical nature.

\than Definition|Let $X$ be a projective $n$-dimensional manifold.
A sequence $Y_q\subset Y_{q+1}\subset\ldots\subset Y_{n-1}\subset Y_n=X$ of
$k$-dimensional algebraic subvarieties $Y_k$ of $X$ is called an ample $q$-flag if
for each $k=q\ld n-1$ there exists an ample Cartier divisor $Z_k$ in the
normalization $\wt Y_{k+1}$, such that $Y_k=\nu_{k+1}(\Supp Z_k)$ as a set,
where $\nu_{k+1}:\wt Y_{k+1}\to Y_{k+1}$ is the normalization map.

We say that a line bundle $L\in\Pic(X)$ is $q$-flag
positive if there exists an ample $q$-flag
\hbox{$Y_q\subset Y_{q+1}\ldots\subset X$} such that $L_{|Y_q}$ is
positive.\fintha

The reason for considering normalizations in the definition of ample
flags is that we want this notion to be invariant by finite maps.
Without taking normalizations, a push forward of a Cartier divisor
would not necessarily be a Cartier divisor. Our main result in this
direction is that the above numerical criterion implies cohomology
vanishing.

\than Theorem|Let $L\in\Pic(X)$. If $L$ is $q$-flag positive,
then $\sigma_+(L)\le n-q$.\fintha

The example of $L=\Oc_{\pb(E)}(1)$ over $X=\pb(E)$ for $E=\Omega^1_S
\otimes\Oc_S(2)$ over a general quartic surface in $\pb_3$ shows that
the converse to the Theorem is not true when $n=3$, $q=2$ (see
Example~5.6). However, we do not have counterexamples in the most
interesting case~$q=1$. In this case we have a partial positive result,
using a recent paper of Campana-Flenner [CF91].

\than Proposition|Let $X=\pb(E)\buildo\pi\over\la C$ be a
$\pb_{n-1}$-bundle over a smooth curve and let $L$ be a line bundle on $X$
with $\sigma_+(L)\le n-1$. Then $L$ is $1$-flag positive. Moreover
there is a base change $F^\star:\wt X=\pb(f^\star E)\to X=\pb(E)$ given
by a finite map $f:\wt C\to C$, such that the pull-back $F^\star L$
admits an ample $1$-flag 
$$Y_1\subset\ldots\subset Y_{n-1}\subset X$$
of the form $Y_i=D_i\cap\ldots\cap D_{n-1}$ with $D_i\ld D_{n-1}$
very ample and intersecting transversally.
\fintha

Our next concern is to study the cone of ``ample curves'', in relation
with effective divisors and ample $1$-flags. To this effect, we
introduce the following notation, which will be used throughout the
paper.

\than Definition|{\rm(i)} Let $X$ be a nonsingular projective variety,
$n=\dim X$. The {\sl Neron-Severi group} of $X$ is by definition the
quotient group $$NS(X)=\big(\Pic(X)/{\equiv}\big)\simeq H^2(X,\zb)\cap
H^{1,1}(X),$$ where $\equiv$ denotes the numerical equivalence of
divisors. We define the real N\'eron-Severi group to be
$NS^1(X)=NS(X)\otimes_\zb\rb$, and we let its dual be $NS_1(X)$. The
Picard number of $X$ is $\rho(X)= \dim_\rb(NS^1(X))=\rank_\zb
NS_\zb(X)$.

\noindent{\rm(i)} We denote by $K_\eff(X)\subset NS^1(X)$ the cone
generated by cohomology classes of effective divisors in~$X$, by
$K_\amp(X)\subset NS^1(X)$ the cone generated by classes of ample
divisors, and by $\ov K_\eff(X)$, $\ov K_\amp (X)$ their closures. In a
parallel way, we define $N_\eff(X)\subset NS_1(X)$ to be the cone
generated by homology classes of effective curves, and we let $\ov
N_\amp(X)\subset NS_1(X)$ be the dual cone of $\ov K_\eff(X)$, i.e.
$\xi\in\ov N_\amp(X)$ if and only if $D\cdot\xi\ge 0$ for all
$D\in K_\eff(X)$. The interior $N_\amp(X)$ of $\ov N_\amp(X)$ will be
called the cone of ample curves.
\fintha

It is well known that $\ov K_\amp(X)$ is the set of classes of nef
divisors, i.e. the dual cone of~$\ov N_\eff(X)$ (see [Ha70]). By
definition $K_\amp(X)$ and $N_\amp(X)$ are open cones.
However, $K_\eff(X)$ and $N_\eff(X)$ are in general neither closed nor
open; the interior $K_\eff^\circ(X)$ is the cone generated by line
bundles of maximum Kodaira-Iitaka dimension $\kappa(L)=\dim X$. The
inclusion \hbox{$K_\amp(X)\subset K_\eff(X)$} yields by duality $\ov
N_\amp(X)\subset\ov N_\eff(X)$, from which we also deduce
$N_\amp(X)\subset N_\eff(X)$. Moreover, the equality $\ov N_\amp(X)=\ov
N_\eff(X)$ occurs if and only if $\ov K_\amp(X)=\ov K_\eff(X)$, i.e.,
if and only if every effective divisor of $X$ is nef. Our main results
in this direction are:

\than Theorem|For an irreducible curve $C\subset X$, consider the
following properties.
\smallskip
\item{\rm(i)} $C$ is the first member $Y_1$ of an irreducible ample
$1$-flag $Y_1\subset\ldots\subset Y_{n-1}\subset X$.\hfill\break
$($We say that a flag is irreducible if all subvarieties $Y_i$ are
irreducible.$)$
\smallskip
\item{\rm(ii)} $\{C\}\in N_\amp(X)$.
\smallskip
\item{\rm(iii)} $\{C\}\in\ov N_\amp(X)$.
\smallskip
\item{\rm(iv)} The normal bundle $N_{C/X}=\Hom_\Oc(\Ic_C/\Ic_C^2,\Oc_C)$ is
ample\hfill\break $($i.e., $\Oc_{\pb(N_{C/X})}(1)$ is ample$)$.
\smallskip
\item{\rm(v)} The normal bundle $N_{C/X}$ is nef $($i.e.,
$\Oc_{\pb(N_{C/X})}(1)$ is nef$)$.
\smallskip
\item{\rm(vi)} The current of integration $[C]$ is weakly cohomologous to a
smooth positive definite form of bidegree $(n-1,n-1)$, i.e. $[C]=u+
\partial R+\ovp\ov R+S$ where $u$ is a smooth positive definite
$(n-1,n-1)$-form with $\partial\ovp u=0$, $R$ is a current of type
$(n-2,n-1)$, and $S$ is a $d$-closed $(n-1,n-1)$-current whose cohomology
class $\{S\}\in H^{n-1,n-1}(X)$ is orthogonal to $NS^1(X)\subset
H^{1,1}(X)$.
\smallskip
\item{\rm(vii)} The current of integration $[C]$ is weakly cohomologous to
a smooth semiposi\-tive form of bidegree $(n-1,n-1)$ $($as in {\rm(vi)},
but with $u\ge 0$ only$)$.
\smallskip
\item{\rm(viii)} There is a family of generically irreducible curves $(C_t)$
covering $X$ such that $C_0=mC$ as a cycle.
\smallskip
\noindent Then we have the following implications:
$$\leqalignno{
&\hbox{\rm(i)}\Rightarrow\hbox{\rm(ii)}\Rightarrow\hbox{\rm(iii)},
&\hbox{\rm(a)}\cr
&\hbox{\rm(i)}\Rightarrow\hbox{\rm(iv)},~\hbox{\sl if additionally
every $Y_i$ is Cartier in $Y_{i+1}$,}&\hbox{\rm(b)}\cr
&\hbox{\rm(iv)}\Rightarrow\hbox{\rm(v)}\Rightarrow\hbox{\rm(iii)},
\quad\hbox{\rm(viii)}\Rightarrow\hbox{\rm(iii)}&\hbox{\rm(c)}\cr
&\hbox{\rm(ii)}\Leftrightarrow\hbox{\rm(vi)}\Rightarrow\hbox{\rm(vii)}
\Rightarrow\hbox{\rm(iii)}&\hbox{\rm(d)}\cr}$$
\fintha

Let $N_\mu(X)$ (or simply $N_\mu$) be the convex cones
generated by all classes of curves defined by Property $(\mu)$,
$1\le\mu\le 8$, and let $\ov N_\mu$ be their closures. Then we have:

\than Theorem|There are the following relations between the
various cones:
$$\ov N_1\subset \ov N_2=\ov N_3=\ov N_6=\ov N_7,\quad
\ov N_8\subset \ov N_5\subset \ov N_3,\quad \ov N_4\subset \ov N_5.$$
In particular, the inclusions $\ov N_1\subset\ov N_4$ and $\ov N_2\subset
\ov N_1$ would imply the equality of all cones. This is indeed the
case if $X$ is a surface.
\fintha

Finally, we investigate the structure of projective $3$-folds with
$\sigma_+(-K_X)=1$. In fact, it is well-known since a long time that
the structure of projective varieties should be described in terms of
the existence of positive, negative or vanishing ``directions'' in the
canonical bundle~$K_X$. In order to understand the negative directions
we put $\sigma(X)=3-\sigma_+(-K_X)$. Then the condition $\sigma(X)\ge 1$
holds precisely if $K_X\notin \ov K_\eff(X)$, which (via deep results
of Mori and Miyaoka on the classification of $3$-folds) just means
that $K_X\notin K_\eff(X)$, i.e. $\kappa(X)=-\infty$. Since $\sigma(X)=\dim X$
if and only if $X$ is a Fano manifold, the invariant $\sigma(X)$ measures
how far $X$ is from being Fano. First we give various examples of
$3$-folds with $\sigma(X)=2\,$; the most interesting example we have is
$X=\pb(\Omega^1_S)$ where $S\subset\pb_3$ is a general quartic surface.
Then we discuss birational properties of~$\sigma(X)$. Among other
results, we show the following

\than Proposition|Let $X$ be a smooth projective $3$-fold with
$-K_X$ big and nef but not ample. Let $\varphi:X\to Y$ be the map
given by the base point free linear system $|-m_0K_X|$ with
$m_0\gg 0$. Then $\sigma(X)=2$ precisely if all nontrivial fibers
of $\varphi$ are of dimension~$1$.
\fintha

An interesting open problem would be to know whether the condition
$\sigma(X)\ge 2$ is invariant under birational maps of flipping type
for singular~$X$, so that this condition would be invariant by all
operations of the Mori program.
\vskip20pt
\medskip\noindent{\bf Acknowledgement}: Our collaboration has been
partially supported by {\sl Procope}, {\sl Europroj} and by the
{\sl DFG Schwerpunktprogramm ``Komplexe Mannigfaltigkeiten''}.

\titre 1. Vanishing of top or bottom cohomology groups|

Let $X$ be a nonsingular projective variety and let $n=\dim X$. We are
interested in line bundles $L$ over $X$ such that all large multiples
$mL$ have zero cohomology groups in a certain range of degrees.
However, the vanishing properties we want to consider should be stable
under small perturbations of~$L$. For this, we make the following
definition.

\than 1.1. Definition|Let $L$ be a holomorphic line bundle over~$X$.
\smallskip
\item{\rm(i)} We let $\sigma_+(L)$ be the smallest integer $q$ with
the following property: there exists an ample divisor $A$ and a
constant $C>0$ such that \hbox{$H^j(X,mL-pA)=0$} for all integers $j>q$
and $m,p\ge 0$, $m\ge C(p+1)$.
\smallskip
\item{\rm(ii)} Similarly, we let $\sigma_-(L)$ be the largest integer $q$
such that there is an ample divisor $A$ and a constant $C>0$ for which
$H^j(X,mL+pA)=0$ when $j<q$ and $m,p\ge 0$, $m\ge C(p+1)$.\smallskip
\fintha

Let $\Fc$ be a coherent sheaf over~$X$ and let $A$ be an ample line bundle.
Then there exist locally free resolutions of the form
$$\cdots\to\bigoplus_{1\le\ell\le m_k}\Oc(-d_{k,\ell}A)\to\cdots\to
\bigoplus_{1\le\ell\le m_0}\Oc(-d_{0,\ell}A)\to\Fc\to 0\leqno(\star)$$
with suitable integers $m_k,d_{k,\ell}\ge 0$. We define the {\sl height} of
$\Fc$ with respect to $A$ to be the integer
$$\hgt_A(\Fc)=\min_{\{{\rm resolutions}~(\star)~{\rm of}~\Fc\}}~~
\max_{0\le k\le n,\,1\le\ell\le m_k}d_{k,\ell}~,~~~~n=\dim X.$$
By taking the tensor product of the resolutions associated with coherent
sheaves $\Fc_1$, $\Fc_2$, we~infer easily
$\hgt_A(\Fc_1\otimes_\Oc\Fc_2)\le\hgt_A(\Fc_1)+\hgt_A(\Fc_2)$.

\than 1.2. Proposition|With the same constant $C$ as in {\rm 1.1$\,$(i)},
we have
$$H^j(X,\Oc(mL)\otimes\Fc)=0$$
for all integers $j>\sigma_+(L)$ and $m\ge C(\hgt_A(\Fc)+1)$.
\fintha

\proof Choose a resolution $(\star)$ which achieves the actual value
of~$\hgt_A(\Fc)$. Let $\Fc_k$ be the image sheaf of the differential of
degree $k$ in~$(\star)$. Then we have short exact sequences
$$0\to\Fc_{k+1}\to\bigoplus_{1\le\ell\le m_k}\Oc(-d_{k,\ell}A)
\to\Fc_k\to 0,~~~~0\le k\le n,$$
and Def.~1.1$\,$(i) yields $H^j(X,mL-d_{k,\ell}A)=0$ for $j>\sigma_+(L)$ and
$m\ge C(d_{k,\ell}+1)$. Since $\Fc_0=\Fc$, this implies inductively
$$H^j(X,\Oc(mL)\otimes\Fc)\simeq\ldots H^{j+k}
(X,\Oc(mL)\otimes\Fc_k)\simeq H^{j+k+1}(X,\Oc(mL)\otimes\Fc_{k+1})$$
for $j>q$ and $m\ge C(\hgt_A(\Fc)+1)$. Taking $k=n=\dim X$, we get the
desired conclusion.\finpr\vskip7pt

Proposition 1.2 implies immediately that the choice of the ample divisor
$A$ in Definition~1.1 is irrelevant: if $A'$ is another ample
divisor, then $\hgt_A(-pA')\le p\,\hgt_A(-A')$, thus the constant $C$
in 1.1$\,$(i) need only be replaced by $C\,\hgt_A(-A')$. On the other hand,
Serre duality gives:

\than 1.3. Duality formula|$\sigma_-(L)=n-\sigma_+(L^\star)$.\fintha

\proof Since $H^j(X,mL+pA)=H^{n-j}(X,mL^\star-pA+K_X)$ and
$\hgt_A(-pA+K_X)\le p+\hgt_A(K_X)$, this group vanishes for $m/(p+1)$ large
and $n-j>\sigma_+(L)$, hence $\sigma_-(L)\ge n-\sigma_+(L)$. On the other
hand, $H^j(X,mL-pA+K_X)=H^{n-j}(X,mL^\star+pA)$ vanishes for
$n-j<\sigma_-(L)$ and $m\ge C(p+1)$. Resolving $\Oc_X$ by direct sums
of negative line bundles of the form $\Oc(-dA+K_X)$ and arguing as
in the proof of Prop.~1.2, we conclude that $H^j(X,mL-pA)$ vanishes for
$j>n-\sigma_-(L)$ and $m/(p+1)$ large, thus $\sigma_+(L)\le
n-\sigma_-(L)$.\finpr\vskip7pt

Note also that our definitions imply $\sigma_-(L)\le\sigma_+(L)$, because
there must be some degree $j\in\{0,1\ld n\}$ such that the groups
$H^j(X,mL-pA)$ are non zero for arbitrary large values of $m\gg p\gg 1$
(the leading term $(mL-pA)^n/n!$ of the Euler characteristic is a non zero
homogeneous polynomial of degree $n$ in $m,p$). Finally, we have
$\sigma_+(L)=\sigma_+(kL)$ for every integer~$k>0$. Hence, for a
$\qb$-divisor~$D$, the integer $\sigma_+(D)$ can be defined as
$\sigma_+(kD)$ for any common denominator $k$ of the coefficients
of~$D$.

\than 1.4. Proposition|For $D\in\Div_\qb(X)$, the integer $\sigma_+(D)$
depends only on the first Chern class~$c_1(D)$. Moreover, the function
$c_1(D)\mapsto\sigma_+(D)$ is upper semicontinuous with respect to the
vector space topology of the real N\'eron-Severi group $NS^1(X)$.\fintha

\proof Fix $D\in\Div_\zb(X)$ and $B_i\in\Div_\zb(X)$, $1\le i\le\rho$, such
that the Chern classes $c_1(B_i)$ define a basis of $NS^1(X)$. Then
for any $D'\in\Div_\qb(X)$, the difference $D'-D$ is numerically equivalent
to a linear combination $\sum\lambda_iB_i$, $\lambda_i\in\qb$, and $c_1(D')$
is close to $c_1(D)$ if and only if $\sum|\lambda_i|$ is small. Let
$k$ be a common denominator of the~$\lambda_i$'s. Then we can write
$$kD'=kD+\sum k\lambda_iB_i+F$$
where $F$ is a numerically trivial integral divisor. Since $\Pic^0(X)$ is
compact, there is a uniform bound $M$ of the height of all numerically trivial
divisors with respect to the given ample divisor~$A$. Thus for $m,p>0$
and $j>\sigma_+(D)$, we get
$$H^j(X,mkD'-pA)=H^j\Big(X,mkD+\sum mk\lambda_iB_i+mF-pA\Big)=0$$
provided that $mk$ is at least equal to
$$C\Big(\hgt_A\big(\sum mk\lambda_iB_i+mF-pA\big)+1\Big)\le
C\Big(\sum mk|\lambda_i|\,\hgt_A(B_i)+M+p+1\Big).$$
If the $\lambda_i$'s are chosen so small that
$C\sum|\lambda_i|\,\hgt_A(B_i)<1/2$, it is then enough to take
$mk\ge 2C(M+p+1)$. This implies $\sigma_+(D')\ge\sigma_+(D)$, whence the
upper semicontinuity of~$\sigma_+(D)$. The fact that $\sigma_+(D)$ depends
solely on $c_1(D)$ is proved similarly (and is even easier).\finpr

\than 1.5. General properties|All varieties $X$, $X'$ involved below are
supposed to be projective and nonsingular. Let $L$ be a holomorphic line
bundle over~$X$.
\smallskip
\item{\rm(i)} $\sigma_+(L)=0$ if and only if $L$ is positive $($i.e.,
ample$)$, and $\sigma_-(L)=n$ if and only if $L$ is negative.
\smallskip
\item{\rm(ii)} $\sigma_-(L)=0$ if and only if $c_1(L)\in\ov K_\eff(X)$,
and $\sigma_+(L)=n$ if and only if $c_1(L^\star)\in\ov K_\eff(X)$.
\smallskip
\item{\rm(iii)} If $f:X'\to X$ is a finite map, then $\sigma_+(f^\star L)=
\sigma_+(L)$ and $\sigma_-(f^\star L)=\sigma_-(L)$.
\smallskip
\item{\rm(iv)} Let $Y\subset X$ be a nonsingular subvariety and let $L$
be a line bundle over~$X$. Then $\sigma_+(L_{|Y})\le\sigma_+(L)$.
\smallskip
\item{\rm(v)} Let $L\to X$ and $L'\to X'$ be holomorphic line bundles. Then
$$\sigma_+(L\stimes L')=\sigma_+(L)+\sigma_+(L').$$
\item{\rm(vi)} Let $L,L'$ be holomorphic line bundles over~$X$. Then
$$\sigma_+(L+L')\le\sigma_+(L)+\sigma_+(L').$$
\smallskip
\fintha

\proof (i) If $\sigma_+(L)=0$, Proposition~1.2 applied to the ideal
sheaves $\Fc=\Ic_{\{x\}}\Ic_{\{y\}}$ of pairs of points shows that
$H^j(X,\Oc(mL)\otimes\Ic_{\{x\}}\Ic_{\{y\}})=0$ for all $m\ge m_0$,
$j>0$ and $x,y\in X$. Hence $\Oc(mL)$ is very ample for $m\ge m_0$, and
$L$ is positive. Conversely, if $L$ is positive, there are integers
$k_0,k_1>0$ such that $k_0L-A>0$ and $k_1L-K_X>0$. Then
$$mL-pA-K_X=(m-pk_0-k_1)L+p(k_0L-A)+k_1L-K_X>0$$
for $m,p\ge 0$ and $m\ge pk_0+k_1$, and the Kodaira vanishing theorem implies
$H^j(X,mL-pA)=0$ for $j>0$. Hence $\sigma_+(L)=0$. The dual case
$\sigma_-(L)=n$ follows by Proposition~1.3.

(ii) By Definition, we have $\sigma_-(L)=0$ if and only if
$H^0(X,mL+pA)\ne0$ for $m,p\ge 0$ and $m/(p+1)$ arbitrary large. The
existence of such sections implies $c_1(L)+{p\over m}c_1(A)
\in K_\eff(X)$, hence $c_1(L)\in\ov K_\eff(X)$ in the limit.
Conversely, the assumption $c_1(L)\in\ov K_\eff(X)$ implies
$\sigma_-(L)=0$. In fact, for every such $L$, we claim that $H^0(X,mL+pA)
\ne0$ for all $m\ge 0$ and $p\ge p_0$, where $p_0$ is chosen so large that
$p_0A-K_X-nH>0$ for any given very ample line bundle~$H$. To see this,
observe that the ample divisors lie in the interior $K_\eff^\circ(X)$
of the cone. Hence we have
$$c_1\big(mL+pA-K_X-(n+\varepsilon)H\big)\in K_\eff^\circ(X)$$
for all $m\ge 0$, $p\ge p_0$ and $\varepsilon<\varepsilon_0$ sufficiently
small. This implies that $mL+pA-K_X-(n+\varepsilon)H$ is linearly
equivalent to an effective $\qb$-divisor $D$ plus a numerically trivial line
bundle~$T$. Hence
$$\Oc(mL+pA)\simeq\Oc(K_X+(n+\varepsilon)H+D+T).$$
Now, fix a point $x_0\notin D$. The line bundle
$G=\Oc((n+\varepsilon)H+D+T)$ can be equipped with
a hermitian metric $h$ with logarithmic poles at $x_0$ and along~$D$, such
that $h$ is nonsingular on $X\ssm (D\cup\{x_0\})$ and has positive
definite curvature everywhere on $X$ in the sense of currents. In fact the
line bundle $\Oc(\varepsilon H+D+T)$ possesses a singular metric with
curvature form equal to $\varepsilon\omega+[D]$, where $\omega$ is a K\"ahler
form in $c_1(H)$ and $[D]$ is the current of integration over~$D\,$; on the
other hand, $\Oc(nH)$ can be equipped with a hermitian metric of semipositive
curvature possessing an isolated pole at $x_0$, namely a metric of the form
$$\Oc(nH)_x\ni\xi\longmapsto|\xi|^2/(\sum|h_j(x)|^2)^n$$
where $(h_j)$ is a basis of the space of sections of $H$ vanishing at~$x_0$.
Since this last metric is not integrable at $x_0$, H\"ormander's $L^2$
existence theorem shows that $mL+pA\sim K_X+G$ has a global section which
does not vanish at~$x_0\,$; see e.g. [De90] for details. Therefore
$H^0(X,mL+pA)\ne0$ and $\sigma_-(L)=0$. The case $\sigma_+(L)=n$ is dual.

(iii) If $f:X'\to X$ is a finite map and $A$ an ample line bundle over~$X$,
then $f^\star A$ is ample on~$X'$. The projection formula yields
$$f_\star\Oc(m\,f^\star L-p\,f^\star A)=\Oc(mL-pA)\otimes f_\star\Oc_{X'}.$$
All higher direct images are zero because $f$ is finite. The Leray spectral
sequence yields
$$H^j(X',m\,f^\star L-p\,f^\star A)=H^j(X,\Oc(mL-pA)\otimes
f_\star\Oc_{X'}).$$
Since the height of $\Oc(-pA)\otimes f_\star\Oc_{X'}$ is bounded by
$p+\hgt_A(f_\star\Oc_{X'})$, we infer from Proposition~1.3 that
$$H^j(X',m\,f^\star L-p\,f^\star A)=0$$
for $j>\sigma_+(L)$ and $m\ge C(p+\hbox{\rm Const})$. Hence
$\sigma_+(f^\star L)\le\sigma_+(L)$. Now, the vanishing of
$H^j(X',m\,f^\star L-p\,f^\star A)$ implies the vanishing of
$H^j(X,mL-pA)$, because $\Oc_X$ is a direct summand in
$f_\star\Oc_{X'}$. Therefore $\sigma_+(f^\star L)\ge\sigma_+(L)$ and we
have equality. The formula for $\sigma_-(f^\star L)$ follows by
duality.

(iv) Let $A$ be an ample divisor on $X$ and let $\Ic_Y$ be the ideal
sheaf of~$Y$. Then $A_{|Y}$ is ample and Proposition~1.3 gives
$$H^j(Y,mL_{|Y}-pA_{|Y})=H^j\big(X,\Oc(mL-pA)\otimes(\Oc_X/\Ic_Y)\big)=0$$
for $j>\sigma_+(L)$ and $m\ge C(p+\hgt_A(\Oc_X/\Ic_Y)+1)$, whence the
desired inequality.

(v) Let $A$, $A'$ be ample divisors on $X$, $X'$ respectively. Then
$A\stimes A'$ is ample on $X\times X'$, and the K\"unneth formula yields
$$H^k(X\times X',m\,L\stimes L'-p\,A\stimes A')=\bigoplus_{i+j=k}
H^i(X,mL-A)\otimes H^j(X',mL'-A').$$
The conclusion follows immediately from this.

(vi) is a straightforward consequence of (iii) and (iv), because the line
bundle $L+L'$ is just the restriction of $L\stimes L'$ to the diagonal
of $X\times X$.\finpr\vskip7pt

We now consider the effect of blowing-up on line bundles with partially
vanishing cohomology groups.

\than 1.6. Proposition|Let $\pi:\wt X\to X$ be a blow-up with smooth
center~$Y\subset X$, and let $E\subset\wt X$ be the exceptional divisor.
Then, for every line bundle $L$ over~$X$, we have the following inequalities.
\smallskip
\item{\rm(i)} $\sigma_+(\pi^\star L)\le\max\big\{\sigma_+(L),\sigma_+(L_{|Y})
+\codim Y-1\big\}\le\sigma_+(L)+\codim Y-1$.
\smallskip
\item{\rm(ii)} $\sigma_-(\pi^\star L)\ge\min\big\{\sigma_-(L),\sigma_-(L_{|Y})
+1\big\}\ge\sigma_-(L)-(\codim Y-1)$.
\smallskip
\item{\rm(iii)} For $k\ge k_0$ large, $\sigma_+(k\,\pi^\star L-E)
\le\sigma_+(L)$.
\fintha

\proof Let $A$ be an ample divisor on~$X$. Then there is an integer $d>0$
such that $\wt A=d\,\pi^\star A-E$ is ample on~$\wt X$. We have
$$H^j\big(\wt X,m\,\pi^\star L-p(d\,\pi^\star A-E)\big)=
H^j\big(\wt X,\pi^\star(L-pdA)+pE\big),$$
and the projection formula implies
$$R^j\pi_\star\Oc\big(\pi^\star(L-pdA)+pE\big)=
\Oc(L-pdA)\otimes R^j\pi_\star\Oc(pE).$$
By a well-known formula, the direct images $R^j\pi_\star\Oc(pE)$ are given by
$$R^j\pi_\star\Oc(pE)=\cases{
   \Ic_Y^{p^-}&for~~$j=0$,\cr
   0\phantom{\Ic_Y^{(p)^-}}&for~~$j\ne 0,r-1$,\cr
   \Fc_p\phantom{\Ic_Y^{(p)^-}}&for~~$j=r-1$,\cr}~~~~r=\codim Y,$$
where $\Fc_p$ is a coherent $\Oc_X/\Ic_Y^{(p-r+1)^+}$-module with
support on~$Y$ (thus equal to $0$ for $p<r$), such that
$$\Fc_p\supset\Fc_{p-1}\supset\ldots\Fc_{p-k}=\Ic_Y^k\Fc_p\supset
\ldots\,,\qquad\Fc_{p-k}/\Fc_{p-k-1}\simeq\det N_Y^\star\otimes
S^{p-k-r}N_Y^\star$$
for $0\le k\le p-r$. By the Leray spectral sequence,
we infer that the groups $H^j\big(\wt X,m\,\pi^\star L-p(d\,\pi^\star A-E)
\big)$ vanish provided that 
$$\eqalign{
&H^j(X,mL-pdA)=0,~~~~\hbox{\rm resp.}\cr
&H^{j-(r-1)}\big(Y,\Oc(mL_{|Y}-pdA_{|Y})\otimes
\det N_Y^\star\otimes S^{p-k-r}N_Y^\star\big)=0\cr}$$
for $k=r,r+1\ld p$. When $m/(p+1)$ is large enough, these groups actually
vanish for $j>\sigma_+(L)$, resp.\ for $j-(r-1)>\sigma_+(L_{|Y})\,$; indeed the
height of the sheaf $\Oc(-pdA)\otimes\det N_Y^\star\otimes S^{p-k-r}N_Y^\star$
is bounded by a constant multiple of~$p$. The first inequality in (i) follows
from this; the second one is then a consequence of 1.5$\,$(iv).
The inequalities in (ii) are equivalent to those in (i) by duality.

(iii) Quite similarly, we find
$$\eqalign{
H^j\big(\wt X,m(k\,\pi^\star L-E)-p(d\,\pi^\star A-E)\big)&=
H^j\big(\wt X,\pi^\star(mkL-pdA)-(m-p)E\big)\cr
&\simeq H^j\big(X,\Oc(mkL-pdA)\otimes\Ic_Y^{m-p}\big)\cr}$$
for $m\ge p$ (in that case, $\Oc(-(m-p)E)$ has no higher direct images).
The height of $\Oc(-pdA)\otimes\Ic_Y^{m-p}$ is bounded by
$pd+(m-p)\hgt_A(\Ic_Y)$, so we get vanishing for $j>\sigma_+(L)$ and
$mk\ge C(pd+(m-p)C'+1)$. The last condition is satisfied for $k\ge CC'+1$ and
$m\ge C(pd+1)$, whence the inequality $\sigma_+(k\,\pi^\star L-E)\le
\sigma_+(L)$ for $k\ge CC'+1$.

\titre 2. Cohomology vanishing and signature properties of the curvature|

Suppose that the line bundle $L$ over $X$ is equipped with a hermitian
metric $L$ and let $\Theta_h(L)={i\over 2\pi}D_{L,h}^2$ be the Chern
curvature form of~$h$. The Andreotti-Grauert theorem implies that
one can deduce coarse vanishing theorems from a knowledge of upper and
lower bounds on the signature of~$\Theta_h(L)$. Here we obtain a slightly
more precise statement by means of the Bochner-Kodaira formula.

\than 2.1. Proposition|Suppose that $L$ has a smooth hermitian metric
$h$ such that the curvature form $\Theta_h(L)$ has at least $q'$ negative
eigenvalues and at least $n-q''$ positive eigenvalues at each point of~$X$,
for some integers $0\le q'\le q''\le n$. Then $q'\le\sigma_-(L)\le
\sigma_+(L)\le q''$.\fintha

\proof This is essentially well-known, but we reproduce briefly the argument
for self-containedness. By duality, it is enough to show the
inequality $\sigma_+(L)\le q''$,
i.e., $H^j(X,mL-pA)=0$ for $j>q''$ and $m\ge C(p+1)$. For this, we apply
the Bochner-Kodaira formula in the case of a nonnecessarily K\"ahler metric
$\omega$ on~$X$ (see e.g. [De84]). Let $u$ be a smooth $(0,j)$-form with
values in a hermitian line bundle~$G$. The Bochner-Kodaira formula implies
an a priori estimate
$$\int_X\big(\Vert\ovp u\Vert^2+\Vert\ovp^\star u\Vert^2\big)dV\ge
\int_X(\gamma_1+\ldots+\gamma_j-C_\omega)\Vert u\Vert^2dV,$$
where $\gamma_1\le\ldots\le\gamma_n$ are the eigenvalues of the curvature
form of $G$ at any point of~$X$, and $C_\omega$ is a constant depending
only on the torsion and Ricci curvature of~$\omega$. In particular, if
$\gamma_1+\ldots+\gamma_j>C_\omega$ everywhere, then $H^j(X,G)=0$.
We select the hermitian metric $\omega$ such that the eigenvalues
$\lambda_1\le\ldots\le\lambda_n$ of $L$ satisfy
$\lambda_{q''+1}=\ldots=\lambda_n=1$ and $\lambda_1\ge-\varepsilon$.
(For this it is enough to take $\omega$ large enough on the negative
eigenspaces of the curvature form $\Theta_h(L)$ and equal to $\Theta_h(L)$
on the positive eigenspaces.) Then, if $\alpha$ is a bound for the
eigenvalues of $\Theta(A)$, the eigenvalues of $G=mL-pA$ satisfy
$\gamma_j\ge m\lambda_j-p\alpha$, hence
$$\gamma_1+\ldots+\gamma_j\ge m\big((j-q'')-q''\varepsilon\big)-pj\alpha.$$
If we choose $j>q''$ and $\varepsilon<1/2q''$, we get
$\gamma_1+\ldots+\gamma_j-C_\omega>m/2-pn\alpha-C_\omega$, hence 
$H^j(X,mL-pA)=0$ for $m\ge 2pn\alpha+2C_\omega$.\finpr
\vskip7pt

In view of the Kodaira ampleness criterion, it is natural to ask whether
the following ``converse'' to Proposition~2.1 holds.

\than 2.2. Problem|Let $L$ be a holomorphic line bundle over a
projective algebraic manifold~$X$ and let $q'=\sigma_-(L)$,
$q''=\sigma_+(L)$. Is there a smooth hermitian metric $h'$ 
$($resp. $h'')$ on $L$ such that $\Theta_{h'}(L)$ has at least $q'$
negative eigenvalues and $\Theta_{h''}(L)$ has at least $n-q''$ positive
eigenvalues at each point$\,$?\fintha

We now study the particular case of line bundles such that the nonzero
cohomology groups arise only in one degree, in relation with the
signature of the curvature form.

\than 2.3. Definition|A line bundle $L$ has cohomology concentrated in
degree $q$ if $\sigma_+(L)=\sigma_-(L)=q$.\fintha

\than 2.4. Definition|A line bundle $L$ has constant signature $(p,q)$
with $p+q\le n$, if there is a smooth hermitian metric $h$ on $L$ such
that $\Theta_h(L)$ has signature $(p,q)$ at every point.\fintha

By Proposition~2.1, if $L$ has signature $(n-q,q)$, then $L$ has
cohomology concentrated in degree~$q$. The converse is not true,
even though we expect Problem~2.2 to have a positive answer. The
trouble is that the metrics $h'$ and $h''$ will in general not coincide
even if we assume that $q'=q''$.

\rem 2.5. Example|An obvious necessary condition for the existence of a
line bundle of constant signature $(n-q,q)$ is that the tangent bundle
$T_X$ splits topologically into a direct sum of complex subbundles of
rank $(n-q)$ and $q$ (the positive and negative eigenspaces with respect
to any fixed hermitian metric on the base). Take for instance a
surface $X$ which is the blow-up of some other surface $X'$ at a point,
and let $L=\pi^\star L'-kE$ where $E$ is the exceptional
divisor, $L'$ is ample on $X'$ and $k\gg 0\,$; it is indeed enough to
take $k$ such that $L\cdot A=\pi^\star L'\cdot A-kE\cdot A<0$ for some
ample divisor $A$ on~$X$. Then neither $mL+pA$ nor $-mL+pA$ have sections
for $m\gg p$ (take e.g. $A=\mu\pi^\star L'-E$ with $\mu$ large enough to
see this easily), hence $\sigma_+(L)=\sigma_-(L)=1$. However $T_X$
does not split topologically into a sum of two line bundles if we take
for instance $X$ to be the blow-up of $\pb_2$ at two points, hence $L$
cannot have constant signature~$(1,1)$.
(This simple example has been communicated to us by A.~Beauville and
F.A.~Bogomolov.)\finpr

By 1.5$\,$(i) and the Kodaira ampleness criterion, Problem~2.2 has an
affirmative answer in the positive or negative definite cases $q=0$,
$q=n$. The nondefinite case, however, seems to be a very hard problem,
and we have very little evidence for~it. The only general indication we
have is that the sign of the determinant is correct in the average, i.e.
that $(-1)^qc_1(L)^n=(-1)^q\int_X\Theta_h(L)^n\ge 0$. (We can even
obtain this integral to be $>0$ when $\dim X\le 3$.)

\than 2.6. Proposition|Suppose that $L$ has cohomology concentrated
in degree~$q$. Let $k$ be the maximal integer with $c_1(L)^k\ne0$.
Then $n-k$ is even and $(-1)^qc_1(L)^k\cdot c_1(G)^{n-k}\ge 0$ for
all $G\in\Pic(X)$, in particular $(-1)^qc_1(L)^n\ge 0$. Moreover,
we have $(-1)^qc_1(L)^n>0$ if $n=\dim X\le 3$.
\fintha

\proof For $m$ large, the Riemann-Roch formula gives
$$0\le h^q(X,mL)=(-1)^q\chi(X,mL)\sim(-1)^qc_1(L)^nm^n/n!,$$
thus $(-1)^qc_1(L)^n\ge 0$. Since $\sigma_+(L)$ and $\sigma_-(L)$
are left unchanged by small perturbations of~$L$, we conclude that
$(-1)^qc_1(L+\varepsilon G)^n\ge 0$ for any $G\in\Pic(X)$ and
$\varepsilon\in\qb$ small. Expanding the $n$-th power and letting
$\varepsilon$ tend to~$0$, we get $(-1)^qc_1(L)^k\cdot(\pm c_1(G))^{n-k}
\ge 0$ for all $G\in\Pic(X)$. Since the product is not
always zero when $G$ runs over~$\Pic(X)$, $n-k$ must be even. Observe
that $k>1$, otherwise $L$ is numerically trivial and
$\sigma_-(L)=0<\sigma_+(L)=n$. If $n=\dim X\le 3$, the only remaining
possibility is $k=n$, whence $(-1)^qc_1(L)^n>0$.\finpr

\rem 2.7. Examples|
\smallskip
\item{(i)} If $L\to X$ is positive, $L'\to X'$ is negative and $n=\dim X$,
$n'=\dim X'$, then $L\stimes L'$ has constant signature $(n,n')$.
\smallskip
\item{(ii)} Let $E$ be a holomorphic vector bundle of rank $r$ over a
$n$-dimensional variety~$Y$, and let $L$ be the tautological line bundle
$L=\Oc_{\pb(E)}(1)$ over $X=\pb(E)$. Then $\sigma_+(L)\le n$.
Moreover, $L$ has cohomology concentrated in degree~$n$, i.e.
$\sigma_+(L^\star)\le r-1$, if and only if $E^\star$ is ample.
\smallskip
\item{(iii)} Let $E\to Y$ be as in~(ii). If $E$ is negative in the
sense of Griffiths, then $L$ has signature $(r-1,n)$ over~$X$. In
particular, Problem~2.2 has a positive answer for $L=\Oc_{\pb(E)}(1)$ when
$E^\star\to Y$ is an ample vector bundle over a curve.
\vskip12pt

\proof (i) is obvious.

(ii) Let $\pi:X\to Y$ be the natural projection. There is an ample line
bundle $H$ on $Y$ such that $A=\Oc_{\pb(E)}(1)\otimes\pi^\star H$
is ample on~$X$. Then, for $m,p\ge0$ and $\varepsilon=\pm 1$, the line bundle
$$mL+\varepsilon pA=\Oc_{\pb(E)}(m+\varepsilon p)\otimes\pi^\star
\Oc(\varepsilon pH)$$
has direct image $\pi_\star\Oc(mL+\varepsilon pA)=S^{m+\varepsilon p}
E\otimes\Oc(\varepsilon pH)$ and zero higher direct images when $m\ge
-\varepsilon p$. The Leray spectral sequence implies, together with Serre
duality,
$$\eqalign{H^j(X,mL+\varepsilon pA)
&\simeq H^j\big(Y,S^{m+\varepsilon p}E\otimes\Oc(\varepsilon pH)\big)\cr
&\simeq H^{n-j}\big(Y,S^{m+\varepsilon p}E^\star\otimes\Oc(-\varepsilon pH)
\otimes K_Y\big).\cr}$$
By taking $\varepsilon=-1$, it follows immediately that $\sigma_+(L)\le n$.
Moreover, if $E^\star$ is ample and $\varepsilon=+1$, the last group vanishes
for $j<n$ and $m/(p+1)$ large, thus $\sigma_-(L)=n$. Conversely, if
$\sigma_-(L)=n$, we have
$H^j(Y,S^{m+p}E^\star\otimes\Oc(-pH)\otimes K_Y)=0$ for $j>0$ and
$m/(p+1)$ large; resolving arbitrary coherent sheaves $\Fc$ by line bundles of
the form $\Oc(-pH)\otimes K_Y$, we conclude that
$H^j(Y,S^mE^\star\otimes\Fc)=0$ for $j>0$ and $m\ge m_0(\Fc)$, hence $E^\star$
is ample.

(iii) Suppose that $E$ carries a hermitian metric $h$ with negative curvature
tensor
$$\Theta_h(E)=\sum_{\scriptstyle 1\le j,k\le n\atop\scriptstyle
1\le\lambda,\mu\le n}c_{jk\lambda\mu}dz_j\wedge d\ov z_k\otimes
e_\lambda^\star\otimes e_\mu$$
in the sense of Griffiths. We suppose here that $(e_\lambda)$ is a local
holomorphic frame of $E$ near a point $x\in X$, which is orthonormal at $x$
and satisfies $\nabla e_\lambda(x)=0$. The Griffiths negativity assumption
means that $\sum c_{jk\lambda\mu}t_j\ov t_kv_\lambda\ov v_\mu<0$ for all
non zero vectors $t=\sum t_j\partial/\partial z_j\in T_{X,x}$ and
$v=\sum v_\lambda e_\lambda\in E_x$. Let $(z_1\ld z_n,w_1\ld w_n)$ be the
normal coordinates on $E^\star$ given by the dual frame $(e_\lambda^\star)$.
A standard calcu\-lation shows that the curvature of $L=\Oc_{\pb(E)}(1)$ at
$[e_\lambda^\star]\in\pb(E)$ is
$$\Theta_h(L)_{[e^\star_\lambda]}=\sum_{1\le j,k\le n}c_{jk\lambda\lambda}
dz_j\wedge d\ov z_k+\sum_{1\le\mu\le r,~\mu\ne\lambda}dw_\mu\wedge d\ov
w_\mu$$
in terms of the coordinates
$(z_1\ld z_n,w_1\ld w_\lambda{=}1\ld w_n)$ on~$\pb(E)$.
Hence the Griffiths negativity of $E$ implies that $\Theta_h(L)$ has
signature $(r-1,n)$, with positive eigenvalues in the fibre directions
and negative eigenvalues in the horizontal directions. Finally,
every ample vector bundle over a curve is positive in the sense of
Griffiths (Umemura), so Conjecture~2.5 holds in that case.\finpr
\vskip7pt

There are only very few known cases of Conjecture~2.5, even in the case
of line bundles $L=\Oc_{\pb(E)}(1)$ with $E^\star$ ample. The following
results are due to A.~Sommese.

\than 2.8. Proposition {\rm(A.~Sommese)}|Conjecture 2.5 holds for
$L=\Oc_{\pb(E)}(1)$ if $E^\star\to Y$ is ample and satisfies one
of the following additional properties:
\smallskip
\item{\rm(i)} $E$ has a strictly pseudoconvex neighborhood $U$
of the zero section such that all slices $E_x\cap U$ are linearly strictly
convex $(U$~is supposed to be relatively compact with smooth boundary in~$E)$.
\smallskip
\item{\rm(ii)} $E$ is seminegative in the sense of Griffiths.
\smallskip
\item{\rm(iii)} $E^\star$ is generated by sections.
\fintha

\noindent{\sl Sketch of the proof (see [So78]).} First observe that the
hypotheses are related by the implications (iii)${}\Rightarrow{}$(ii)${}
\Rightarrow{}$(i). The implication \hbox{(iii)${}\Rightarrow{}$(ii)}
is standard, while \hbox{(ii)${}\Rightarrow{}$(i)} is checked as follows.
Since $E^\star$ is ample, there is a stricly
plurisubharmonic Finsler metric on~$E$, i.e., a smooth strictly
plurisubharmonic function $F$ on $E\ssm\{0\}$ such that
$F(\lambda\xi)=|\lambda|\,F(\xi)$.  On the other hand, the Griffiths
seminegativity means that there is a weakly plurisubharmonic hermitian
norm $\Vert\xi\Vert$. Then, for $\varepsilon>0$ small, the Finsler
metric $\Vert\xi\Vert+\varepsilon F(\xi)$ is strictly plurisubharmonic
and fibrewise strictly convex; thus $U=\{\xi\in
E\,;\,\Vert\xi\Vert+\varepsilon F(\xi)<1\}$ satisfies~(i).

Now, assuming that (i) holds, Sommese [So78] has made an explicit calculation
of the Levi form of the dual neighborhood
$$U^\circ=\big\{\xi^\star\in E^\star_x\,;\,x\in X,~F^\star(\xi^\star):=
\sup_{\xi\in E_x\cap U}|\xi^\star(\xi)|<1\big\}.$$
It follows that the Finsler metric $F^\star(\xi^\star)$ on $E^\star$ has a
Levi form $i\partial\ovp F^\star$ of signature~$(r,n)$, where $r={}$rank$(E)$
and~$n=\dim Y$ (in particular, $U^\circ$ is a \hbox{$(n+1)$-convex}
neighborhood of the zero section in~$E^\star$). Then $F^\star$
can be also seen as a hermitian metric on $L^\star=\Oc_{\pb(E)}(-1)$ whose
curvature form has signature $(n,r-1)$ on~$\pb(E)$.\finpr\vskip7pt

It is interesting to observe that Conjecture~2.5 is related to several
natural questions in analytic geometry. In particular, it would yield
a considerably simpler proof of the regularization theorem for closed
positive $(1,1)$-currents established in~[De92]. More importantly, it
would yield a positive answer to the following conjecture made
by M.~Schneider~[Sch73].

\than 2.9. Conjecture {\rm (M.~Schneider)}|Let $Y\subset X$ be a
nonsingular subvariety with ample normal bundle~$N_Y$. Then, denoting
$k=\codim Y$, the complement $X\ssm Y$ is $k$-convex in the sense of
Andreotti-Grauert.\fintha

\than 2.10. Proposition|Conjecture~$2.9$ holds provided that Problem~$2.2$
has an affirmative solution in the case of line bundles of the form
$L=\Oc_{\pb(N^\star_Y)}(1)$. Especially, $X\ssm Y$ is $k$-convex if $N_Y$
is ample and generated by sections.\fintha

\proof Let $\wt X\to X$ be the blow-up of $X$ with center~$Y$, let $E$
be the exceptional divisor and $n=\dim X$. Then $X\ssm Y\simeq\wt X\ssm E$
and $\smash{\Oc(E)_{|E}\simeq N_{E/\wt X}\simeq\Oc_{\pb(N_Y^\star)}(-1)}$
has a metric of signature $(n-k,k-1)$ by Conjecture~2.5. Therefore,
$\Oc(E)$ can be equipped with a metric of signature $(n-k+1,k-1)$ in a
neighborhood of~$E$. (Take a smooth extension of the metric to $\wt X$ and
multiply if necessary by a factor of the form $\exp(-Cd(z,E)^2)$ in order
to produce a positive curvature eigenvalue in the normal directions to~$E$.)
If $\sigma$ is the canonical section of $\Oc(E)$, then
$-\log||\sigma(z)||$ is an exhaustion function on $\smash{\wt X}\ssm E$,
and its Levi form has signature~$(n-k,k-1)$ in a neighborhood
of~$E$. It follows that $\smash{\wt X}\ssm E$ is $k$-convex.\finpr
\vskip15pt

\titre 3. Ample flags and partial cohomology vanishing|

Proposition 1.4 shows that $\sigma_+(L)$ is a numerical invariant of~$L$.
For instance, the ampleness of~$L$ (i.e., the vanishing of $\sigma_+(L)$)
is characterized by the well-known Nakai-Moishezon criterion: $L$ is ample
if and only if $L^p\cdot Y>0$ for every $p$-dimensional subvariety
$Y\subset X$. We expect that there is such a nice numerical criterion
characterizing the invariant~$\sigma_+(L)$. First, we introduce the
notion of an ample flag of algebraic subvarieties.

\than 3.1. Definition|Let $X$ be a projective $n$-dimensional manifold.
A sequence $Y_q\subset Y_{q+1}\subset\ldots\subset Y_{n-1}\subset Y_n=X$ of
$k$-dimensional algebraic subvarieties of $X$ is called an ample $q$-flag if
for each $k=q\ld n-1$ there exists an ample Cartier divisor $Z_k$ in the
normalization $\wt Y_{k+1}$, such that $Y_k=\nu_{k+1}(\Supp Z_k)$ as a set,
where $\nu_{k+1}:\wt Y_{k+1}\to Y_{k+1}$ is the normalization map.
\fintha

\vbox{\than 3.2. Definition|We say that $L\in\Pic(X)$ is $q$-flag
positive $($resp. negative$)$ if there exists an ample $q$-flag
\hbox{$Y_q\subset Y_{q+1}\ldots\subset X$} such that $L_{|Y_q}$ is
positive $($negative$)$.\fintha}

The reason for considering normalizations in the definition of ample
flags is that we want this notion to be invariant by finite maps.
Without taking normalizations, a push forward of a Cartier divisor
would not necessarily be a Cartier divisor. The invariance by finite
maps can be stated as follows. 

\than 3.3. Proposition|Let $f:X'\to X$ be a finite map and let
$L\in\Pic(X)$. Then $L$ is $q$-flag positive if and only if $f^\star L$
is $q$-flag positive. Moreover, if $(Y'_k)_{q\le k\le n}$ is an ample
$q$-flag for~$f^\star L$, then $(f(Y'_k))_{q\le k\le n}$ is an ample
$q$-flag for $L$, and if $(Y_k)_{q\le k\le n}$ is an ample $q$-flag
for $L$, then there exists an ample $q$-flag $(Y'_k)_{q\le k\le n}$
for~$f^\star L$ such that $Y_k=f(Y'_k)$ $($however $Y'_k$ may be
smaller than $f^{-1}(Y_k))$.
\fintha

\proof First suppose that $f^\star L$ is $q$-flag positive and that
$(Y'_k)_{q\le k\le n}$ is an ample \hbox{$q$-flag} with~$L_{|Y'_q}$ ample. 
Then we set $Y_k=f(Y'_k)$. This gives a commutative diagram of
normalizations 
$$\leqalignno{
\wt Y'_{k+1}&\buildo{\displaystyle\wt f}\over\la\wt Y_{k+1}\phantom{\bigg)}\cr
\nu'_{k+1}\big\downarrow~~&\phantom{\la}~~~~\big\downarrow \nu_{k+1}&(\star)\cr
Y'_{k+1}&\buildo{\smash{\displaystyle f}}\over\la Y_{k+1}.\cr}$$
Let $Z'_k$ be a Cartier divisor in $\wt Y'_{k+1}$ such that
$Y'_k=\nu'_{k+1}(\Supp Z'_k)$ set theoretically. We define $Z_k$ to be
the push forward $Z_k=\wt f_\star Z'_k$. Recall that if $U$ is a small open
set in $\wt Y_{k+1}$ and $g'$ a generator of $\Oc(-Z'_k)$ on $\wt Y'_{k+1}
\cap\wt f^{-1}(U)$, then $g=\wt f_\star g'$ defined by
$g(x):=\prod_{x'\in\wt Y'_{k+1}}g'(x')$ is a generator of $\Oc(-Z_k)$
on~$U\,$; note that $g$ is actually holomorphic on $\wt Y_{k+1}$ because
$\wt Y_{k+1}$ is a normal space. Hence $Z_k$ is a Cartier divisor, and
we have
$$\nu_{k+1}(Z_k)=\nu_{k+1}(\wt f(Z'_k))=f(\nu'_{k+1}(Z'_k))=f(Y'_k)=Y_k.$$
Finally, $\Oc(Z_k)$ is ample on $Y_{k+1}$. Indeed, for $m$ large enough
and for all $x_1\ne x_2$ in $\wt Y_{k+1}$, there are sections
$h'\in H^0(\wt Y'_{k+1},\Oc(mZ'_k))$ which vanish at a given point of a
fibre $\wt f^{-1}(x_1)$, but do not vanish at the other points nor
at the points of $\wt f^{-1}(x_2)$. Hence $h=\wt f_\star h'\in
H^0(\wt Y_{k+1},\Oc(mZ_k))$ vanishes at $x_1$ but does not vanish at~$x_2$.
This implies that $\Oc(Z_k)$ is ample on~$\wt Y_{k+1}$.

Now, suppose that $L$ is $q$-flag positive and that $(Y_k)_{q\le k\le n}$
is an ample $q$-flag with~$L_{|Y_q}$ ample. Let $Z_k$ be
a Cartier divisor in $\wt Y_{k+1}$ with $Y_k=\nu_{k+1}(\Supp Z_k)$.
We construct an ample flag $(Y'_k)_{q\le k\le n}$ in $X'$ such that
$f(Y'_k)=Y_k$, defining $Y'_k$ by backward induction on~$k$. We set
of course $Y'_n=X'$. If $Y'_{k+1}$ has already been constructed,
we get a commutative diagram of normalizations $(\star)$ as above. We then
define $Z'_k:=\wt f^\star(Z_k)$ and $Y'_k:=\nu'_{k+1}(\Supp Z'_k)$ as a set.
Since the pull-back of an ample Cartier divisor is an ample Cartier divisor,
it is clear that $(Y'_k)_{q\le k\le n}$ is an ample $q$-flag. Moreover,
$(f^\star L)_{|Y'_q}$ is the pull-back of the ample line bundle $L_{|Y_q}$,
hence $(f^\star L)_{|Y'_q}$ is ample.\finpr

\than 3.4. Theorem|Let $L\in\Pic(X)$. If $L$ is $q$-flag positive,
then $\sigma_+(L)\le n-q$, and thus $\sigma_-(L^\star)\ge q$.\fintha

\proof Let $(Y_k)_{q\le k\le n}$ be an ample $q$-flag such that $L_{|Y_q}$
is ample, and let $Z_k$ be an ample Cartier divisor in the normalization
$\wt Y_{k+1}$ of $Y_k$ with $Y_k=\nu_{k+1}(\Supp Z_k)$. The inclusions
$j_{k,\ell}:Y_k\hookrightarrow Y_\ell$ can be lifted as finite maps
$\wt j_{k,\ell}:\wt Y_k\to Y_\ell$. For each multi-index $\alpha=
(\alpha_k\ld\alpha_{n-1})$ of nonnegative integers, we set
$$G^\alpha=\wt j_{k,k+1}^\star\Oc(\alpha_kZ_k)\otimes\ldots\otimes
\wt j_{k,n}^\star\Oc(\alpha_{n-1}Z_{n-1}).$$
Since $\Oc(Z_\ell)$is ample on~$\wt Y_{\ell+1}$, we conclude that
$G^\alpha$ is nef on~$\wt Y_k$ and ample if $\alpha\ne 0$. Let
$A$ be an ample line bundle on~$X$, and let $\Fc$ be an arbitrary
coherent sheaf on~$\wt Y_k$. We prove by induction on $k=q,q+1\ld n$
the following property:
$$\left\{\eqalign{
&H^i(\wt Y_k,\Oc(mL-pA)_{|\wt Y_k}\otimes G^\alpha\otimes\Fc)=0\cr
&\hbox{\rm for all}~~i>k-q,~\alpha\in\nb^{n-k},~m\ge C(\Fc)(p+1).\cr}
\right.\leqno(\Pc_k)$$
Here we write $\Oc(mL-pA)_{|\wt Y_k}=\wt j_{k,n}^\star\Oc(mL-pA)$ for
the simplicity of notations. The desired conclusion is just statement
$(\Pc_n)$ in the case $G^\alpha=\Fc=\Oc_X$. Observe that it is sufficient
to prove $(\Pc_k)$ in the special case $\Fc=\Oc_{\wt Y_k}$, since we may
otherwise take resolutions of $\Fc$ by locally free sheaves of the form
$\Oc(-dA)^{\oplus N}$, and argue by backward induction on $i$ as in the
proof of Proposition~1.2.

{\sl First step: $k=q$}\pointir By our assumption, $\Oc(L)_{|\wt Y_q}=
\wt j_{q,n}^\star\Oc(L)$ is ample. Using a resolution of
$\Oc(-A)_{|\wt Y_q}$ of the form
$$\Oc(-d_s L)_{|\wt Y_q}^{\oplus N_s}\to\cdots\to
\Oc(-d_1 L)_{|\wt Y_q}^{\oplus N_1}\to\Oc(-d_0 L)_{|\wt Y_q}^{\oplus N_0}
\to\Oc(-A)_{|\wt Y_q}\to 0$$
and raising it to the power $p$ to get a resolution of $\Oc(-pA)_{|\wt
Y_q}$, we are led to prove the vanishing property
$$H^i(\wt Y_q,\Oc((m-d_{s_1}-\cdots-d_{s_p})L)_{|\wt Y_k}\otimes G^\alpha)=0,
~~~~\forall i>0,~s_j\le n,$$
for $m'=m-d_{s_1}-\cdots-d_{s_p}\ge m-pd_n\ge m_0$ independent of~$\alpha$.
Hence we merely have to consider the groups $H^i(\wt Y_q,\Oc(mL)\otimes
G^\alpha)$. By the Kodaira-Serre vanishing theorem and the ampleness of $L$
on $\wt Y_q$, we know that these groups vanish for $m\ge m_0(\alpha)$. We have
to check that $m_0(\alpha)$ can be taken independent of~$\alpha$. (This would
be of course a straightforward consequence of the precise vanishing theorem
if $\wt Y_q$ were smooth.) For this, we use the ampleness of $\Oc(Z_\ell)$
to obtain resolutions of $\Oc_{\wt Y_q}$ of the form
$$\Oc(-\delta_{n,\ell}Z_\ell)_{|\wt Y_q}^{\oplus N_{n,\ell}}\to\cdots\to
\Oc(-\delta_{1,\ell}Z_\ell)_{|\wt Y_q}^{\oplus N_{1,\ell}}\to
\Oc(-\delta_{0,\ell}Z_\ell)_{|\wt Y_q}^{\oplus N_{0,\ell}}
\to\Oc_{\wt Y_q}\to 0$$
with $\delta_{n,\ell}>\ldots>\delta_{1,\ell}>\delta_{0,\ell}>0$, for each
$\ell=k\ld n-1$. If $\alpha_\ell\ge\delta_{n,\ell}$, we take the tensor
product of this resolution with~$\Oc(mL)\otimes G^\alpha$. This
reduces the vanishing of $H^i(\wt Y_q,\Oc(mL)\otimes G^\alpha)$, $i>0$,
to the vanishing of all analogous groups with $\alpha'=(\alpha_q\ld
\alpha_{\ell}-d_{s,\ell}\ld\alpha_{n-1})$ in place of $\alpha$. Hence
we are reduced inductively to the case where $\alpha_\ell<\delta_{n,\ell}$
for all~$\ell$, and we may take $m_0=\max_{\{\alpha;\,\alpha_\ell<
\delta_{n,\ell}\}}m_0(\alpha)$.

{\sl Second step}\pointir Suppose that property $(\Pc_k)$ holds for some
index~$k$ in the range $\{q\ld n-1\}$. Observe that $G^\alpha$
is the restriction to $\wt Y_k$ of an invertible sheaf
defined on $\wt Y_{k+1}$, namely
$$\Oc(\alpha_kZ_k)\otimes G^\beta~~~~\hbox{\rm with}~~
\alpha=(\alpha_k,\beta),~~\beta=(\alpha_{k+1}\ld\alpha_{n-1}).$$
The exact sequence
$$0\la\Oc(-Z_q)\la\Oc_{\wt Y_{k+1}}\la\Oc_{Z_k}\la 0$$
defines a (possibly non reduced) scheme structure on~$Z_k$. Taking the
tensor product with $\Oc(\alpha_kZ_k)\otimes G^\beta$ and the associated
cohomology sequence, we get the exact sequence
$$\leqalignno{
&H^{i-1}\big(Z_k,\Oc(mL-pA)_{|Z_k}\otimes\Oc(\alpha_kZ_k)_{|Z_k}
\otimes G^\beta_{|Z_k}\big)\cr
&\qquad\la H^i\big(\wt Y_{k+1},\Oc(mL-pA)_{|\wt Y_{k+1}}\otimes
\Oc((\alpha_k-1)Z_k)\otimes G^\beta\big)&(\dagger)\cr
&\qquad\qquad\la H^i\big(\wt Y_{k+1},\Oc(mL-pA)_{|\wt Y_{k+1}}\otimes
\Oc(\alpha_kZ_k)\otimes G^\beta\big)\cr
&\qquad\qquad\qquad\la H^i\big(Z_k,\Oc(mL-pA)_{|Z_k}\otimes\Oc(\alpha_kZ_k
)_{|Z_k}\otimes G^\beta_{|Z_k}\big).\cr}$$

\than Claim 3.5|For every subscheme $S\subset Z_k\subset \wt Y_{k+1}$
and every coherent sheaf $\Fc$ on~$S$, we have
$$H^i\big(S,\Oc(mL-pA)_{|S}\otimes\Oc(\alpha_kZ_k)_{|S}\otimes G^\beta_{|S}
\otimes\Fc\big)=0$$
for all $i>k-q\ge 0$, $\alpha\in\nb^{n-k}$ and $m\ge C(\Fc)(p+1)$.
\fintha

We apply this to $S=Z_k$ itself and $\Fc=\Oc_S=\Oc_{Z_k}$. Then $(\dagger)$
implies that the groups
$$H^i\big(\wt Y_{k+1},\Oc(mL-pA)_{|\wt Y_{k+1}}\otimes\Oc(\alpha_kZ_k)
\otimes G^\beta\big)$$
are independent of $\alpha_k$ for all $i>k+1-q$, $\beta\in\nb^{n-k-1}$ and
$m\ge C(p+1)$. However, these groups vanish for $\alpha_k$ large since
$\Oc(Z_k)$ is ample on~$\wt Y_{k+1}$. Hence they vanish for $\alpha_k=0$
and for all $i>k+1-q$, $\beta\in\nb^{n-k-1}$ and $m\ge C(p+1)$.
Therefore property $(\Pc_{k+1})$ holds.

{\sl Third step: $(\Pc_k)$ implies Claim 3.5}\pointir We prove the
claim by induction on~$\dim S$, the result being obvious if~$\dim S=0$.
In fact, it is sufficient to prove the claim when $S$ is reduced.
Otherwise, let $\Nc$ be the sheaf of nilpotent elements of the
structure sheaf~$\Oc_S\,$; then $(\Nc^j\Fc)$ defines a finite
filtration of $\Fc$ and the quotients $\Nc^j\Fc/\Nc^{j+1}\Fc$ are
coherent sheaves over~$S_\red$. Therefore, if the vanishing property
3.5 holds on $S_\red$, it also holds on $S$ with $C(\Fc)=\max_j
C(\Nc^j\Fc/\Nc^{j+1}\Fc)$. We can also suppose $\Fc=\Oc_S$, otherwise
we are reduced to this case by taking resolutions of $\Fc$ by locally
free sheaves of the form $\Oc(-dA)_{|S}^{\oplus N})$.

Now, suppose that $S$ is reduced and $\Fc=\Oc_S$. The map $\mu=\nu_{k+1|S}:
S\to Y_k$ can be lifted to a map $\wt\mu:\wt S\to\wt Y_k$. This gives a
commutative diagram
$$\eqalign{
\wt S&\buildo{\displaystyle\wt\mu}\over\la
\wt Y_k\buildo{\displaystyle\wt j_{k,k+1}}\over\la\wt Y_{k+1}
\phantom{\bigg)}\cr
\nu\big\downarrow&\phantom{\la}~~~\big\downarrow \nu_k
\phantom{\la}~~~~\big\downarrow \nu_{k+1}\cr
S&\buildo{\displaystyle\mu}\over\la     
Y_k\buildo{\displaystyle j_{k,k+1}}\over\lhra Y_{k+1}\cr}$$
where the vertical arrows are the normalization maps. Moreover, the diagram
commutes with the diagonal injection $S\hookrightarrow\wt Y_{k+1}$. 

Since we have vanishing on $\wt Y_k$ by hypothesis $(\Pc_k)$ and
since $\wt\mu$ is a finite map, the Leray spectral sequence implies
$$\eqalign{
H^i\big(\wt S,\Oc(mL&{}-pA)_{|\wt S}\otimes
\Oc(\alpha_kZ_k)_{|\wt S}\otimes G^\beta_{|\wt S}\big)\cr
&=H^i\big(\wt Y_k,\Oc(mL-pA)_{|\wt Y_k}\otimes\Oc(\alpha_kZ_k)_{|\wt Y_k}
\otimes G^\beta_{|\wt Y_k}\otimes\wt\mu_\star\Oc_{\wt S}\big)=0\cr}$$
for $i>k-q$, $\alpha\in\nb^{n-k}$ and $m\ge C(p+1)$. Taking the direct
image by~$\nu$, this gives
$$H^i\big(S,\Oc(mL-pA)_{|S}\otimes\Oc(\alpha_kZ_k)_{|S}\otimes
G^\beta_{|S}\otimes\nu_\star\Oc_{\wt S}\big)=0.\leqno(\dagger\dagger)$$
Let $\Ic\subset\Oc_S$ be the sheaf of universal denominators of
$\nu_\star\Oc_{\wt S}$, i.e., the largest ideal $\Ic\subset\Oc_S$ such that
$\Ic\cdot\nu_\star\Oc_{\wt S}\subset\Oc_S$. The support of $\Oc_S/\Ic$
coincides with the sub\-va\-riety of non normal points in~$S$. Hence, for
$d>0$ large enough, $\Oc(dA)_{|S}\otimes\Ic$ has a global section $u$
whose set of zeros is a subvariety $S'\subset S$ with $\dim S'<\dim S$
at every point. We get an exact sequence
$$0\la\Oc(-dA)_{|S}\otimes\nu_\star\Oc_{\wt S}\buildo u\times\over\la
\Oc_S\la\Fc\la 0,$$
where $\Fc$ is supported on~$S'$. By the induction hypothesis on $\dim S$
we have
$$H^i\big(S',\Oc(mL-pA)_{|S'}\otimes\Oc(\alpha_kZ_k)_{|S'}\otimes
G^\beta_{|S'}\otimes\Fc\big)=0$$
for $i>k-q$ and $m\ge C(\Fc)(p+1)$, while $(\dagger\dagger)$ implies
$$H^i\big(S,\Oc(mL-pA)_{|S}\otimes\Oc(\alpha_kZ_k)_{|S}\otimes
G^\beta_{|S}\otimes\Oc(-dA)_{|S}\otimes\nu_\star\Oc_{\wt S}\big)=0$$
for $i>k-q$ and $m\ge C(p+d+1)$. The above exact sequence shows that the
cohomology groups of the central term also vanish in degrees $i>k-q$, as
asserted by Claim~3.5.\finpr\vskip8pt

Theorem~3.4 leads in a natural way to the following problem.

\than 3.6. Problem|Given a line bundle $L\in\Pic(X)$ such that
$\sigma_+(L)\le n-q$, under which conditions can one conclude that
$L$ is $q$-flag positive~?\fintha

If no further assumptions are made, the answer is in general negative
by Example~5.6 below (in which case we have $n=3$ and $q=2$).
However, we have no counterexample in the most interesting case~$q=1$.
Since the problem seems very difficult to attack in full generality, we
will content ourselves with rather special cases.

\than 3.7. Lemma|Let $X$ be a projective manifold and let $D_1\ld D_{n-1}
\subset X$ be irreducible divisors, such that the $\Oc(D_i)$ are nef and
$Y_i=D_i\cap\ldots\cap D_{n-1}$ is \hbox{$i$-dimensional}. Let $L$ be a line
bundle with $L\cdot Y_1>0$. Then there is an ample $1$-flag for~$L$ of the
form $Y'_i=D'_i\cap\ldots\cap D'_{n-1}$ with all $D'_j$ being very ample,
nonsingular and intersecting transversally.
\fintha

\proof Fix an ample divisor $H$ on~$X$. Choosing smooth generic members
$D'_i\in|k(mD_i+H)|$, $k\gg 0$, $m\gg 0$, we get the desired $1$-ample
flag for $L$. (Note that $L\cdot Y'_i\sim (km)^{n-1}L\cdot Y_1$ as
$k,m\to+\infty$.)\finpr

\than 3.8. Proposition|Let $X=\pb(E)\buildo\pi\over\la C$ be a
$\pb_{n-1}$-bundle over a smooth curve and let $L$ be a line bundle on $X$
with $\sigma_+(L)\le n-1$. Then $L$ is $1$-flag positive. Moreover
there is a base change $F^\star:\wt X=\pb(f^\star E)\to X=\pb(E)$ given
by a finite map $f:\wt C\to C$, such that the pull-back $F^\star L$
admits an ample $1$-flag 
$$Y_1\subset\ldots\subset Y_{n-1}\subset X$$
of the form $Y_i=D_i\cap\ldots\cap D_{n-1}$ with $D_i\ld D_{n-1}$
very ample and intersecting transversally.
\fintha

\proof All line bundles over $\pb(E)$ are of the form
$L=\Oc_{\pb(E)}(k)\otimes\pi^\star G$ for some $G\in\Pic(C)$. If
$k=0$, then $G$ must have positive degree and the conclusion is obtained
by taking an arbitrary ample flag such that $Y_1$ is mapped onto $C$
by~$\pi$. Hence suppose $k\ne0$. After taking a finite covering $f:\wh
C\to C$, we may assume that $G=\Oc_C$, replacing $E$ by
$f^\star(E\otimes G^{1/k})$. By Proposition~3.3, the conclusion is
invariant by finite maps, hence we may replace $C$ by $\wh C$ and assume
$L=\Oc_{\pb(E)}(\pm 1)$.

In the case $L=\Oc_{\pb(E)}(1)$, we take $Y_1\subset\ldots\subset
Y_{n-1}$ to be a flag of linear subspaces in a fibre
$Y_{n-1}=\pb(E_x)$. Then $D_{n-1}=\pb(E_x)$ and $D_2\ld D_{n-2}$ can be
taken to be suitable $\pb_{n-2}$-subbundles of~$\pb(E)$ defined by
sections of an ample line bundle $\Oc_{\pb(E)}(1)\otimes\pi^\star H$.
Note that $D_{n-1}$ is just nef, but Lemma~3.7 can be applied to get an
ample $1$-flag.

The only nontrivial case is $L=\Oc_{\pb(E)}(-1)$. Our assumption
$\sigma_+(L)\le n-1$ implies that $E^\star$ is ample (cf.~2.7$\,$(ii)).
By a result of Campana and Flenner [CF91], there is a finite map
$f:\wt C\to C$, a very ample line bundle $A$ on $\wt C$ and a surjective
morphism
$$A^{\oplus N}\la f^\star E^\star\la0.$$
Replacing $C$ by $\wt C$ and $E$ by $f^\star E$, we may assume that we
have $f={}$Id. Then $E^\star\otimes A^\star$ is generated by sections
and we get bundle-monomorphisms
$$\varphi_j:A^{\oplus(n-1)}\lhra E^\star,$$
sending the sum of factors $A$ of indices $m\in\{1\ld n\}\ssm\{j\}$ in
$A^{\oplus N}$ into~$E^\star$. (Otherwise, change $A^{\oplus N}\to E$ by a
constant linear transformation acting on $A^{\oplus N}$ to achieve this
property.) These monomorphisms define
exact sequences
$$0\la K\la E\buildo\varphi_j^\star\over\la (A^\star)^{\oplus(n-1)}\la 0
\leqno(\star\star)$$
and divisors
$$D_j:=\pb\big((A^\star)^{\oplus(n-1)}\big)\subset\pb(E).$$
These divisors intersect transversally. We set
$$Y_j:=D_j\cap\ldots\cap D_{n-1}.$$
(a) First, we check that $\Oc(D_j)$ is nef in~$\pb(E)$. Since $\Oc(D_j)=
\Oc_{\pb(E)}(1)\otimes\pi^\star(K^\star)$, we have to show that
$E\otimes K^\star $ is nef. From $(\star\star)$, we get
$$0\la\Oc\la E\otimes K^\star\la (A^\star\otimes K^\star)^{\oplus(n-1)}\la0.$$
Hence it is enough to show that $A^\star\otimes K^\star$ is nef. However, this
is clear because $K^\star$ is a quotient of $E^\star$ and thus of
$A^{\oplus N}$.

\noindent(b) Now, we have to check that $L\cdot Y_1>0$. First suppose that
$n=2$, thus $Y_1=D_1$. Then
$$\eqalign{L\cdot D_1
&=c_1\big(\Oc_{\pb(E)}(-1)\big)\cdot
\Big(c_1\big(\Oc_{\pb(E)}(1)\big)+\pi^\star c_1(K^\star)\Big)\cr
&=-c_1(E)+c_1(K)=c_1(A)>0\cr}$$
thanks to $(\star\star)$ and to the relations $h^2-\pi^\star c_1(E)h+
\pi^\star c_2(E)=0$, $\pi_\star h=1$ satisfied by
$h=c_1\big(\Oc_{\pb(E)}(1)\big)$.

\noindent(c) In the general case $n>2$, we have
$$Y_1=\pb(A^\star)\subset Y_2=\pb(A^{\star\oplus 2})\subset\pb(E),~~~~
L_{|Y_2}=\Oc_{\pb(A^{\star\oplus 2})}(-1),$$
where the inclusions are defined by the projections
$E\to A^\star$ (resp. $E\to A^{\star\oplus 2}$) on the $n$-th factor
(resp. on the first and $n$-th factor). This reduces the problem to
the case of the rank $2$ bundle $E=(A^\star)^{\oplus2}$.
By applying case (b), we thus get $L\cdot Y_1>0$, as desired.\finpr

\titre 4. The cone of ample curves|

In this section, we concentrate ourselves on the study of the cone of
``ample curves'', in relation with effective divisors and ample
$1$-flags.

\than 4.1. Theorem|For an irreducible curve $C\subset X$, consider the
following properties.
\smallskip
\item{\rm(i)} $C$ is the first member $Y_1$ of an irreducible ample
$1$-flag $Y_1\subset\ldots\subset Y_{n-1}\subset X$.\hfill\break
$($We say that a flag is irreducible if all subvarieties $Y_i$ are
irreducible.$)$
\smallskip
\item{\rm(ii)} $\{C\}\in N_\amp(X)$.
\smallskip
\item{\rm(iii)} $\{C\}\in\ov N_\amp(X)$.
\smallskip
\item{\rm(iv)} The normal bundle $N_{C/X}=\Hom_\Oc(\Ic_C/\Ic_C^2,\Oc_C)$ is
ample\hfill\break $($i.e., $\Oc_{\pb(N_{C/X})}(1)$ is ample$)$.
\smallskip
\item{\rm(v)} The normal bundle $N_{C/X}$ is nef $($i.e.,
$\Oc_{\pb(N_{C/X})}(1)$ is nef$)$.
\smallskip
\item{\rm(vi)} The current of integration $[C]$ is weakly cohomologous to a
smooth positive definite form of bidegree $(n-1,n-1)$, i.e. $[C]=u+
\partial R+\ovp\ov R+S$ where $u$ is a smooth positive definite
$(n-1,n-1)$-form with $\partial\ovp u=0$, $R$ is a current of type
$(n-2,n-1)$, and $S$ is a $d$-closed $(n-1,n-1)$-current whose cohomology
class $\{S\}\in H^{n-1,n-1}(X)$ is orthogonal to $NS^1(X)\subset
H^{1,1}(X)$.
\smallskip
\item{\rm(vii)} The current of integration $[C]$ is weakly cohomologous to
a smooth semiposi\-tive form of bidegree $(n-1,n-1)$ $($as in {\rm(vi)},
but with $u\ge 0$ only$)$.
\smallskip
\item{\rm(viii)} There is a family of generically irreducible curves $(C_t)$
covering $X$ such that $C_0=mC$ as a cycle.
\smallskip
\noindent Then we have the following implications:
$$\leqalignno{
&\hbox{\rm(i)}\Rightarrow\hbox{\rm(ii)}\Rightarrow\hbox{\rm(iii)},
&\hbox{\rm(a)}\cr
&\hbox{\rm(i)}\Rightarrow\hbox{\rm(iv)},~\hbox{\sl if additionally
every $Y_i$ is Cartier in $Y_{i+1}$,}&\hbox{\rm(b)}\cr
&\hbox{\rm(iv)}\Rightarrow\hbox{\rm(v)}\Rightarrow\hbox{\rm(iii)},
\quad\hbox{\rm(viii)}\Rightarrow\hbox{\rm(iii)}&\hbox{\rm(c)}\cr
&\hbox{\rm(ii)}\Leftrightarrow\hbox{\rm(vi)}\Rightarrow\hbox{\rm(vii)}
\Rightarrow\hbox{\rm(iii)}&\hbox{\rm(d)}\cr}$$
\fintha

\proof The implications $\hbox{\rm(ii)}\Rightarrow\hbox{\rm(iii)}$, 
$\hbox{\rm(iv)}\Rightarrow\hbox{\rm(v)}$ and $\hbox{\rm(vi)}\Rightarrow
\hbox{\rm(vii)}$ are obvious.

\noindent (a) We only have to show $\hbox{\rm(i)}\Rightarrow\hbox{\rm(ii)}$.
Let $C=Y_1$ be the first member of the ample irreducible flag  
$$Y_1\subset Y_2\subset\ldots\subset Y_{n-1}\subset X.$$
Let $D\in\ov K_\eff(X)$. We first check $D\cdot C\ge 0$, so as to
conclude that $C\in\ov N_\amp(X)$. Otherwise $L=\Oc(-D)$ satisfies
$L\cdot C>0$, i.e. $L_{|C}$ is ample. By Theorem~3.4 we conclude that
$\sigma_+(L)\le n-1$, hence $D\notin\ov K_\eff(X)$ by 1.5~(ii),
contradiction. This shows that the cone $N_1$ generated by all
curves $C=Y_1$ is contained in $\ov N_\amp(X)$.

Now, we check that $C$ cannot be a boundary point of $\ov N_\amp(X)$.
Let $H$ be a very ample line bundle on~$X$ and let
$(Z_k)_{1\le k\le n-1}$ be Cartier divisors in $\wt Y_{k+1}$ such that
$Y_k=\nu_{k+1}(\Supp Z_k)$ (see Def.~3.1). There is an integer $m_1>0$
such that $m_1Z_1-H_{|\wt Y_2}$ is very ample on~$\wt Y_2$. Select an
irreducible divisor $Z'_1\in|m_1Z_1-H_{|\wt Y_2}|$. Since the
$1$-cycle $(\nu_2)_\star(Z_1)$ is a multiple $p_1Y_1$, we conclude
that $Y'_1=(\nu_2)_\star(Z'_1)\equiv m_1p_1Y_1-(H\cdot Y_2)\in N_1$. Quite
similarly, we have irreducible very ample divisors
$Z'_k\sim m_kZ_k-H_{|\wt Y_{k+1}}$ and $k$-dimensional cycles
$Y'_k=(\nu_{k+1})_\star(Z'_k)\equiv m_kp_kY_k-H\cdot Y_{k+1}$. Moreover
$$H^{k-1}\cdot Y'_k\equiv m_kp_kH^{k-1}\cdot Y_k-H^k\cdot Y_{k+1}\in\ov N_1$$
because $H$ is very ample and $Y'_k$ is a member of a $k$-flag.
By multiplying the above line by $m_{k+1}p_{k+1}\ldots m_{n-1}p_{n-1}$ and
adding everything together for $k=1,\ldots,n-1$, we find
$$m_1p_1\ldots m_{n-1}p_{n-1}Y_1-H^{n-1}\in\ov N_1\subset\ov N_\amp(X).$$
However $H^{n-1}$ is in the open cone~$N_\amp(X)$. Therefore
$\{Y_1\}\in N_\amp(X)$ and $N_1\subset N_\amp(X)$.
 
\noindent(b) Next we prove $\hbox{\rm(i)}\Rightarrow\hbox{\rm(iv)}$ in
case every $Y_i$ is Cartier in $Y_{i+1}$. This is done easily by
induction using the exact sequences of vector bundles 
$$0\longrightarrow N_{C/Y_i}\longrightarrow N_{C/Y{i+1}}
\longrightarrow N_{Y_i/Y_{i+1}|C}\longrightarrow 0.$$

\noindent (c) The implication $\hbox{\rm(v)}\Rightarrow\hbox{\rm(iii)}$
is more complicated. Let $C\subset X$ be an irreducible curve such that
$N_{C/X}$ is nef. Let $D\subset X$ be an irreducible hypersurface.
It is sufficient to prove $D\cdot C\ge 0 $ if $C\subset D$. Let
$\Ic$  be the ideal defining $C$ in $X$, $\Jc$ the ideal
defining $C$ in $D$. Then there is an exact sequence 
$$\Oc_X(-D)_{|C}=N_{D/X|C}^\star\longrightarrow\Ic/\Ic^2
\longrightarrow\Jc/\Jc^2\longrightarrow 0.$$
If $C\not\subset\Sing(D)$, the map $\alpha:N_{D/X|C}^\star
\longrightarrow\Ic/\Ic^2$ is non-zero, hence injective. Thus 
$$\alpha^\star:N_{C/X}\longrightarrow N_{D/X|C}
$$
is generically surjective. Since $\Im\alpha^\star $ is again nef  
(observe that $\pb(\Im\alpha^\star)\subset\pb((\Ic/\Ic^2)^\star)$
and $\Oc_{\pb(\Im\alpha^\star)}(1)=\Oc_{\pb((\Ic/\Ic^2)^\star)}(1) 
_{|\pb(\Im\alpha^\star)}$), it follows clearly that $\deg(N_{D/X|C})
=D\cdot C\ge 0$. If however $\alpha=0 $ we have slightly to modify
the above arguments. We want to "generalize" the diagram 
$$\eqalign{
&\Ic/\Ic^2\cr
\alpha\nearrow&\qquad\searrow\cr
N_{D/X|C}^\star~&\smash{\buildo{\textstyle\gamma}\over\longrightarrow}~
\Omega_{X|C}^1\cr}$$
where $\gamma$ was given just by taking the linear
part of the power series expansion along $D$ of a local equation $f$ for~$D$.
Suppose now that $C\subset\Sing(D)$, so that $\alpha=0$. Let $m$
be the vanishing order of $f$ along $C$. By taking the homogeneous part
of degree $m$ in~$f$, we obtain a map
$$\wt\gamma:N_{D/X|C}^\star
\longrightarrow S^m(\Omega_{X|C}^1).$$
As before $\wt\gamma$ factors over
$$\wt\alpha:N_{D/X|C}^\star\longrightarrow S^m(\Ic/\Ic^2).$$
Since $S^m((\Ic/\Ic^2)^\star)=S^m(\Ic/\Ic^2)^\star $ is again~nef,
we conclude in the same way as before that $D\cdot C\ge 0$.

\medskip\noindent 
The implication $\hbox{(viii)}\Rightarrow\hbox{(iii)}$ is again easy.
Let $(C_t)$ be a family of curves covering $X$ such that $C_0=mC$ as
cycles. It is sufficient to prove that for every effective divisor~$D$ we
have $D\cdot C\ge 0$. But this is obvious: we choose $t$ general, such that
$C_t\not\subset\Supp(D)$.

\medskip\noindent 
(d) Finally, the implications $\hbox{(vi)}\Rightarrow\hbox{(ii)}$ and
$\hbox{(vii)}\Rightarrow\hbox{(iii)}$ follow from the following fact:
if $D\in \ov K_\eff(X)$ and $[C]\sim u$ weakly, $u\ge 0$, then
$$D\cdot C=\int_D u\ge 0,$$
with strict inequality if $D\ne 0$ and $u>0$. We now prove the converse
implication $\hbox{(ii)}\Rightarrow\hbox{(vi)}$ as a consequence of
the Hahn-Banach theorem.

Let $\Dc'_{1,1}$ be the space of currents of bidegree $(1,1)$ equipped
with the weak topology. In $\Dc'_{1,1}$ we consider the closed convex
cone of positive currents $\Dc^+_{1,1}$. Let $\omega$ be a K\"ahler
metric on~$X$. It is well-known that the intersection of $\Dc^+_{1,1}$
with the hyperplane of currents $T$ such that $\int_X T\wedge\omega^{n-1}
=1$ is weakly compact, so $\Dc^+_{1,1}$ has a compact base.
On the other hand, we pick a class $\{C\}\in N_\amp(X)$ and consider
the vector space $\Ec_C$ of $d$-closed currents $T$ such that the
cohomology class $\{T\}$ belongs to $NS^1(X)$ and satisfies
$\{T\}\cdot\{C\}=0$. We claim that $\Dc^+_{1,1}\cap\Ec_C=\{0\}$.
In fact the set of classes in $NS^1(X)$ represented by positive currents
is precisely $\ov K_\eff(X)$ (see e.g. [De92], Prop.~6.1$\,$(vi)),
hence a class $\Dc^+_{1,1}\cap\Ec_C$ is the set of classes
$\{T\}\in\ov K_\eff(X)$ such that $\{T\}\cdot\{C\}=0$, but this set
reduces to $\{0\}$ by our assumption $\{C\}\in N_\amp(X)$.
The Hahn-Banach theorem implies that there is a linear form $u$
on $\Dc'_{1,1}$ such that $u$ is positive on $\Dc^+_{1,1}\ssm\{0\}$
and vanishes on~$\Ec$. From the positivity property, we conclude that
$u$ is a positive definite smooth $(n-1,n-1)$-form on~$X$. On the other
hand, since $\Ec_C$ contains $\partial\ovp\Dc'_{0,0}$, we have $\partial
\ovp u=0$. In fact $\Ec_C$ is the set of $d$-closed $(1,1)$-currents
whose classes are orthogonal to $\rb\{C\}\oplus NS^1(X)^\perp
\subset H^{n-1,n-1}(X)$. Hence there is a class $\lambda\{C\}+\{S\}$
with a constant $\lambda\in\rb$ and $\{S\}\in NS^1(X)^\perp$ such that
$\{u-\lambda[C]-S\}$ vanishes on all $d$-closed $(1,1)$-currents.
But this means precisely that there is a $(n-2,n-1)$-current $R$
such that $u-\lambda[C]-S=\partial R+\ovp\ov R$. Now $\lambda$
must be positive, otherwise $u>0$ would have nonpositive intersection
product with effective divisors. Hence we get the desired conclusion
(vi) by multiplying $u$, $S$, $R$ with $\lambda^{-1}$.\finpr

\rem 4.2. Remark|In general neither (vi) nor (ii) implies (v). For
example take a Fano $3$-fold $X$ with $b_2=1$ and index~$1$. Then
$\{C\}\in N_\amp(X) $ (resp. $[C]$ is cohomologous to a smooth
positive definite form of bidegree $(2,2)$ for every curve $C\subset X$,
however $N_{C/X}$ might not be nef. Take e.g. a line in $X$,
this is a smooth rational curve $C$ with $N_{C/X}=\Oc\oplus
\Oc(-1)$ or $\Oc(1)\oplus\Oc(-2)$.

\noindent However we may hope that (vi) or (ii) imply (v) on the level
of cones, e.g., given $\alpha\in N_\amp(X) $ which is represented by
curves, then one can find $C$ with $\alpha=\lambda \{C\}$ with
$\lambda\in\qb^+$ such that $N_{C/X}$ is nef (or even ample). An
implication $\hbox{(viii)}\Rightarrow\hbox{(v)}$ is true generically as
we show in Theorem~4.3, however there seems to be no hope to prove
$\hbox{(iv)}\Rightarrow\hbox{(viii)}$ or $\hbox{(v)}\Rightarrow
\hbox{(viii)}$ despite a concrete counterexample seems still not to
exist, at least to our knowledge.

\medskip\noindent Let $N_\mu(X)$ (or simply $N_\mu$) be the convex cones
generated by all classes of curves in 4.1$(\mu)$, $1\le\mu\le 8$,
and let $\ov N_\mu$ be their closures. Then we have:

\than 4.3. Theorem|There are the following relations between the
various cones: $\ov N_1\subset \ov N_2=\ov N_3=\ov N_6=\ov N_7,\quad
\ov N_8\subset \ov N_5\subset \ov N_3,\quad \ov N_4\subset \ov N_5$.
\fintha

\proof The equalities $\ov N_2=\ov N_3$ and $\ov N_6=\ov N_7$ are obvious.
Moreover $\ov N_2=\ov N_6$, $\ov N_1\subset\ov N_2$ and $\ov N_5\subset
\ov N_3$ follow from~4.1.

\noindent The only remaining case is $\ov N_8 \subset \ov N_5$. Let $(C_t)$
be a covering family of irreducible curves. We show that for general~$t$,
$N_{C_t/X}$ is nef. Consider the graph of the family  
$$\eqalign{
Z&\buildo{\textstyle p}\over\longrightarrow X\cr
q\big\downarrow&\cr
T&\cr}$$
We may assume $\dim T=n-1$, hence $p$ is generically finite. Let $\Ic$
be the ideal of $C_t$ in $X$, $\Jc$ the ideal of $C_t$ in~$Z$. As
$C_t$ is a fiber of~$q$, $(\Jc/\Jc^2)^\star$ is nef; it is even trivial
for generic~$t$. Now the canonical map 
$$p^\star(\Omega^1_{X|C_t})\longrightarrow\Omega^1_{Z|C_t}$$
is generically surjective. Thus for generic~$t$, $N_{C_t/Z }$ embeds
into $p^\star(N_{C_t/X})$ and consequently --both sheaves being of
same rank-- $N_{C_t/X}$ is nef.\finpr

In this context, the special case $q=1$ of Problem~3.6 can be restated
in a slightly stronger form:

\than 4.4. Proposition|The following three statements are equivalent:
\smallskip
\item{\rm(i)} A line bundle $L\in\Pic(X)$ satisfies $\sigma_+(L)\le n-1$
if and only if there is an irreducible ample $1$-flag
$Y_1\subset\ldots\subset Y_{n-1}\subset X$ such that $L_{|Y_1}$ is ample.
\smallskip
\item{\rm(ii)} $\ov N_1(X)=\ov N_2(X)\,;$
\smallskip
\item{\rm(iii)} $N_1(X)=N_2(X)$.
\fintha

\proof Thanks to 1.5$\,$(ii), Statement~4.4(i) can be restated:
$\{L^\star\}\notin\ov K_\eff(X)$ if and only if there is an irreducible
ample $1$-flag $Y_1\subset\ldots\subset Y_{n-1}\subset X$ with
$L_{|Y_1}$ ample. Changing $L$ into $L^\star$ and taking the negation,
this becomes: $\{L\}\in\ov K_\eff(X)$ if and only if for every irreducible
ample $1$-flag $Y_1\subset\ldots\subset Y_{n-1}\subset X$, then $L\cdot
Y_1\ge 0$. By duality, this means precisely that the dual cone
$\ov N_\amp(X)$ of $\ov K_\eff(X)$ is the closure $\ov N_1(X)$ of the convex
cone generated by all curves~$Y_1$. This is actually equivalent
to $N_\amp(X)=N_1(X)$, because we have shown in the Proof of 4.1(a)
that $N_1(X)$ does not contain boundary points of $\ov N_\amp(X)$.
\finpr

\than 4.5. Proposition|Let $X$ be a projective surface. Then all cones
$\ov N_\mu$ coincide, in particular Statements $4.4\,${\rm(i,ii,iii)}
hold on~$X$.
\fintha

\proof By definition, we have $K_\eff(X)=N_\eff(X)$ in dimension~$2$. Thus
by duality, $N_\amp(X)=K_\amp(X)$. This means that $N_\amp(X)$ is
generated by the irreducible curves $Y_1$ which are ample divisors
in~$X$, i.e. $N_1=N_2$. Now, an irreducible ample curve has an ample
normal bundle $\Oc(C)_{|C}$, hence $N_1\subset N_4$. By Theorem~4.3, this
implies the equality of all cones~$\ov N_\mu$.\finpr

\rem 4.6. Remarks and open problems|\smallskip
\item{(1)} Observe that the inclusions $\ov N_2\subset\ov N_1\subset\ov N_4$
and $\ov N_1\subset\ov N_8$ would imply the equality of all cones~$\ov N_\mu$.
The first inclusion seems to be the most difficult step.

\item{(2)} The comparison between $\ov N_1$ and $\ov N_4$ is not quite clear,
however the (a priori) slightly smaller cone $\ov N_1'$  consisting of
classes of those $C$ which are the first members of an ample irreducible flag
$$Y_1\subset\ldots\subset Y_{n-1}\subset X$$
such that every $Y_i$ is Cartier in $Y_i$, is contained in $\ov N_4$.

\item{} {\sl Problem}\/: Is $\ov N_1=\ov N_1'\,?$ Is at least
$\ov N_1\subset\ov N_4\,$?

\item{(3)} Is always $\ov N_4=\ov N_5\,$?

\item{(4)} What is the relation between $\ov N_8$ and $\ov N_1\,$? Given a
covering family $(C_t)$ of irreducible curves in~$X$, it is at least true
that we obtain a flag of irreducible varieties
$$Y_1\subset\ldots\subset  Y_{n-1}\subset X$$ such that the
deformations of $Y_i$ fill up $Y_{i+1}\,$.

\rem 4.7. Remark|Exceptional curves are far from being ample. This is made
precise by the following statement. Let $X$ be a projective manifold,
$f:X\longrightarrow Y$ be a modification such that $Y$ carries a big
line bundle (e.g. Y is smooth or normal projective). Let $C\subset X$
be an irreducible curve with $\dim f(C)=0$. Then $\{C\}\not\in
\ov N_\amp(X)$.

At this point, one should note that $N_\amp(X)$ is in general
not generated by complete intersection curves of the type
$H_1\cdot\ldots\cdot H_{n-1}$ with divisors $H_i$ that are ample.
This is shown by the following example:

\rem 4.8. Example|Let $X=\pb(E)$ over $\pb_1$, with
$E=\Oc\oplus\Oc(-1)\oplus\Oc(-2)$. Let $p$ be the projection. Assume that
$N_\amp(X)$ is generated by curves $H_1\cdot H_2$ with $H_i$ ample
on~$X$. Let $L=\Oc_{\pb(E)}(1)\otimes p^\star(\Oc(\lambda))$ with
$\lambda\in\qb$. Then for every $\lambda<0$, we have
$\{L\}\notin\ov K_\eff(X)$. We will show that for $\lambda\ge-1$ there cannot
exist $H_1$, $H_2$ ample such that $L\cdot H_1\cdot H_2<0$.

Write $H_i=\Oc_{\pb(E)}(1)\otimes p^\star(\Oc(a_i))$ as
$\qb$-divisors, after possibly rescaling $H_i$ by a positive factor.
Since $H_i=\Oc_{\pb(E\otimes\Oc(a_i))}(1)$, $H_i$ is ample if and only
if $a_i>2$. Denoting $h=c_1(\Oc_{\pb(E)}(1))$, we have $c_1(E)=-3$, thus
$h^3+3h^2=0$ and $p_\star h^2=1$. This implies
$$L\cdot H_1\cdot H_2=(h+\lambda)(h+a_1)(h+a_2)=p_\star\big
(h^3+(a_1+a_2+\lambda)h^2\big)=a_1+a_2+\lambda-3>\lambda+1.$$
Hence we must have $L\cdot H_1\cdot H_2>0$ for $\lambda\ge -1$,
although $\{L\}\notin\ov K_\eff(X)$ for $\lambda<0$. This example also shows
that $f$ cannot be taken to be the identity map in the conclusion of
Theorem~3.8.\finpr

\medskip We now discuss Problem~3.6 in the case $\sigma_+ (L)\le n-1$
for $3$-folds and only for the most important line bundle, namely the
anti-canonical bundle $-K_X$. The condition $\sigma_+(-K_X)\le 2$ is
equivalent to $K_X\notin\ov N_\eff(X)$ which in turn just means
$\kappa (X)=-\infty$, i.e. $X$ is uniruled ([Mo88]). Instead of $X$ being
smooth, we deal even with normal projective varieties having at most
terminal singularities~([KMM87]). 

\than 4.9. Proposition|Let $X$ be a normal projective $\qb$-Gorenstein
$3$-fold, $\varphi:X\to Y$ a surjective holomorphic map to a
normal projective variety $Y$ of dimension at most~$2$, such that
$-K_{X|\varphi^{-1}(y)}$ is ample for general $y\in Y$. Then there exists 
$H$ ample on $X$ with $K_X\cdot H^2<0$. In particular there exists an
irreducible ample $1$-flag $Y_1\subset Y_2\subset X$ such that
$-K_{X|Y_1}$ is ample.
\fintha

\proof If $\dim Y=0$, then $-K_X$ is ample and there is nothing to prove.
So assume $\dim Y\ge 0$ and fix an ample divisor $H_0$ on~$X$. If
$K_X\cdot H_0^2<0$, we are done. Thus we assume $K_X\cdot H_0^2\ge 0$.
Choose $L\in\Pic(Y)$ ample and define
$$H=H_0+\varphi^\star(mL)$$
with $m\gg 0$ to be specified in a moment. Clearly $H$ is ample. We have 
$$K_X\cdot H^2=K_X\cdot H_0^2+2 K_X\cdot H_0\cdot\varphi^\star (mL)+K_X
\cdot(\varphi^\star(mL))^2.$$
(a) If $\dim Y=2$, then $K_X\cdot\varphi^\star (L)^2<0$, since $L\cdot L$
is cohomologous to a multiple of a generic fibre and $-K_{X|\varphi^{-1}(y)}$
is ample on such fibres. Hence for $m\gg 0$, $K_X\cdot H^2<0$.

\noindent (b) If $\dim Y=1$, we obtain $\varphi^\star(mL)^2=0$.  But now
$$K_X\cdot H_0\cdot\varphi^\star(mL)<0,$$
hence again $K_X\cdot H^2<0$ for $m\gg 0$. This finishes the proof of~4.9,
the existence of the ample $1$-flag being an obvious consequence of the
main statement.

\than 4.10. Corollary|Let $X$ be a normal projective $\qb$-Gorenstein
$3$-fold with at most terminal singularities. Assume $\kappa(X)=-\infty$.
If one of the following conditions is satisfied, there
exists an ample divisor $H$ on $X$ with $K_X\cdot H^2<0\,$:
\smallskip
\item{\rm(a)} The dimension of the image of the Albanese map
$\alpha$ of $X$ is 2 $($or, more generally, there exists a surjective
map of $X$ to an irreducible reduced surface whose desingularisation
has Kodaira dimension at least $0)$.
\smallskip
\item{\rm (b)} There exists a modification $\varphi:X\to X'$ onto a normal
projective $\qb$-Gorenstein $3$-fold $X'$ such that $X'$ admits a
map onto $Y$ as in~$4.9$.
\smallskip
\item{\rm(c)} $\kappa(-K_X)\ge 0$.
\fintha

\proof (a) The general fiber of $\alpha$ is $\pb_1$, by Iitaka's conjecture
$C_{3,2}$ or the uniruledness of~$X$, hence we can apply~4.9.

\noindent (b) By 4.9 there exists $H'$ ample on $X'$ such that
$K_{X'}\cdot H^{\prime2}$. We conclude $K_X\cdot\varphi^\star(H')^2<0$,
hence, approximating the nef and big divisor $\varphi^\star(H')$ by ample
$\qb$-divisors, we obtain our claim.

\noindent c) Take $Z\in|-mK_X|$ and any ample divisor~$H$. Then clearly
$Z\cdot H^2>0$, hence our claim follows.

\rem 4.11. Remark|By the minimal model theory [Mo88] every smooth
projective 3-fold $X$ with $\kappa(X)=-\infty$ is birational to a
normal projective variety $X'$ having at most terminal singularities
which admits a map $\varphi : X'\longrightarrow Y$ with $-K_X'$ being
$\varphi$-ample. Hence there exists $H'$ ample on $X'$ such that
$K_X'\cdot H^{\prime2}<0$. However we cannot easily conclude the same
for~$X$. The difficulty is the following: $X'$ arises from $X$ by a
sequence of contractions of prime divisors and by flips. For
contractions of prime divisors everything is settled by 4.10$\,$(b). What
is totally unclear and maybe even false is the following:

\rem 4.12. Problem|Assume $X$ and $X'$ are normal projective varieties
having at most  terminal singularities. Let $\varphi:X\longrightarrow X'$
be a birational rational map of flipping type (in particular
isomorphic in codimension 1). Assume that there exists $H'$ ample on
$X'$ such that $K_{X'}\cdot H'^2<0$. Does there exist an ample divisor $H$
on $X$ with the analogous property$\,$? Slightly more modest would be to
ask for an ample 1-flag for $-K_X$ if there is one for~$-K_{X'}$.

\titre 5. Threefolds with $\sigma_-(K_X)=2$|

Our goal here is to study $3$-folds which have a noneffective
canonical divisor $K_X$. To this end, we introduce the following
definition.

\than 5.1. Definition|Let $X$ be a normal projective variety,
assume $X$ to be $\qb$-Gorenstein. Choose $m\in\nb$ such that $mK_X$ is
Cartier. Then we define:
$$\sigma(X)=\sigma_-(mK_X).$$
\fintha

Note that $\sigma(X)$ does not depend on the choice of $m$ (we could also
take the minimal $m$ such that $mK_X$ is Cartier). We have $0\le\sigma(X)\le
n=\dim X\,$; and $\sigma(X)\ge 1$ is equivalent to saying that $K_X\notin
\ov N_\eff(X)$. Moreover $\sigma(X)=n$ iff $-K_X$ is ample, i.e. $X$
is~$\qb$-Fano. Thus $\sigma(X)$ measures how far $X$ is from being Fano. We
now concentrate on $3$-folds. Then for $X$ smooth, $\sigma(X)\ge 1$ iff
$\kappa(X)=-\infty$ (see [Mi87], [Mo88]), thus it is interesting to
investigate $3$-folds with $\sigma(X)=2$.

\rem 5.2. Example|Let $X$ be a smooth projective $3$-fold,
$\varphi:X\to Y$ a surjective holomorphic map onto a compact Riemann
surface $Y$ such that $-K_X$ is~$\varphi$-ample. Then $\sigma(X)\ge 2$
(in particular, if $g(Y)\ge 1$, then $\sigma(X)=2)$.

\proof Fix $A$ ample on~$X$. Since $-K_X$ is $\varphi$-ample, we have
$R^i\varphi_\star(-mK_X-pA)=0$, for $i\ge 1$ and $m\ge C(p+1)$. Hence
$$H^2(X,-mK_X-pA)=H^2(Y,\varphi_\star(-mK_X-pA))=0$$
for those~$m,p$, i.e. $\sigma_+(-K_X)\le 1$. (Of course here we do not
need $\dim X=3$.)\finpr

Examples of maps $\varphi$ arising in Example~5.2 are contractions
of extremal rays; note also that the proof still works if $X$ is
only normal and $\qb$-Gorenstein.

Now we turn to fibrations over surfaces. Since this seems to be rather
complicated in general, we restrict ourselves to the case of
$\pb_1$-bundles. Recall that for a $\pb_m$-bundle $X=\pb(E)\buildo\pi
\over\la Y$ over a projective manifold $Y$ we have the formula
$$K_X=\pi^\star(K_Y+\det E)\otimes\Oc_{\pb(E)}(-m-1).$$

\rem 5.3. Problem|Let $X=\pb(E)\buildo\pi\over\longrightarrow S$
be a $\pb_1$-bundle over a smooth projective surface~$S$. Under which
conditions is $\sigma(X)=2\,$?

This question will be considered in (5.5), (5.6). Note that in the Fano
case $\sigma(X)=3$, the situation is not so difficult to understand:
$S$ has automatically to be a del Pezzo surface and the condition on
$E$ can be expressed in the following way: $E\otimes(\det E\otimes
K_S)^{-1/2}$ has to be ample. At least for $S=\pb_2$ all those $E$ are
classified (see [SW90]).

\rem 5.4. Examples|\smallskip
\item{(1)} Let $X=\pb_1\times S$ with a projective surface~$S$. Then
$\sigma(X)=\sigma(S)+1$, hence $\sigma(X)\ge 2$ iff $\kappa(S)=-\infty\,$,
with equality iff $S$ is not Fano.
\smallskip
\item{(2)} Now assume $X=C\times S$ with $g(C)\ge 1$. Then $\sigma(X)=
\sigma(S)$, hence $\sigma(X)=2$ iff $S$ is Fano. In fact,
let $A=A_1\stimes A_2$ with $A_1$ ample on~$C$, $A_2$ ample on~$S$. Then
$$\eqalign{
H^q(X,mK_X+pA)&=H^q\big(X,\Oc(mK_C+pA_1)\stimes\Oc(mK_S+pA_2)\big)\cr
&=H^0(X,\Oc(mK_C+pA_1))\otimes H^q(S,\Oc(mK_S+pA_2))\cr}$$
by K\"unneth, hence $\sigma_-(K_X)=\sigma_-(K_S)$.
\smallskip
\item{(3)} Let $X=\pb\big(\Oc_{\pb_2}(1)\oplus\Oc_{\pb_2}(n)\big)$,
$n\in\nb$. Then
$$\eqalign{
&\sigma(X)=3\quad\hbox{\rm iff$\quad n=1$ or $2$,}\cr
&\sigma(X)=2\quad\hbox{\rm iff$\quad n=3$.}\cr}$$
(See 5.5 below for more general calculations).

\rem 5.5. Remarks|Let $X=\pb(E)\to S$ be a $\pb_1$-bundle over
a projective surface~$S$. We may assume $E$ to be ample. In order to compute
$\sigma(X)$ we have to consider $H^1(X,mK_X+pA)$ for some ample $A$. We
choose $A=\Oc_{\pb(E)}(1)$. Then
$$H^1(X,mK_X+pA)\simeq H^0(S,S^{2m-p-2}(E^\star)\otimes(\det E)^{m-1}
\otimes K_S^m).$$
Thus $\sigma(X)=2$ implies (and, if $X$ is not Fano, is in fact
equivalent to)
$$H^0(S,S^{2m-p-2}(E^\star)\otimes(\det E)^{m-1}\otimes K_S^m)=0,\quad
m\ge C(p+1).\leqno(\star)$$
Consider the $\qb$-vector bundle $E_0=E\otimes(\det E)^{-1/2}$, so that
$E_0^\star\simeq E_0$ and $c_1(E_0)=0$. Then $(\star)$ is equivalent to
$$H^0(S,S^{2m-p-2}E_0\otimes(\det E)^{p/2}\otimes K_S^m)=0,\quad
m\ge C(p+1).\leqno(\star\star)$$
This should be considered as a kind of stability property of~$E$. For
instance, if $\kappa(S)\ge 0$, then $(\det E)^{p/2}\otimes K_S^m$ has
sections for all $p\ge p_0$ and $m\ge 0$, so $(\star\star)$ immediately
implies that
$$H^0(S,S^kE_0)=0,\quad \forall k>0.$$
(Note that $E_0$ is not necessarily a ``real'' vector bundle if $\det E$
is not divisible by $2$ in $\Pic(S)\,$; in that case we only consider the
case of even integers $k$ so that $S^kE_0$ is actually a vector bundle.)

\noindent(a) Assume that $c_1(E)^2>4\,c_2(E)$, $S$ minimal and
$\kappa(S)\ge 0$. Then $\sigma(X)=1$.

\proof If $\sigma(X)=2$, then by $(\star\star)$ with $m=(k+p+2)/2$ and
$p$ chosen such that $(\det E)^{p/2}\otimes K_S^{(p+2)/2}$ has sections,
we conclude (at least for even $k$) that
$$H^0\big(S,S^k(E_0\otimes K_S^{1/2})\big)=0,\forall k\ge 0.\leqno(+)$$
By Riemann-Roch, $\chi\big(S^k(E_0\otimes K_S^{1/2})\big)\sim
(c_1^2-c_2)k^3/6$ where $c_i=c_i(E_0\otimes K_S^{1/2})$. By assumption
$c_1^2(E)>4\,c_2(E)$ and $K_S^2\ge 0$, hence
$$c_1^2-c_2={3\over 4}K_S^2-c_2(E_0)={3\over 4}K_S^2+{1\over 4}
c_1(E)^2-c_2(E)>0.$$
Thus Riemann-Roch implies
$$H^2\big(S,S^k(E_0\otimes K_S^{1/2})\big)\ne 0$$
for $k$ large. By duality we obtain 
$$H^0\big(S,S^k(E_0\otimes K_S^{-1/2})\otimes K_S\big)\ne 0.$$
This contradicts $(+)$.

\noindent(b) We observe also that $\sigma(X)=2$ implies $\kappa(K_S+
\det E^\star)=-\infty$, and even that $K_S+\det E^\star\notin\ov N_\eff(S)$.
Furthermore the proof of a) yields ${3\over 4}K_S^2\le c_2(E_0)$
if $\sigma(X)=2$. Our conclusion is that the condition ``$\sigma(X)=2$''
becomes more and more difficult to achieve when $K_S$ becomes more positive.
Hence the question arises whether there exists any $\pb_1$-bundle $X$ over
a surface $S$ with $\kappa(S)\ge 0$ having $\sigma(X)=2$.

\rem 5.6. Example|Let $S\subset\pb_3$ be a general quartic surface.
Then $S$ is a $K3$ surface and has Picard number $\rho(S)=1$.
Let $X=\pb(\Omega^1_S)\to S$. We are going to show that
$\sigma(X)=2$.

By (5.5) it is sufficient to prove that
$$H^0(S,S^{2m-p-2}(\Omega^1_S)\otimes A^p)=0,\quad
m\ge C(p+1)\leqno(\star)$$
where $A$ is a sufficiently ample line bundle (so that $E=\Omega^1_S\otimes A$
in the context of~5.5; note that $\det\Omega^1_S=\Oc_S$, hence
$E_0=\Omega^1_S$).  We may take for instance $A=\Oc_S(2)=
\Oc_{\pb_3}(2)_{|S}$.
Now $(\star)$ is easily verified by considering the standard exact
sequences
$$\eqalign{
0&\la\Oc_S(-4)\la\Omega^1_{\pb_3|S}\la\Omega^1_S\la 0,\cr
0&\la\Oc_S\la\Oc_S(1)^{\oplus 4}\la T_{\pb_3|S}\la 0\quad
\hbox{\rm(Euler sequence),}\cr
0&\la\Omega^1_{\pb_3|S}\la\Oc_S(-1)^{\oplus 4}\la\Oc_S\la 0\quad
\hbox{\rm(dual sequence),}\cr}$$
and their $k$-th symmetric powers
$$\eqalign{
0&\la\Oc_S(-4)\otimes S^{k-1}\Omega^1_{\pb_3|S}\la S^k\Omega^1_{\pb_3|S}
\la S^k\Omega^1_S\la 0,\cr
0&\la S^k\Omega^1_{\pb_3|S}\la\bigoplus\Oc_S(-k)\la\bigoplus\Oc_S(1-k)
\la 0.\cr}$$
In fact we find for every $k\in\nb$ a constant $C_k$ such that
$$H^0\big(S,S^m\Omega^1_X\otimes\Oc_X(k)\big)=0,\quad m\ge C_k.$$
Note that property $(\star)$ for an arbitrary $2$-bundle $E$ with $c_1(E)=0$
on a $K3$ surface with $\rho(S)=1$ is just slightly stronger than the
condition that all $S^mE$ are stable. Namely, $(\star)$ can be rephrased
by saying that $\Oc_{\pb(E)}(1)\notin\ov N_\eff(\pb(E))$, while
stability of $S^mE$ means that $\Oc_{\pb(E)}(m)$ is never effective.
So it seems that 5.6 should work for most Hermite-Einstein bundles for
which $S^mE$ never splits off a direct summand.

Now we use this example to show that Problem~3.6 has not always a
positive answer. In fact we prove
$$\hbox{\rm If $Y\subset X$ is ample, then $K_X^2\cdot Y<0$,}
\leqno(\star\star)$$
whereas $\sigma_+(K_X)=1$. We can write
$$\Oc_X(Y)=
\Oc_{\pb(\Omega^1_S)}(\alpha)\otimes\pi^\star\Oc_S(\beta),$$
with $\alpha,\beta\in\zb$. Then certainly $\alpha>0$, and also $\beta>0$
since $\Omega^1_S$ is not ample. Replacing $Y$ by some multiple, we may
assume the existence of a smooth curve $C\in|\Oc_S(\beta)|$. Since
$$-K_X=\Oc_{\pb(\Omega^1_S)}(2),$$
we have (abbreviating $\Oc_{\pb(\Omega^1_S)}(2)$ by $\Oc_\pb(2)$):
$$\eqalign{
K_X^2\cdot Y&=4\,c_1\big(\Oc_{\pb}(1)\big)^2\big(\alpha\,c_1(\Oc_\pb(1))+
\pi^\star C\big)\cr
&=4\alpha\,c_1\big(\Oc_{\pb}(1)\big)^3+4\,c_1\big(\Oc_{\pb(\Omega^1_{S|C})}
(1)\big)^2.\cr}$$
Since
$$c_1\big(\Oc_{\pb}(1)\big)^3=c_1(\Omega^1_S)^2-c_2(\Omega^1_S)=-c_2(S)=-24$$
and
$$c_1\big(\Oc_{\pb(\Omega^1_{S|C})}(1)\big)^2=c_1(\Omega^1_{S|C})=0,$$
our claim $(\star\star)$ follows.

Now we discuss birational properties of $\sigma(X)$.

\than 5.7. Proposition|Let $X$ and $Y$ be normal projective varieties of 
dimension $3$ with
at most terminal singularities. Let $\varphi:X\to Y$ be a birational map
with exceptional set $E$ such that $-K_X$ is $\varphi$-ample. Then:
\item{\rm(a)} $\sigma(Y)=2\quad\Longrightarrow\quad\sigma(X)=2$\quad
if $\dim\varphi(E)=0\,;$
\item{\rm(b)} $\sigma(X)=2\quad\Longrightarrow\quad\sigma(Y)\ge 2.$
\fintha

\proof Fix an ample divisor $A$ on $Y$. Let $E_i$ be the (automatically
$1$-codimensional) components of~$E$. For simplicity of notations we may
assume $X$ and $Y$ Gorenstein (otherwise substitute $K$ by $mK$). Write
$$K_X=\varphi^\star(K_Y)+\sum\lambda_iE_i.$$
Because $Y$ has terminal singularities, all $\lambda_i$ are $>0$. Since
$-K_X$ is $\varphi$-ample, so is $-\sum\lambda_iE_i$. Hence we find $p_0$
such that
$$B=\varphi^\star(p_0A)-\sum\lambda_iE_i$$
is ample on $X$. Substituting $A$ by $p_0A$, we may assume $p_0=1$.

Let us first prove (b). So assume $\sigma(X)=2$. Then
$$H^q(X,-mK_X-pB)=0,\quad m\ge C(p+1),~~q\ge 2.$$
Since
$$\eqalign{
&-mK_X-pB=-\varphi^\star(mK_Y+pA)-\sum(m-p)\lambda_iE_i\quad\hbox{\rm and}\cr
&R^q\varphi_\star\big(\Oc\big(\sum(p-m)\lambda_iE_i\big)\big)=0\cr}$$
for $p<m$ and $q>0$ ($-\sum\lambda_iE_i$ being $\varphi$-ample), we conclude
via the Leray spectral sequence that
$$H^q(Y,-mK_Y-pA)=0,\quad m\ge C(p+1),~~q\ge 2,$$
hence $\sigma(Y)\ge 2$. For (a) assume $\sigma(Y)=2$, so
$$H^q(Y,-mK_Y-pA)=H^{3-q}(Y,(m+1)K_Y+pA)=0,\quad m\ge C(p+1),~~q\ge 2.$$
We observe that
$$\varphi_\star(-mK_X-pB)=\Oc(-mK_Y-pA)\otimes\varphi_\star\Big(\Oc\big(
{\textstyle\sum}\,(p-m)\lambda_iE_i\big)\Big).$$
Since $p<m$, we see that $\varphi_\star\big(\Oc\big(\sum(p-m)\lambda_iE_i
\big)\big)$ is an ideal sheaf $\Ic$ with Supp$(\Oc/\Ic)=\varphi(E)$, hence
with finite support. Thus
$$H^q\big(Y,\varphi_\star(-mK_X-pB)\big)=H^q(Y,-mK_Y-pA),~~q\ge 2.$$
By the Leray spectral sequence, we conclude as before that
$$H^q(X,-mK_X-pB)=H^q(Y,-mK_Y-pA)=0,\quad m\ge C'(p+1),~~q\ge 2.$$
Hence $\sigma(X)\ge 2$. If finally $\sigma(X)=3$, we have
$$H^1\big(Y,\varphi_\star(-mK_X-pB)\big)=0,$$
hence $H^1(Y,-mK_Y-pA)=0$ and consequently $\sigma(Y)=3$.\finpr

\rem 5.8. Remark|If in particular $X$ is a projective $3$-fold with at most
terminal singularities and $\varphi:X\to Y$ is a birational contraction of
an extremal ray whose exceptional set is not of codimension $\ge2$, then
Proposition~5.7 applies.

\rem 5.9. Problem|Let $X$, $Y$ be projective $3$-folds with only terminal
singularities and let $\varphi:X\merto Y$ be a birational morphism
of flipping type. Is $\sigma(X)\ge 2$ iff $\sigma(Y)\ge 2\,$?

If this is true, then by Mori [Mo88] we conclude the following: if
$X$ and $Y$ are smooth projective $3$-folds (or $\qb$-factorial with at
most terminal singularities) and if $f:X\to Y$ is a birational morphism,
then $\sigma(X)=2$ implies $\sigma(Y)\ge 2$.

\rem 5.10. Example|We will show that in Prop.~5.7$\,$(a) the conclusion
$\sigma(X)=2$ no longer holds if $\dim\varphi(E)=1$. Let $Y=\pb_3\,$, let
$D$ be a smooth complete intersection curve of type $(a,a)$ with
$a\ge 7$. Let $\varphi:X\to Y$ be the blow-up of $Y$ along~$D$.

\than Claim|We have $\sigma(X)=1$.\fintha

\proof Let $A$ and $B$ as in the proof of Prop.~5.7, so $A=\Oc_{\pb_3}(p_0)$
and $B=\varphi^\star A-E$ for some integer~$p_0$ (In this situation, the
exceptional divisor $E$ has just one component with $\lambda=1$.) We will
show that
$$H^2(X,-mK_X)\ne0$$
for $m\gg 0$. In fact we have an exact sequence
$$\eqalign{
0\to\Ic_D^{m}&{}\otimes\Oc(-mK_Y)\to
\Oc(-mK_Y)\to\Oc(-mK_Y)_{|D_{m}}\kern-1.65pt\to 0,\cr
&~|\,|\cr
\varphi_\star(-m&K_X)\cr}$$
where $D_{m}$ is the $(m-1)$-th infinitesimal neighborhood of $D$
defined by $\Ic_D^{m}$. Since $H^q(Y,-mK_Y)=0$ for $q>0$ and all $m>0$,
it is sufficient to prove
$$H^1(D_{m},-mK_Y)\ne 0,\quad m\gg 0.$$
Since
$$\chi (D_m, \Oc (4m)) = \sum_{0\le\nu\le m-1}\chi
(D, S^\nu N_C^\star\otimes\Oc _D(4m)\big)$$
it is sufficient to prove that this last sum is negative for $m\gg 0$.
By Riemann-Roch we have 
$$\eqalign{
&\sum_{0\le\nu\le m-1}\chi(D, S^\nu N_C^\star\otimes\Oc_D(4m)\big)={}\cr
&\qquad\qquad{}=\sum_{0\le\nu\le m-1}\big((1-g)(\nu+1)+
(\nu+1)a^2(4m-\nu a)\big)\cr
&\qquad\qquad{}=(1-g+4ma^2)m(m+1)/2\,{}-{}\,a^3m(m^2-1)/3.\cr}$$
The leading term of this cubic polynomial in $m$ is 
$$a^2(2-a/3)m^3$$
which is negative by our assumption $a\ge7$.
\finpr

\than 5.11. Proposition|Let $X$ be a smooth projective $3$-fold. Assume
$-K_X$ to be big and nef but not ample. Let $\varphi:X\to Y$ be the map
given by the base point free linear system $|-m_0K_X|$ with
$m_0\gg 0$. Then:
$$\sigma(X)=2~~\hbox{\sl iff all nontrivial fibers
of $\varphi$ are $1$-dimensional.}$$
\fintha

\proof By the base point free theorem, some multiple $-m_0K_X$ is globally
generated and since $-K_X$ is big, the associated map $\varphi:X\to Y$ is
a modification onto a normal projective variety~$Y$.

\noindent(1) Assume that $\varphi$ has at least one $2$-dimensional fiber~$F$.

In this case, we will show that $H^2(X,-mK_X-pA)\ne 0$, $p\gg 0$, where
$A$ is an arbitrary fixed ample divisor on~$X$. For this we use the Leray
spectral sequence
$$E^{p,q}_2=H^p\big(Y,R^q\varphi_\star\Oc(-mK_X-pA)\big).$$
First consider
$E^{0,2}_2=H^0\big(Y,R^2\varphi_\star\Oc(-mK_X-pA)\big)$. Since $\sigma_-(mK_X)
=\sigma_-(K_X)$ we may assume $m_0=1$ in our considerations, hence
$K_X=\varphi^\star(K_Y)$. So
$$E^{0,2}_2=H^0\big(Y,R^2\varphi_\star\Oc(-pA)\otimes\Oc(-mK_Y)\big).$$
We show that $R^2\varphi_\star\Oc(-pA)\ne 0$ in order to conclude
$E^{0,2}_2\ne 0$. Let $\{x\}=\varphi(F)$. Then rk$(R^2\varphi_\star\Oc(-pA)_x)
\ge h^2(F,-pA_{|F})$. Here we equip $F$ with the reduced structure. Since
$F$ is Cohen-Macaulay, we have
$$H^2(F,-pA_{|F})\simeq H^0(F,K_F\otimes pA_{|F}),$$
which is nonzero for all $p\gg 0$. Hence $E^{0,2}_2\ne 0$ for $p\gg 0$
and all~$m$. Next consider
$E^{2,1}_0=H^2\big(Y,R^1\varphi_\star\Oc(-pA)\otimes\Oc(-mK_Y)\big)$. Since
$\dim\Supp R^1\varphi_\star\Oc(-pA)=0$, $E^{2,1}_2=0$. Hence
$E^{0,2}_2\simeq E^{0,2}_3\simeq E^{0,2}_\infty$ and thus
$$H^2(X,-mK_X-pA)\ne 0,\quad p\gg 0.$$

\noindent(2) Now assume that all fibers of $\varphi$ have dimension $1$.

Then take a general smooth ample (hyper)surface $H\subset X$. It will meet
every fiber in a finite set. Hence $-K_X\cdot C>0$ for every curve
$C\subset H$, otherwise $C$ would be contracted to a point by~$\varphi$.
Furthermore $K_X^2\cdot H>0$ since $-K_X$ is big and nef on the general
member of the linear system $|mH|$, $m\gg 0$. Hence $-K_{X|H}$ is ample
by Nakai-Moishezon. Theorem~3.4 implies $\sigma_-(K_X)\ge 2$, and a strict
inequality cannot occur because $-K_X$ is not ample.\finpr

\rem 5.12. Remark|The proof of 5.11 can be easily modified to make it work
for all $\qb$-Gorenstein normal projective varieties~$X$. In that case $F$
might not be Cohen-Macaulay, so we possibly have to pass to the normalization
which is Cohen-Macaulay.
\vskip30pt
\centerline{\bf References}
\vskip5pt
\eightpoint
\lettre KMM87|
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\article De92|Demailly, J.P|Regularization of closed positive currents and
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\article De93|Demailly, J.P|A numerical criterion for very ample line
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\livre Ha70|Hartshorne, R|Ample subvarieties of algebraic varieties|Lecture
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\article KMM87|Kawamata, Y.; Matsuda, K.; Matsuki, K|Introduction to the
minimal model program|Adv.\ Stud.\ Pure Math.|10|1987|283-360|
\article Mi87|Miyaoka, Y|The Chern classes and Kodaira dimension of a minimal
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\article Mo88|Mori, S|Flip theorem and the existence of minimal models for 
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\article Sch73|Schneider, M|\"Uber eine Vermutung von Hartshorne|Math.\
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\article So80|Sommese, A.J|Concavity theorems|Math.\ Ann.|235|1978|37-53|
\article SW90|Szurek, M.; Wi\'sniewski, J|On Fano manifolds which are
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\vskip25pt
\tenpoint\parindent=0cm
{\bf Jean-Pierre Demailly}\hfill\break
Universit\'e de Grenoble I\hfill\break
Institut Fourier, BP 74\hfill\break
U.R.A. 188 du C.N.R.S.\hfill\break
38402 Saint-Martin d'H\`eres, France\hfill\break {\sl e-mail: demailly@grenet.fr}
\vskip6pt
{\bf Thomas Peternell}\hfill\break
Universit\"at Bayreuth\hfill\break
Mathematisches Institut\hfill\break
D-95440 Bayreuth, Deutschland\hfill\break {\sl e-mail: thomas.peternell@uni-bayreuth.de} \vskip6pt
{\bf Michael Schneider}\hfill\break
Universit\"at Bayreuth\hfill\break
Mathematisches Institut\hfill\break
D-95440 Bayreuth, Deutschland\hfill\break {\sl e-mail: michael.schneider@uni-bayreuth.de} \vskip15pt
(July 23, 1993)
\end

