% Article "Effective Bounds for Very Ample Line Bundles"
% Submitted to Inventiones Mathematicae
% J.P. Demailly, Universit\'e de Grenoble I, Saint Martin d'H\`eres, France
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  \MAINTITLE={Effective Bounds for Very Ample Line Bundles}

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  \AUTHOR={Jean-Pierre Demailly}
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    \MAIL={Jean-Pierre Demailly,
           Universit\'e de Grenoble I, Institut Fourier,
           BP74, F--38400 Saint-Martin d'H\`eres, France\hfill
           {\it e-mail}: demailly\at fourier.ujf-grenoble.fr}
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  \INSTITUTE={Universit\'e de Grenoble I, Institut Fourier}
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  \DATE={Received January 30, 1995, accepted May 18, 1995}
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  \SUMMARY={Let $L$ be an ample line bundle on a non singular projective
$n$-fold~$X$. It is first shown that $2K_X+mL$ is very ample for
$m\ge 2+{3n+1\choose n}$. The proof develops an original idea of Y.T.~Siu
and is based on a combination of the Riemann-Roch theorem together with an
improved Noetherian induction technique for the Nadel multiplier ideal
sheaves. In the second part, an effective version of the big Matsusaka
theorem is obtained, refining an earlier version of Y.T.~Siu: there is
an explicit polynomial bound $m_0=m_0(L^n,L^{n-1}\cdot K_X)$ of
degree${}\le n3^n$ in the arguments, such that $mL$ is very ample for
$m\ge m_0$. The refinement is obtained through a new sharp upper bound
for the dualizing sheaves of algebraic varieties embedded in projective
space.}

%
% \KEYWORDS={ positive or ample line bundle --  nef line bundle -- 
% Fujita conjecture -- Kawamata-Viehweg and Nadel vanishing theorems --
% effective very ampleness -- dualizing sheaf --  big Matsusaka theorem }
% or
% \SUBCLASS={ 32J25, 32L20, 14C20, 14F17 }
%
\DEDICATION={Dedicated to Professor Reinhold Remmert on the occasion of his
retirement}

\maketitle

\titlea{0. Introduction}
In the last six or seven years, considerable progress has been achieved
in the understanding of adjoint linear systems $|K_X+mL|$ associated with
an ample line bundle $L$ on a smooth projective manifold~$X$. When $X$
is a surface, I.~Reider [Rei88] obtained a quasi-optimal criterion for
the global generation and very ampleness of $K_X+L$, showing in particular
that $K_X+3L$ is always generated by global sections and $K_X+4L$ very
ample. Around the same period, T.~Fujita [Fuj87] raised the following
interesting conjecture.

\begstat{(0.1) Conjecture (Fujita)} Let $X$ be a smooth projective
$n$-fold over $\bbbc$ and let $L$ be an ample line bundle on~$X$.
Then $K_X+(n+1)L$ is generated by global sections and $K_X+(n+2)L$
is very ample.
\endstat

One of the first results proved in dimension $n\ge 3$ is the very
ampleness of $2K_X+12n^nL$, using an analytic method based on the
solution of a Monge-Amp\`ere equation (see [Dem93]). Slightly later,
J.~Koll\'ar [Kol93] obtained an effective version of the base point
free theorem, while a major step was made in small dimension by L.~Ein
and R.~Lazarsfeld [EL93], with the solution of the global generation
part of Fujita's conjecture for $n=3$. Other related works are [EL92],
[Fuj94], [EKL94], [Ein94] (see also [Laz93] and [Dem94] for survey
expositions). Recently, Y.T.~Siu [Siu94a] introduced a simple algebraic
method for proving the very ampleness of $2K_X+mL$. His method is based
on a combination of the Riemann-Roch formula with the Kawamata-Viehweg
vanishing theorem, in the generalized form given by A.~Nadel [Nad89].
Our first goal is to develop a more efficient Noetherian induction
process for the Nadel multiplier sheaves associated with singular
hermitian metrics, along the lines of Siu's method. The new induction
process is simpler and allows us to refine further Siu's original
bounds. In the sequel the intersection numbers of $L$ over
$d$-dimensional subvarieties $Y\subset X$ are denoted
$$L^d\cdot Y=\int_Y c_1(L)^d.$$
We say that $L$ is numerically effective (nef for short) if $L\cdot C
\ge 0$ for every algebraic curve $C\subset X$. By [Dem90], $L$ is nef if
and only if for each $\varepsilon>0$ there is a hermitian metric
$h_\varepsilon$ on $L$ of curvature $\Theta_{h_\varepsilon}(L)\ge
-\varepsilon\omega$, where $\omega$ is a given K\"ahler form on~$X$.

\begstat{(0.2) Theorem} Let $X$ be a smooth projective $n$-fold and let
$L$ be an ample line bundle over~$X$. Then
\smallskip
\item{\rm a)} $2K_X+mL$ is very ample for $m\ge 2+{3n+1\choose n}\,;$
\smallskip
\item{\rm b)} $2K_X+L$ generates simultaneous jets of order $s_1\ld s_p$
at arbitrary points $x_1\ld x_p\in X$ provided that the intersection numbers
$L^d\cdot Y$ of $L$ over all \hbox{$d$-dimensional} algebraic subsets $Y$ of
$X$ satisfy
$$L^d\cdot Y>{2^{d-1}\over\lfloor n/d\rfloor^d}
\sum_{1\le j\le p}{(n+1)(4n+2s_j+1)-2\choose n},\qquad 1\le d\le n.$$
\smallskip
\item{\rm c)} $m(K_X+(n+2)L)$ is very ample for $m\ge{3n+1\choose n}-2n$.
\smallskip\noindent
All results still hold true by adding any nef line bundle $G$ to the
line bundles under consideration.
\endstat

Our method of proof is sharp enough to yield as a by-product the
well-known result that $K_X+(n+1)L$ is numerically effective if $L$ is
ample (a result originally proved as a consequence of Mori theory). A
basic problem would be to find an analogue of Th.~(0.2\,a,\,b) with
$K_X$ in place of~$2K_X$. For the global generation question, the
answer has been settled in the affirmative recently
by U.~Angehrn and Y.T.~Siu [AS94], who showed that $K_X+{1\over2}(n^2+n+2)L$
is always generated by global sections; their method is again based on
Nadel's vanishing theorem, using a different idea for the construction of
the required singular hermitian metrics. The result of Angehrn-Siu implies
that $K_X+2n(K_X+{1\over 2}(n^2+n+2)L)$ is very ample for $n\ge 2$ (by the
elementary observation that $K_X+2nF$ is always very ample if $F$ is
ample and generated by sections); the bound obtained in (0.2\,c) can
then be improved into $m\ge n^3$. In a related paper [Siu94b], Y.T.~Siu
obtains a variant of (0.2$\,$b) in which the numerical condition for
$L^d\cdot Y$ is replaced by $(L^d\cdot Y)^{1/d}>2n\sum_{1\le j\le p}
{3n+2s_j-3\choose n}+2pn\,$; this bound, which has a rather involved proof,
is sharper than ours for $d\le O(\ln(n))$ but weaker for larger values
of~$d$. At the time these lines are written, it seems to be unknown whether
there is a bound $m_0(n)$ depending only on the dimension such that
$K_X+mL$ is very ample for $m$ larger than $m_0(n)$. Also it seems to
be unknown whether polynomial bounds $m_0(n)$ exist for $2K_X+mL$ (the
bound given by (0.2\,a) is of the order of magnitude of $(27/4)^n$ and
seems to be the best presently known).
\medskip

Another important question is to find effective bounds $m_0$ such that
$mL$ becomes very ample for $m\ge m_0$. From a theoretical point of
view, this problem is solved by Matsusaka [Mat72] and
Koll\'ar-Matsusaka [KoM83]. Their result states that there is a bound
$m_0=m_0(n,L^n,L^{n-1}\cdot K_X)$ depending only on the dimension and
on the first two coefficients $L^n$ and $L^{n-1}\cdot K_X$ in the
Hilbert polynomial of~$L$. Unfortunately, the original proof does not
tell much on the actual dependence of $m_0$ in terms of these
coefficients. In a ground-breaking paper [Siu93], Y.T.~Siu introduced
new techniques leading to effective bounds for~$m_0$. The published
version of [Siu93] incorporates an induction argument which we
developped in collaboration with the author after the preprint version
circulated, enabling us to obtain much better final estimates. Our goal
in the last sections \S\,3,\,4 is to present a further substantial
refinement of this method. The main point is that a crucial technical
lemma used in [Siu93] to deal with dualizing sheaves can be made
optimal by using a different idea based on the Ohsawa-Takegoshi $L^2$
extension theorem~[OT87].

\begstat{(0.3) Theorem} Let $H$ be a very ample line bundle on a
projective algebraic manifold~$X$, and let $Y\subset X$ be a
$p$-dimensional irreducible algebraic subvariety. Denote by $\omega_Y$
the $L^2$ dualizing sheaf of~$Y$. If $\delta=H^p\cdot Y$ is the degree
of $Y$ with respect to $H$, the sheaf
$\cHom\big(\omega_Y,\cO_Y((\delta-p-2)H)\big)$ has a nontrivial
section.
\endstat

Using this sharp ``upper estimate'' on dualizing sheaves and some other
refinements of the inductive method explained in [Siu93], we obtain the
following improved bounds.

\begstat{(0.4) Theorem} If $L$ is an ample line bundle on a projective
$n$-fold $X$, then $mL$ is very ample for
$$m\ge m_0=C_n~(L^n)^{3^{n-2}}\left(n+2+{L^{n-1}\cdot K_X\over L^n}
\right)^{3^{n-2}(n/2+3/4)+1/4}$$
where $C_n$ depends only on~$n$, e.g.,
$$C_n=(2n)^{(3^{n-1}-1)/2}\left({3n+1\choose n}-2n\right)^{3^{n-2}(n/2+3/4)+
1/4}.$$
\endstat

The bound (0.4) turns out to be essentially optimal for $n=2$ (apart
from a small multiplicative constant), as was shown recently by
Fern\'andez del Busto [FdB94] by means of Reider's theorem and an
example of Gang Xiao. Our bound is probably not optimal for $n\ge 3$,
and we strongly believe that there should exist an optimal bound of the
form $C_n(L^n)^{a_n}(n+2+L^{n-1}\cdot K_X/L^n)^{b_n}$, involving
exponents $a_n$, $b_n$ of the order of magnitude of $n$ or $n^2$
instead of $n3^n$.

\titlea{1. Nadel's Vanishing Theorem}
We recall here briefly a few basic ideas developped in~[Dem90,\,93], which
will be equally useful in this paper.
Let $X$ be a projective algebraic manifold equipped with a K\"ahler
metric~$\omega$, and let $F$ be a holomorphic line bundle over~$X$.
We assume that $F$ is equipped with a (possibly singular) hermitian
metric~$h$. In each open set $U$ where $F_{\restriction U}\simeq
U\times\bbbc$ is trivial, the metric $h$ is given by a weight $\varphi$
such that $\|\xi\|_h=|\tau(\xi)|\E^{-\varphi(x)}$ for all $\xi\in F_x$,
where $\tau:F_{\restriction U}\to\bbbc$ is the trivialization map.
If $\varphi$ is supposed to be locally integrable on~$U$, the curvature
form of $F$ can be defined to be the
closed $(1,1)$-current $\Theta_h(F)={\I\over\pi}\partial\dbar\varphi$.
Here, we will only consider the case of nonnegative curvature currents
$\Theta_h(F)\ge 0$, i.e., we suppose that the weights $\varphi$ are
plurisubharmonic. Following Nadel [Nad89], we associate to $\varphi$
the ideal sheaf
$$\cI(\varphi)=\big\{f\in\cO_{X,x}\,;\,\exists W\ni x,~\int_W|f|^2
\E^{-2\varphi}dV_\omega<+\infty\big\}\leqno(1.1)$$
where $dV_\omega=\omega^n/n!$ is the K\"ahler volume form and $W$ is an
arbitrary open neighborhood of~$x$. Of course, $\cI(\varphi)$ does not
depend on the choice of the trivialization, and thus we get a global ideal
sheaf $\cI(h)$ on $X$ depending only on~$h$. By [Nad89] and [Dem93],
$\cI(h)$ is a coherent ideal sheaf in $\cO_X$, and we have the following
fundamental vanishing theorem.

\begstat{(1.2) Nadel vanishing theorem} Assume that $\Theta_h(F)\ge
\varepsilon\omega$ for some $\varepsilon>0$. Then
$$H^q\big(X,\cO(K_X+F)\otimes\cI(h)\big)=0\qquad
\hbox{for all~ $q\ge 1$.}$$
\endstat

The proof is a straightforward consequence of the Bochner-Kodaira-Nakano
identity ([AN54], [Nak55]) and of H\"ormander's $L^2$
estimates for the $\dbar$ operator (see [H\"or65], [AV65], [Nad89], [Dem93]).
In the present paper, we only need ``algebraic'' metrics $h$ of the form
$$\|\xi\|_h^2={|\tau(\xi)|^2\over\big(\sum_{1\le j\le N}|\tau^\mu
(\sigma_j(x))|^2\big)^{1/\mu}}\leqno(1.3)$$
where $\sigma_1\ld\sigma_N\in H^0(X,\mu F)$ are non zero algebraic sections
of $\mu F=F^{\otimes\mu}$, and $\tau^\mu$ is the local trivialization
of~$F^{\otimes\mu}$ induced by a local trivialization $\tau$ of~$F$.
The corresponding weight is
$$\varphi={1\over 2\mu}\log\big(\sum_{1\le j\le N}|\tau^\mu
(\sigma_j(x))|^2\big).\leqno(1.4)$$
In this case, (1.2) is equivalent to the Kawamata-Viehweg vanishing
([Kaw82], [Vie82]), and the proof can be reduced to the usual Kodaira
vanishing theorem by purely algebraic means. Now, recall that
the Lelong number of a plurisubharmonic function $\varphi$ at
a point $x$ is $\nu(\varphi,x)=\lim_{r\to 0}\sup_{B(x,r)}\varphi/\log r$.
In the special case (1.4) under consideration, we simply have
$$\nu(\varphi,x)={1\over m}\min_{1\le j\le N}\ord_x(\sigma_j)$$
where $\ord_x(\sigma_j)$ is the vanishing order of $\sigma_j$ at~$x$.

\begstat{(1.5) Corollary} Let $(X,\omega)$, $F$, $h$ and $\varphi$ be as in
$(1.2)$ and let $x_1\ld x_N$ be isolated points in the zero
variety~$V(\cI(\varphi))$. Then there is a surjective map
$$H^0(X,K_X+F)\lraww\bigoplus_{1\le j\le N}\cO(K_X+F)_{x_j}\otimes
\big(\cO_X/\cI(h)\big)_{x_j}.$$
In particular, if $\nu(\varphi,x_j)\ge n+s_j$, then $H^0(X,K_X+F)$ generates
simultaneously all jets of order $s_j$ at~$x_j$.
\endstat

\begproof{} Consider the long exact sequence of cohomology associated to the
short exact sequence $0\to\cI(\varphi)\to\cO_X\to\cO_X/\cI(\varphi)\to 0$
twisted by $\cO(K_X+F)$, and apply Th.~(1.2) to obtain the vanishing of
the first $H^1$ group. The asserted surjectivity property follows. The last
statement follows from the fact that $\nu(\varphi)\ge n+s$ implies
$\cI(h)_x\subset\gm_x^{s+1}$. Indeed, we then have
$$\varphi(z)\le(n+s)\log|z-x|+O(1),\qquad \E^{-\varphi(z)}\ge
c|z-x|^{-(n+s)},\quad c>0,$$
as is obvious in the ``algebraic case'' (in general, the inequality follows
from the standard logarithmic convexity property of plurisubharmonic
functions).\qed
\endproof

\begstat{(1.6) Remark} {\rm As is well known, Corollary (1.6) can be
proved by a direct application of H\"ormander's $L^2$ estimates,
namely by solving a $\dbar$-equation \hbox{$\dbar u=\sum\dbar(\psi_jP_j)$}
for forms of type $(n,1)$, where $P_j$ is a finite holomorphic
Taylor expansion achieving the desired jet at~$x_j$, and where $\psi_j$
is a cut-off function with support in a neighborhood of~$x_j$.
In this way, we see that Cor.~(1.6) still holds if we only have
$\Theta_h(F)\ge 0$ and $\Theta_h(F)\ge\varepsilon\omega$ in a
neighborhood of each~$x_j$.}
\endstat

\titlea{2. Some Results Around the Fujita Conjecture}
This section is devoted to a proof of various results related to the
Fujita conjecture. The main ideas occuring here are inspired by a very
recent work of Y.T.~Siu [Siu94a]. His method, which is algebraic in
nature and quite elementary, consists in a combination of the
Riemann-Roch formula together with Nadel's vanishing theorem (in fact,
only the algebraic case is needed, thus the original Kawamata-Viehweg
vanishing theorem would be sufficient). In the sequel, $X$ denotes a
projective algebraic $n$-dimensional manifold. The first observation is
the following well-known consequence of the Riemann-Roch formula.

\begstat{(2.1) Special case of Riemann-Roch} Let $\cJ\subset\cO_X$ be a
coherent ideal sheaf on $X$ such that the subscheme $Y=V(\cJ)$ has
dimension $d$ $($with possibly some lower dimensional components$)$.
Let $[Y]=\sum\lambda_j[Y_j]$ be the effective algebraic cycle of dimension
$d$ associated to the $d$ dimensional components of~$Y$ $($taking
into account multiplicities $\lambda_j$ given by the ideal $\cJ)$. Then
for any line bundle $F$, the Euler characteristic
$$\chi(Y,\cO(F+mL)_{\restriction Y})=\chi(X,\cO(F+mL)\otimes\cO_X/\cJ)$$
is a polynomial $P(m)$ of degree $d$ and leading coefficient
$L^d\cdot[Y]/d!$
\endstat

The second fact is an elementary lemma about numerical polynomials
(polynomials with rational coefficients, mapping $\bbbz$ into $\bbbz$).

\begstat{(2.2) Lemma} Let $P(m)$ be a numerical polynomial of degree $d>0$
and leading coefficient $a_d/d!$, $a_d\in\bbbz$, $a_d>0$. Suppose that
$P(m)\ge 0$ for $m\ge m_0$. Then
\smallskip
\item{\rm a)} For every integer $N\ge0$, there exists $m\in[m_0,m_0+Nd]$
such that~\hbox{$P(m)\ge N$}.
\smallskip
\item{\rm b)} For every $k\in\bbbn$, there exists $m\in[m_0,m_0+kd]$ such
that \hbox{$P(m)\ge a_dk^d/2^{d-1}$}.
\smallskip
\item{\rm c)} For every integer $N\ge 2d^2$, there exists
$m\in[m_0,m_0+N]$ such that \hbox{$P(m)\ge N$}.
\vskip0pt
\endstat

\begproof{} a) Each of the $N$ equations $P(m)=0$, $P(m)=1$,
$\ldots$, $P(m)=N-1$ has at most $d$ roots, so there must be an integer
$m\in[m_0,m_0+dN]$ which is not a root of these.
\smallskip\noindent
b) By Newton's formula for iterated differences $\Delta P(m)=P(m+1)-P(m)$,
we get
$$\Delta^dP(m)=\sum_{1\le j\le d}(-1)^j{d\choose j}P(m+d-j)=a_d,
\qquad\forall m\in\bbbz.$$
Hence if $j\in\big\{0,2,4\ld 2\lfloor d/2\rfloor\big\}\subset[0,d]$ is the
even integer achieving the maximum of $P(m_0+d-j)$ over this finite set,
we find
$$2^{d-1}P(m_0+d-j)=\left({d\choose 0}+{d\choose 2}+\ldots\right)P(m_0+d-j)
\ge a_d,$$
whence the existence of an integer $m\in[m_0,m_0+d]$ with $P(m)\ge
a_d/2^{d-1}$. The case $k=1$ is thus proved. In general, we apply the
above case to the polynomial $Q(m)=P(km-(k-1)m_0)$, which has leading
coefficient $a_dk^d/d!$
\smallskip\noindent
c) If $d=1$, part a) already yields the result. If $d=2$, a look at the
parabola shows that
$$\max_{m\in[m_0,m_0+N]}P(m)\ge\cases{
a_2N^2/8&if $N$ is even,\cr
a_2(N^2-1)/8&if $N$ is odd;\cr}$$
thus $\max_{m\in[m_0,m_0+N]}P(m)\ge N$ whenever $N\ge 8$. If $d\ge 3$,
we apply b) with $k$ equal to the smallest integer such
that $k^d/2^{d-1}\ge N$, i.e.\ $k=\lceil 2(N/2)^{1/d}\rceil$, where
$\lceil x\rceil\in\bbbz$ denotes the round-up of $x\in\bbbr$.
\hbox{Then $kd\le (2(N/2)^{1/d}+1)d\le N$} whenever $N\ge 2d^2$, as a short
computation shows.\qed
\endproof

We now apply Nadel's vanishing theorem pretty much in the same way as
Siu [Siu94a], but with substantial simplifications in the technique and
improvements in the bounds. Our method yields simultaneously a simple
proof of the following basic result.

\begstat{(2.3) Theorem} If $L$ is an ample line bundle over
a projective $n$-fold~$X$, then $K_X+(n+1)L$ is nef.
\endstat

By using Mori theory and the base point free theorem ([Mor82],
[Kaw84]), one can even show that $K_X+(n+1)L$ is semiample, i.e., there
exists a positive integer $m$ such that $m(K_X+(n+1)L)$ is generated by
sections (see [Kaw85] and [Fuj87]). The proof rests on the observation
that $n+1$ is the maximal length of extremal rays of smooth projective
$n$-folds. Our proof of (2.3) is different and will be given simultaneously
with the proof of Th.~(2.4) below.

\begstat{(2.4) Theorem} Let $L$ be an ample line bundle and let $G$ be
a nef line bundle on a projective $n$-fold~$X$. Then the following
properties hold.
\smallskip
\item{\rm a)} $2K_X+mL+G$ generates simultaneous jets of order
$s_1\ld s_p\in\bbbn$ at arbitrary points $x_1\ld x_p\in X$, i.e., there
is a surjective map
$$H^0(X,2K_X+mL+G)\lraww\bigoplus_{1\le j\le p}\cO(2K_X+mL+G)\otimes
\cO_{X,x_j}/\gm_{X,x_j}^{s_j+1},$$
provided that $\displaystyle m\ge 2+\sum_{1\le j\le p}{3n+2s_j-1\choose
n}$.\hfill\break In particular $2K_X+mL+G$ is very ample for
$\displaystyle m\ge 2+{3n+1\choose n}$.
\smallskip
\item{\rm b)} $2K_X+(n+1)L+G$ generates simultaneous jets of order
$s_1\ld s_p$ at arbitrary points $x_1\ld x_p\in X$ provided that
the intersection numbers $L^d\cdot Y$ of $L$ over all
$d$-dimensional algebraic subsets $Y$ of $X$ satisfy
$$L^d\cdot Y>{2^{d-1}\over\lfloor n/d\rfloor^d}\sum_{1\le j\le p}
{3n+2s_j-1\choose n}.$$
\endstat

\begproof{} The proofs of (2.3) and (2.4\,a,\,b) go along the
same lines, so we deal with them simultaneously (in the case of
(2.3), we simply agree that $\{x_1\ld x_p\}=\emptyset$). The idea is
to find an integer (or rational number) $m_0$ and a singular hermitian
metric $h_0$ on $K_X+m_0L$ with strictly positive curvature current
$\Theta_{h_0}\ge\varepsilon\omega$, such that $V(\cI(h_0))$ is
$0$-dimensional and the weight $\varphi_0$ of $h_0$ satis\-fies
$\nu(\varphi_0,x_j)\ge n+s_j$ for all~$j$. As $L$ and $G$ are nef,
$(m-m_0)L+G$ has for all $m\ge m_0$ a metric $h'$ whose
curvature $\Theta_{h'}$ has arbitrary small negative part (see
[Dem90]), e.g., $\Theta_{h'}\ge -{\varepsilon\over 2}\omega$. Then
$\Theta_{h_0}+\Theta_{h'}\ge{\varepsilon \over 2}\omega$ is again
positive definite. An application of Cor~(1.5) to
$F=K_X+mL+G=(K_X+m_0 L)+((m-m_0)L+G)$ equipped with the metric
$h_0\otimes h'$ implies the existence of the desired sections in
$K_X+F=2K_X+mL+G$ for $m\ge m_0$.

Let us fix an embedding $\Phi_{|\mu L|}:X\to\bbbp^N$, $\mu\gg 0$, given
by sections $\lambda_0\ld\lambda_N\in H^0(X,\mu L)$, and let $h_L$ be
the associated metric on $L$ of positive definite curvature form
$\omega=\Theta(L)$. In order to obtain the desired metric $h_0$ on
$K_X+m_0 L$, we fix $a\in\bbbn^\star$ and use a double induction
process to construct singular metrics $(h_{k,\nu})_{\nu\ge 1}$ on
$aK_X+b_kL$ for a non increasing sequence of positive integers $b_1\ge
b_2\ge\ldots\ge b_k\ge\ldots\,$. Such a sequence much be stationary and
$m_0$ will just be the stationary limit $m_0=\lim b_k/a$. The metrics
$h_{k,\nu}$ are taken to satisfy the following properties:
\smallskip
\item{$\alpha\!)$} $h_{k,\nu}$ is an algebraic metric of the form
$$\|\xi\|_{h_{k,\nu}}^2={|\tau_k(\xi)|^2\over
\big(\sum_{1\le i\le\nu,\,0\le j\le N}\big|\tau_k^{(a+1)\mu}(\sigma_i^{a\mu}
\cdot\lambda_j^{(a+1)b_k-am_i})\big|^2\big)^{1/(a+1)\mu}},$$
defined by sections $\sigma_i\in H^0(X,(a+1)K_X+m_iL)$, $m_i<{a+1\over a}b_k$,
$1\le i\le\nu$, where $\xi\mapsto\tau_k(\xi)$ is an arbitrary local
trivialization of $aK_X+b_kL\,$; note that $\sigma_i^{a\mu}\cdot
\lambda_j^{(a+1)b_k-am_i}$ is a section of
$$a\mu((a+1)K_X+m_iL)+((a+1)b_k-am_i)\mu L=(a+1)\mu(aK_X+b_kL).$$
\item{$\beta\kern-1pt)$\kern-1pt} $\ord_{x_j}(\sigma_i)\ge(a+1)(n+s_j)$ for
all $i,j\,$;
\smallskip
\item{$\gamma)$} $\cI(h_{k,\nu+1})\supset\cI(h_{k,\nu})$ and
$\cI(h_{k,\nu+1})\ne\cI(h_{k,\nu})$ whenever the zero variety
$V(\cI(h_{k,\nu}))$ has positive dimension.
\medskip\noindent
The weight $\varphi_{k,\nu}={1\over 2(a+1)\mu}\log\sum\big|\tau_k^{(a+1)\mu}
(\sigma_i^{a\mu}\cdot\lambda_j^{(a+1)b_k-am_i})\big|^2$ of $h_{k,\nu}$ is
plurisubharmonic and the condition $m_i<{a+1\over a}b_k$ implies
$(a+1)b_k-am_i\ge 1$, thus the difference
$\varphi_{k,\nu}-{1\over2(a+1)\mu}\log\sum|\tau(\lambda_j)|^2$ is also
plurisubharmonic. Hence $\Theta_{h_{k,\nu}}(aK_X+b_kL)={\I\over\pi}
\partial\dbar\varphi_{k,\nu}\ge{1\over(a+1)}\omega$. Moreover, condition
$\beta)$ clearly implies $\nu(\varphi_{k,\nu},x_j)\ge a(n+s_j)$. Finally,
condition $\gamma)$ combined with the strong Noetherian property of coherent
sheaves ensures that the sequence $(h_{k,\nu})_{\nu\ge 1}$ will finally
produce a zero dimensional subscheme $V(\cI(h_{k,\nu}))$. We agree that
the sequence $(h_{k,\nu})_{\nu\ge 1}$ stops at this point, and we denote
by $h_k=h_{k,\nu}$ the final metric, such that $\dim V(\cI(h_k))=0$.

For $k=1$, it is clear that the desired metrics $(h_{1,\nu})_{\nu\ge 1}$
exist if $b_1$ is taken large enough (so large, say, that
$(a+1)K_X+(b_1-1)L$ generates jets of order $(a+1)(n+\max s_j)$ at every
point; then the sections $\sigma_1\ld\sigma_\nu$ can be chosen with
$m_1=\ldots=m_\nu=b_1-1$). Suppose that the metrics $(h_{k,\nu})_{\nu\ge 1}$
and $h_k$ have been constructed and let us proceed with the construction
of $(h_{k+1,\nu})_{\nu\ge 1}$. We do this again by induction on~$\nu$,
assuming that $h_{k+1,\nu}$ is already constructed and that $\dim
V(\cI(h_{k+1,\nu}))>0$. We start in fact the induction with $\nu=0$, and
agree in this case that $\cI(h_{k+1,0})=0$ (this would correspond to an
infinite metric of weight identically equal to $-\infty$). By Nadel's
vanishing theorem applied to $F_m=aK_X+mL=(aK_X+b_kL)+(m-b_k)L$ with
the metric $h_k\otimes(h_L)^{\otimes m-b_k}$, we get
$$H^q(X,\cO((a+1)K_X+mL)\otimes\cI(h_k))=0\qquad
\hbox{\rm for $q\ge 1$, $m\ge b_k$.}$$
As $V(\cI(h_k))$ is $0$-dimensional, the sheaf $\cO_X/\cI(h_k)$ is a
skyscraper sheaf, and the exact sequence $0\to\cI(h_k)\to\cO_X\to
\cO_X/\cI(h_k)\to 0$ twisted with the invertible sheaf $\cO((a+1)K_X+mL)$
shows that
$$H^q(X,\cO((a+1)K_X+mL))=0\qquad \hbox{\rm for $q\ge 1$, $m\ge b_k$.}$$
Similarly, we find
$$H^q(X,\cO((a+1)K_X+mL)\otimes\cI(h_{k+1,\nu}))=0\qquad
\hbox{\rm for $q\ge 1$, $m\ge b_{k+1}$}$$
(also true for $\nu=0$, since $\cI(h_{k+1,0})=0$), and when
\hbox{$m\ge\max(b_k,b_{k+1})=b_k$}, the exact sequence $0\to\cI(h_{k+1,\nu})
\to\cO_X\to\cO_X/\cI(h_{k+1},\nu)\to 0$ implies
$$H^q(X,\cO((a+1)K_X+mL)\otimes\cO_X/\cI(h_{k+1,\nu}))=0\qquad
\hbox{\rm for $q\ge 1$, $m\ge b_k$.}$$
In particular, since the $H^1$ group vanishes, every section $u'$ of
$(a+1)K_X+mL$ on the subscheme $V(\cI(h_{k+1,\nu}))$ has an extension
$u$ to $X$. Fix a basis $u_1'\ld u_N'$ of the sections on
$V(\cI(h_{k+1,\nu}))$ and take arbitrary extensions $u_1\ld u_N$ to
$X$. Look at the linear map assigning the collection of jets of order
$(a+1)(n+s_j)-1$ at all points $x_j$
$$u=\sum_{1\le j\le N}a_ju_j\longmapsto
\bigoplus J^{(a+1)(n+s_j)-1}_{x_j}(u).$$
Since the rank of the bundle of $s$-jets is ${n+s\choose n}$, the target
space has dimension
$$\delta=\sum_{1\le j\le p}{n+(a+1)(n+s_j)-1\choose n}.$$
In order to get a section $\sigma_{\nu+1}=u$ satisfying condition $\beta)$
with non trivial restriction $\sigma'_{\nu+1}$ to $V(\cI(h_{k+1,\nu}))$, we
need at least $N=\delta+1$ independent sections $u'_1\ld u'_N$. This
condition is achieved by applying Lemma~(2.2) to the numerical polynomial
$$\eqalign{P(m)
&=\chi(X,\cO((a+1)K_X+mL)\otimes\cO_X/\cI(h_{k+1,\nu}))\cr
&=h^0(X,\cO((a+1)K_X+mL)\otimes\cO_X/\cI(h_{k+1,\nu}))\ge 0,\qquad
m\ge b_k.\cr}$$
The polynomial $P$ has degree $d=\dim V(\cI(h_{k+1,\nu}))>0$.
We get the existence of an integer $m\in[b_k,b_k+\eta]$ such that
$N=P(m)\ge\delta+1$ with some explicit integer $\eta\in\bbbn$
(for instance $\eta=n(\delta+1)$ always works by (2.2\,a), but we will also
use the other possibilities to find an optimal choice in each case).
Then we find a section $\sigma_{\nu+1}\in H^0(X,(a+1)K_X+mL)$
with non trivial restriction $\sigma'_{\nu+1}$ to $V(\cI(h_{k+1,\nu}))$,
vanishing at order $\ge(a+1)(n+s_j)$ at each point $x_j$. We just set
$m_{\nu+1}=m$, and the condition $m_{\nu+1}<{a+1\over a}b_{k+1}$ is
satisfied if $b_k+\eta<{a+1\over a}b_{k+1}$. This shows that we can take
inductively
$$b_{k+1}=\left\lfloor{a\over a+1}(b_k+\eta)\right\rfloor+1.$$
By definition, $h_{k+1,\nu+1}\le h_{k+1,\nu}$, hence $\cI(h_{k+1,\nu+1})
\supset\cI(h_{k+1,\nu})$. We necessarily have $\cI(h_{k+1,\nu+1})\ne
\cI(h_{k+1,\nu})$, for $\cI(h_{k+1,\nu+1})$ contains the ideal sheaf
associated with the zero divisor of $\sigma_{\nu+1}$, whilst
$\sigma_{\nu+1}$ does not vanish identically on~$V(\cI(h_{k+1,\nu}))$.
Now, an easy computation shows that the iteration of
$b_{k+1}=\lfloor{a\over a+1}(b_k+\eta)\rfloor+1$ stops at $b_k=a(\eta+1)+1$ for
any large initial value $b_1$. In this way, we obtain a metric $h_\infty$
of positive definite curvature on $aK_X+(a(\eta+1)+1)L$, with
$\dim V(\cI(h_\infty))=0$ and $\nu(\varphi_\infty,x_j)\ge a(n+s_j)$ at
each point~$x_j$.
\endproof

\begproof{of (2.3).} In this case, the set $\{x_j\}$ is taken to be
empty, thus $\delta=0$. By (2.2\,a), the condition $P(m)\ge 1$ is
achieved for some $m\in[b_k,b_k+n]$ and we can take $\eta=n$. As $\mu L$ is
very ample, there exists on $\mu L$ a metric with an isolated logarithmic
pole of Lelong number $1$ at any given point~$x_0$ (e.g., the algebraic
metric defined with all sections of $\mu L$ vanishing at~$x_0$). Hence
$$F_a'=aK_X+(a(n+1)+1)L+n\mu L$$
has a metric $h_a'$ such that $V(\cI(h_a'))$ is zero dimensional and
contains $\{x_0\}$. By Cor~(1.5), we conclude that
$$K_X+F_a'=(a+1)K_X+(a(n+1)+1+n\mu)L$$
is generated by sections, in particular $K_X+{a(n+1)+1+n\mu\over a+1}L$
is nef. As $a$ tends to $+\infty$, we infer that $K_X+(n+1)L$ is nef.\qed
\endproof

\begproof{of (2.4\,a).} Here, the choice $a=1$ is sufficient for our purposes.
Then
$$\delta=\sum_{1\le j\le p}{3n+2s_j-1\choose n}.$$
If $\{x_j\}\ne\emptyset$, we have $\delta+1\ge{3n-1\choose n}+1\ge 2n^2$
for $n\ge 2$. Lemma (2.2\,c) shows that $P(m)\ge\delta+1$ for some
$m\in[b_k,b_k+\eta]$ with $\eta=\delta+1$. We can start in fact the
induction procedure $k\mapsto k+1$ with $b_1=\eta+1=\delta+2$, because the
only property needed for the induction step is the vanishing
property
$$H^0(X,2K_X+mL)=0\qquad\hbox{for $q\ge 1$, $m\ge b_1$},$$
which is true by the Kodaira vanishing theorem and the ampleness
of $K_X+b_1L$ (here we use Fujita's result (2.3), observing that $b_1>n+1$).
Then the recursion formula $b_{k+1}=\lfloor{1\over 2}(b_k+\eta)\rfloor+1$
yields $b_k=\eta+1=\delta+2$ for all $k$, and (2.4\,a) follows.\qed
\endproof

\begproof{of (2.4\,b).} Quite similar to (2.4\,a), except that we take
$\eta=n$, $a=1$ and $b_k=n+1$ for all~$k$. By Lemma (2.2\,b), we have
$P(m)\ge a_dk^d/2^{d-1}$ for some integer $m\in[m_0,m_0+kd]$,
where $a_d>0$ is the coefficient of highest degree in~$P$. By Lemma (2.1)
we have $a_d\ge\inf_{\dim Y=d}L^d\cdot Y$. We take $k=\lfloor n/d\rfloor$. The
condition $P(m)\ge\delta+1$ can thus be realized for some $m\in[m_0,m_0+kd]
\subset[m_0,m_0+n]$ as soon as
$$\inf_{\dim Y=d}L^d\cdot Y~\lfloor n/d\rfloor^d/2^{d-1}>\delta,$$
which is equivalent to the condition given in (2.4\,b).\qed
\endproof

Theorem~(0.2\,a) is a special case of Th.~(2.4\,a). Theorem (0.2\,b)
can be derived from (2.4\,b) by using the following simple lemma.

\begstat{(2.5) Lemma} Assume that for some integer $\mu\in\bbbn^\star$
the line bundle $\mu F$ gene\-rates simultaneously all jets of order
$\mu(n+s_j)+1$ at any point $x_j$ in a subset $\{x_1\ld x_p\}\subset
X$. Then $K_X+F$ generates simultaneously all jets of order $s_j$ at~$x_j$.
\endstat

\begproof{} Take the algebraic metric on $F$ defined by a basis of
sections $\sigma_1\ld\sigma_N$ of $\mu F$ which vanish at order
$\mu(n+s_j)+1$ at all points~$x_j$. Since we are still free to choose
the homogeneous term of degree $\mu(n+s_j)+1$ in the Taylor expansion
at~$x_j$, we find that $x_1\ld x_p$ are isolated zeroes of
$\bigcap\sigma_j^{-1}(0)$. If $\varphi$ is the weight of the metric
of $F$ near~$x_j$, we thus have $\varphi(z)\sim (n+s_j+{1\over\mu})
\log|z-x_j|$ in suitable coordinates. We replace $\varphi$ in a
neighborhood of $x_j$ by
$$\varphi'(z)=\max\big(\varphi(z)\,,\,|z|^2-C+(n+s_j)\log|z-x_j|\big)$$
and leave $\varphi$ elsewhere unchanged (this is possible by taking
$C>0$ very large). Then $\varphi'(z)=|z|^2-C+(n+s_j)\log|z-x_j|$ near
$x_j$, in particular $\varphi'$ is strictly plurisubharmonic
near~$x_j$. In this way, we get a metric $h'$ on $F$ with semipositive
curvature everywhere on $X$, and with positive definite curvature on a
neighborhood of $\{x_1\ld x_p\}$. The conclusion then follows from
Cor.~(1.5) and Rem.~(1.6).\qed
\endproof

\begproof{of Theorem (0.2\,b).} By Lemma~(2.5) applied with $F=K_X+L$
and $\mu=n+1$, the desired jet generation of $2K_X+L$ occurs if
$(n+1)(K_X+L)$ generates jets of order $(n+1)(n+s_j)+1$ at~$x_j$. By
Lemma~(2.5) again with $F=aK_X+(n+1)L$ and $\mu=1$, we see by backward
induction on $a$ that we need the simultaneous generation of jets of
order $(n+1)(n+s_j)+1+(n+1-a)(n+1)$ at~$x_j$. In particular, for
$2K_X+(n+1)L$ we need the generation of jets of order
$(n+1)(2n+s_j-1)+1$. Theorem (2.4\,b) yields the desired condition.\qed
\endproof

\begproof{of Theorem (0.2\,c).} Apply Th.~(2.4\,a) with
$G'=a(K_X+(n+1)L)+G$, so that
$$2K_X+mL+G'=(a+2)(K_X+(n+2)L)+(m-2n-4-a)L+G,$$
and take $m=a+2n+4\ge 2+{3n+1\choose n}$.\qed
\endproof
 
\titlea{3. An Estimate for $L^2$ Dualizing Sheaves}
If $Y$ is a complex $p$-dimensional analytic space with arbitrary
singularities, we define the $L^2$ dualizing sheaf of~$Y$ to be the sheaf of
holomorphic $p$-forms $u$ on the regular part $Y_\reg$ which are
locally $L^2$ near $Y_\sing$, that is, for any open set $W\subset Y$,
$$\Gamma(W,\omega_Y)=\big\{u\in\Gamma(W\cap Y_\reg,\Omega^p_{Y_\reg})\,;\,
\forall x\in W,\,\exists V\ni x,~\int_{V\cap Y_\reg}\I^{p^2}u\wedge\ovl u
<+\infty\big\},$$
where $V$ is an arbitrary neighborhood of~$x$. It is easily seen that
$\omega_Y$ is the direct image of the dualizing sheaf $K_{\wt Y}$ of
a desingularization of $Y$, thus $\omega_Y$ is a coherent sheaf on~$Y$
($\omega_Y$ is just the usual dualizing sheaf of algebraic geometers).
Then we have the following optimal ``upper estimate'' for $\omega_Y$.

\begstat{(3.1) Theorem} Let $H$ be a very ample line bundle on a
projective algebraic manifold~$X$, and let $Y\subset X$ be a
$p$-dimensional irreducible algebraic subvariety. If $\delta=H^p\cdot Y$
is the degree of $Y$ with respect to $H$, the sheaf
\hbox{$\cHom\big(\omega_Y,\cO_Y((\delta-p-2)H)\big)$} has a nontrivial
section.
\endstat

Observe that if $Y$ is a smooth hypersurface of degree $\delta$ in
$(X,H)=(\bbbp^{p+1},\cO(1))$, then $\omega_Y=\cO_Y(\delta-p-2)$ and the
estimate is optimal. On the other hand, if $Y$ is a smooth complete
intersection of multidegree $(\delta_1\ld\delta_r)$ in $\bbbp^{p+r}$,
then $\delta=\delta_1\ldots\delta_r$ whilst
$\omega_Y=\cO_Y(\delta_1+\ldots+\delta_r-p-r-1)\,$; in this case, Th.~(3.1)
is thus very far from being sharp.

\begproof{} Let $X\subset\bbbp^N$ be the embedding given by $H$, so
that $H=\cO_X(1)$. There is a linear projection
$\bbbp^n\dasharrow\bbbp^{p+1}$ whose restriction $\pi:Y\to\bbbp^{p+1}$
to $Y$ is a finite and regular birational map of $Y$ onto an algebraic
hypersurface $Y'$ of degree $\delta$ in $\bbbp^{p+1}$.  Let $s\in
H^0(\bbbp^{p+1},\cO(\delta))$ be the polynomial of degree $\delta$
defining $Y'$. We claim that for any small Stein open set
$W\subset\bbbp^{p+1}$ and any $L^2$ holomorphic $p$-form $u$ on $Y'\cap
W$, there is a $L^2$ holomorphic $(p+1)$-form $\wt u$ on $W$ with
values in $\cO(\delta)$ such that $\wt u_{\restriction Y'\cap
W}=u\wedge ds$. In fact, this is precisely the conclusion of the
Ohsawa-Takegoshi extension theorem [OT87], [Ohs88] (see also [Man93]
for a more general version); one can also invoke more standard local
algebra arguments (see Hartshorne [Har77], Th.~III-7.11). As
$K_{\bbbp^{p+1}}=\cO(-p-2)$, the form $\wt u$ can be seen as a section
of $\cO(\delta-p-2)$ on~$W$, thus the sheaf morphism $u\mapsto u\wedge
ds$ extends into a global section of
$\cHom\big(\omega_{Y'},\cO_{Y'}(\delta-p-2)\big)$. The pull-back by
$\pi^\star$ yields a section of $\cHom\big(\pi^\star\omega_{Y'},
\cO_Y((\delta-p-2)H)\big)$. Since $\pi$ is finite and generically
$1:1$, it is easy to see that $\pi^\star\omega_{Y'}=\omega_Y$. The
Theorem follows.\qed
\endproof

\titlea{4. An Effective Version of Matsusaka's Big Theorem}
Let $L$ be an ample line bundle on a projective algebraic manifold~$X$.
We look for an explicit value of $m_0$ such that $mL$ is very ample
for $m\ge m_0$. As in [Siu93], our starting point is the following lemma.

\begstat{(4.1) Lemma} Let $F$ and $G$ be nef line bundles over~$X$.
If $F^n>n\,F^{n-1}\cdot G$, all large positive multiples $k(F-G)$,
$k\ge k_0$, have non trivial sections.
\endstat

\begproof{} This is a special case of the holomorphic Morse inequalities
(see [Dem85], [Tra95], [Siu93], [Ang95]). Here is a simple proof,
following a suggestion of F.~Catanese. We can suppose that $F$ and $G$
are very ample (other\-wise, we replace $F$ and $G$ by $pF+A$ and $pG+A$
with $A$ very ample and large enough, and $p>0$ very large). Then
$\cO(k(F-G))\simeq\cO(kF-G_1-\ldots-G_k)$ for arbitrary members $G_1\ld
G_k$ in the linear system $|G|$, and the Lemma follows from
Riemann-Roch by looking at the restriction morphism
$H^0(X,\cO(kF))\to\bigoplus H^0(G_j, \cO(kF_{\restriction G_j})$.\qed
\endproof

\begstat{(4.2) Corollary} Let $F$ and $G$ be nef line bundles over~$X$.
If $F$ is big and $m>n\,F^{n-1}\cdot G/F^n$, then $\cO(mF-G)$ can be
equipped with a $($possibly singular$)$ hermitian metric $h$ with positive
definite curvature form $\Theta_h(mF-G)\ge\varepsilon\omega$, $\varepsilon>0$,
for some K\"ahler metric~$\omega$.
\endstat

\begproof{} In fact, if $A$ is ample and $\varepsilon\in\bbbq_+$ small
enough, Lemma~(4.1) implies that some multiple $k(mF-G-\varepsilon A)$ has a
section. Let $E$ be the divisor of this section and let $\omega=\Theta(A)
\in c_1(A)$ be a K\"ahler form. Then $mF-G\equiv\varepsilon A+{1\over k}E$
can be equipped with a singular metric $h$ of curvature form
$\Theta_h(mF-G)=\varepsilon\Theta(A)+{1\over k}[E]\ge\varepsilon\omega$.\qed
\endproof

We now consider the question of obtaining a nontrivial section in $mL$.
The idea, more generally, is to obtain a criterion for the ampleness of
$mL-B$ when $B$ is nef. In this way, one
is able to subtract from $mL$ any undesirable multiple of $K_X$ which
otherwise gets added to $L$ by the application of Nadel's vanishing
theorem (for this, we simply replace $B$ by $B$ plus a multiple of
$K_X+(n+1)L$). 

\begstat{(4.3) Proposition} Let $L$ be an ample line bundle over a projective
$n$-fold $X$ and let $B$ be a nef line bundle over~$X$. Then $K_X+mL-B$
has a nonzero section for some integer $m$ such that\smallskip
\centerline{$\displaystyle m\le n\,{L^{n-1}\cdot B\over L^n}+n+1$.}
\endstat

\begproof{} Let $m_0$ be the smallest integer${}>n\,{L^{n-1}\cdot B\over
L^n}$. Then $m_0L-B$ can be equip\-ped with a singular hermitian metric
$h$ of positive definite curvature. By Nadel's vanishing theorem, we have
$$H^q(X,\cO(K_X+mL-B)\otimes\cI(h))=0\qquad\hbox{for $q\ge 1$,}$$
thus $P(m)=h^0(X,\cO(K_X+mL-B)\otimes\cI(h))$ is a polynomial for
$m\ge m_0$. Since $P$ is a polynomial of degree $n$ and is not identically
zero, there must be an integer $m\in[m_0,m_0+n]$ which is not a root.
Hence there is a nontrivial section in
$$H^0(X,\cO(K_X+mL-B))\supset H^0(X,\cO(K_X+mL-B)\otimes\cI(h))$$
for some $m\in[m_0,m_0+n]$, as desired.\qed
\endproof

\begstat{(4.4) Corollary} If $L$ is ample and $B$ is nef, $mL-B$ has a
nonzero section for some integer
$$m\le n\Big({L^{n-1}\cdot B+L^{n-1}\cdot K_X\over L^n}+n+1\Big).$$
\endstat

\begproof{} By Fujita's result (2.3\,a), $K_X+(n+1)L$ is nef. We can thus
replace $B$ by $B+K_X+(n+1)L$ in the result of Prop.~(4.3).
Corollary (4.4) follows.\qed
\endproof

\begstat{(4.5) Remark} \rm We do not know if the above Corollary is
sharp, but it is certainly not far from being so. Indeed, for $B=0$, the
initial constant $n$ cannot be replaced by anything smaller than
$n/2\,$: take $X$ to be a product of curves $C_j$ of large genus $g_j$ and
$B=0$; our bound for $L=\cO(a_1[p_1])\otimes\ldots\otimes\cO(a_n[p_n])$
to have $|mL|\neq\emptyset$ becomes $m\le\sum(2g_j-2)/a_j+n(n+1)$, which
fails to be sharp only by a factor $2$ when $a_1=\ldots=a_n=1$ and
$g_1\gg g_2\gg\ldots\gg g_n\to+\infty$. On the other hand, the additive
constant $n+1$ is already best possible when $B=0$ and $X=\bbbp^n$.\qed
\endstat

So far, the method is not really sensitive to singularities (Lemma (4.1)
is still true in the singular case as is easily seen by using a
desingularization of $X$). The same is true with Nadel's vanishing Theorem
(1.2), provided that $K_X\otimes\cI(h)$ is replaced by the sheaf $\omega_X(h)$
of $n$-forms which are locally $L^2$ near $X_\sing$ with respect to the
weight $e^{-\varphi}$ of $h$ (according to that notation, the $L^2$
dualizing sheaf $\omega_X$ is associated with $\varphi=0$ or with any
nonsingular weight~$\varphi$). Then Prop.~(4.3) can be generalized as

\begstat{(4.6) Proposition} Let $L$ be an ample line bundle over a
projective $n$-fold $X$ and let $B$ be a nef line bundle over~$X$. For every
$p$-dimensional $($reduced$)$ algebraic subvariety $Y$ of~$X$,
there is an integer
$$m\le p{L^{p-1}\cdot B\cdot Y\over L^p\cdot Y}+p+1$$
such that the sheaf $\omega_Y\otimes\cO_Y(mL-B)$ has a nonzero section.\qed
\endstat

By an appropriate induction process based on the above results, we can
now improve Siu's effective version of the Big Matsusaka Theorem
[Siu93]. Our version depends on a constant $\lambda_n$ such that
$m(K_X+(n+2)L)+G$ is very ample for $m\ge\lambda_n$ and every nef line
bundle~$G$. Theorem (0.2\,c) shows that $\lambda_n\le{3n+1\choose n}-2n$,
and a similar argument involving the recent results of Angehrn-Siu
[AS94] implies $\lambda_n\le n^3-n^2-n-1$ for $n\ge 2$.
Of course, it is expected that $\lambda_n=1$ in view of the Fujita
conjecture. 

\begstat{(4.7) Effective version of the Big Matsusaka Theorem} Let $L$ and
$B$ be nef line bundles on a projective $n$-fold~$X$. Assume that $L$
is ample and set $H=\lambda_n(K_X+(n+2)L)$. Then $mL-B$ is very ample for
$$m\ge(2n)^{(3^{n-1}-1)/2}{(L^{n-1}\cdot(B+H))^{(3^{n-1}+1)/2}
(L^{n-1}\cdot H)^{3^{n-2}(n/2-3/4)-1/4}\over(L^n)^{3^{n-2}(n/2-1/4)+1/4}}.$$
In particular $mL$ is very ample for
$$m\ge C_n~(L^n)^{3^{n-2}}\left(n+2+{L^{n-1}\cdot K_X\over L^n}
\right)^{3^{n-2}(n/2+3/4)+1/4}$$
with $C_n=(2n)^{(3^{n-1}-1)/2}(\lambda_n)^{3^{n-2}(n/2+3/4)+1/4}$.
\endstat

\begproof{} We use Th.~(3.1) and Prop.~(4.6) to construct inductively a
sequence of (non necessarily irreducible) algebraic subvarieties
$X=Y_n\supset Y_{n-1}\supset\ldots\supset Y_2\supset Y_1$ such that
$Y_p=\bigcup_jY_{p,j}$ is $p$-dimensional, and $Y_{p-1}$ is obtained
for each $p\ge 2$ as the union of zero sets of sections
$$\sigma_{p,j}\in H^0(Y_{p,j},\cO_{Y_{p,j}}(m_{p,j}L-B))$$
with suitable integers $m_{p,j}\ge 1$. We proceed by induction on
decreasing values of the dimension~$p$, and find inductively upper
bounds $m_p$ for the integers $m_{p,j}$. 

By Cor.~(4.4), an integer $m_n$ for $m_nL-B$ to have a section
$\sigma_n$ can be found with
$$m_n\le n\,{L^{n-1}\cdot(B+K_X+(n+1)L)\over L^n}
\le n\,{L^{n-1}\cdot(B+H)\over L^n}.$$
Now suppose that the sections $\sigma_n$, $\ldots$, $\sigma_{p+1,j}$ have
been constructed. Then we get inductively a $p$-cycle $\wt
Y_p=\sum\mu_{p,j}Y_{p,j}$ defined by $\wt Y_p={}$ sum of zero divisors
of sections $\sigma_{p+1,j}$ in $\wt Y_{p+1,j}$, where the mutiplicity
$\mu_{p,j}$ on $Y_{p,j}\subset Y_{p+1,k}$ is obtained by multiplying
the corresponding multiplicity $\mu_{p+1,k}$ with the vanishing order
of $\sigma_{p+1,k}$ along $Y_{p,j}$. As cohomology classes, we find
$$\wt Y_p\equiv\sum(m_{p+1,k}L-B)\cdot (\mu_{p+1,k}Y_{p+1,k})\le
m_{p+1} L\cdot \wt Y_{p+1}.$$
Inductively, we thus have the numerical inequality
$$\wt Y_p\le m_{p+1}\ldots m_n L^{n-p}.$$
Now, for each component $Y_{p,j}$, Prop.~(4.6) shows that there exists
a section of $\omega_{Y_{p,j}}\otimes\cO_{Y_{p,j}}(m_{p,j}L-B)$ for some
integer
$$m_{p,j}\le p{L^{p-1}\cdot B\cdot Y_{p,j}\over L^p\cdot Y_{p,j}}+p+1
\le pm_{p+1}\ldots m_n\,L^{n-1}\cdot B+p+1.$$
Here, we have used the obvious lower bound $L^{p-1}\cdot Y_{p,j}\ge 1$
(this is of course a rather weak point in the argument). The degree of
$Y_{p,j}$ with respect to $H$ admits the upper bound
$$\delta_{p,j}:=H^p\cdot Y_{p,j}\le
m_{p+1}\ldots m_n H^p\cdot L^{n-p}.$$
We use the Hovanski-Teissier concavity inequality
$$(L^{n-p}\cdot H^p)^{1\over p}(L^n)^{1-{1\over p}}\leq L^{n-1}\cdot H$$
([Hov79], [Tei79,~82], see also [Dem93]) to express our bounds in terms of
the intersection numbers $L^n$ and $L^{n-1}\cdot H$ only. We then get
$$\delta_{p,j}\le m_{p+1}\ldots m_n{(L^{n-1}\cdot H)^p\over (L^n)^{p-1}}.$$
By Th.~(3.1), there is a nontrivial section in
$$\cHom\big(\omega_{Y_{p,j}},\cO_{Y_{p,j}}((\delta_{p,j}-p-2)H)\big).$$
Combining this section with the section in $\omega_{Y_{p,j}}\otimes
\cO_{Y_{p,j}}(m_{p,j}L-B)$ already cons\-truc\-ted, we get a section of
$\cO_{Y_{p,j}}(m_{p,j}L-B+(\delta_{p,j}-p-2)H)$ on~$Y_{p,j}$. Since we do
not want $H$ to appear at this point, we replace $B$ with
$B+(\delta_{p,j}-p-2)H$ and thus get a section $\sigma_{p,j}$ of
$\cO_{Y_{p,j}}(m_{p,j}L-B)$ with some integer $m_{p,j}$ such that
$$\eqalign{m_{p,j}
&\le pm_{p+1}\ldots m_n\,L^{n-1}\cdot(B+(\delta_{p,j}-p-2)H)+p+1\cr
&\le p\,m_{p+1}\ldots m_n\,\delta_{p,j}\,L^{n-1}\cdot(B+H)\cr
&\le p\,(m_{p+1}\ldots m_n)^2{(L^{n-1}\cdot H)^p\over(L^n)^{p-1}}\,
L^{n-1}\cdot(B+H).\cr}$$
Therefore, by putting $M=n\,L^{n-1}\cdot(B+H)$, we get the recursion relation
$$m_p\le M\,{(L^{n-1}\cdot H)^p\over (L^n)^{p-1}}\,(m_{p+1}\ldots m_n)^2\qquad
\hbox{for $2\le p\le n-1$},$$
with initial value $m_n\le M/L^n$. If we let $(\ovl m_p)$ be the sequence
obtained by the same recursion formula with equalities instead of
inequalities, we get $m_p\le\ovl m_p$ with $\ovl m_{n-1}=
M^3(L^{n-1}\cdot H)^{n-1}/(L^n)^n$ and
$$\ovl m_p={L^n\over L^{n-1}\cdot H}\,\ovl m_{p+1}^2\ovl m_{p+1}$$
for $2\le p\le n-2$. We then find inductively
$$m_p\le\ovl m_p=M^{3^{n-p}}{(L^{n-1}\cdot H)^{3^{n-p-1}(n-3/2)+1/2}\over
(L^n)^{3^{n-p-1}(n-1/2)+1/2}}.$$
We next show that $m_0L-B$ is nef for
$$m_0=\max\big(m_2\,,\,m_3\ld m_n\,,\,m_2\ldots m_n\,L^{n-1}\cdot B\big).$$
In fact, let $C\subset X$ be an arbitrary irreducible curve. Either
$C=Y_{1,j}$ for some $j$ or there exists an integer $p=2\ld n$ such that
$C$ is contained in $Y_p$ but not in~$Y_{p-1}$. If $C\subset
Y_{p,j}\ssm Y_{p-1}$, then $\sigma_{p,j}$ does not vanish identically
on~$C$. Hence $(m_{p,j}L-B)_{\restriction C}$ has nonnegative degree and
$$(m_0L-B)\cdot C\ge(m_{p,j}L-B)\cdot C\geq 0.$$
On the other hand, if $C=Y_{1,j}\,$, then
$$(m_0L-B)\cdot C\ge m_0-B\cdot\wt Y_1\ge m_0-m_2\ldots m_n\,L^{n-1}\cdot
B\ge 0.$$
By the definition of $\lambda_n$ (and the proof of (0.2\,c) that such a
constant exists), $H+G$ is very ample for every nef line bundle~$G$, in
particular $H+m_0L-B$ is very ample. We thus replace again $B$ with $B+H$.
This has the effect of replacing $M$ with $M=n\,\big(L^{n-1}\cdot(B+2H)\big)$
and $m_0$ with
$$m_0=\max\big(m_n\,,\,m_{n-1}\ld m_2\,,\,m_2\ldots m_n\,L^{n-1}\cdot(B+H)
\big).$$
The last term is the largest one, and from the estimate on $\ovl m_p\,$, we
get
$$\eqalignno{m_0
&\le M^{(3^{n-1}-1)/2}{(L^{n-1}\cdot H)^{(3^{n-2}-1)(n-3/2)/2+(n-2)/2}
(L^{n-1}\cdot(B+H))\over(L^n)^{(3^{n-2}-1)(n-1/2)/2+(n-2)/2+1}}\cr
&\le(2n)^{(3^{n-1}-1)/2}{(L^{n-1}\cdot(B+H))^{(3^{n-1}+1)/2}
(L^{n-1}\cdot H)^{3^{n-2}(n/2-3/4)-1/4}\over(L^n)^{3^{n-2}(n/2-1/4)+1/4}}
\cr}$$
\qed
\endproof

\begstat{(4.8) Remark} \rm In the surface case $n=2$, one can take
$\lambda_n=1$ and our bound yields $mL$ very ample for
$$m\ge 4\,{(L\cdot(K_X+4L))^2\over L^2}.$$
If one looks more carefully at the proof, the initial constant $4$ can be
replaced by~$2$. In fact, it has been shown recently by Fern\'andez del
Busto that $mL$ is very ample for
$$m>{1\over 2}\left[{(L\cdot(K_X+4L)+1)^2\over L^2}+3\right],$$
and an example of G.~Xiao shows that this bound is essentially optimal
(see [FdB94]).
\endstat

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\endref

\end

