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\centerline{\fourteenbf
$\hbox{\fourteenbf L}^{\hbox{\tenbf 2}}$ methods and effective 
results in algebraic geometry} 
\vskip6mm
\centerline{\bf Jean-Pierre Demailly}
\medskip
\centerline{\bf Institut Fourier, Universit\'e de Grenoble I}
\medskip
\centerline{\bf F-38402 Saint Martin d'H\`eres, France}
\vskip10mm

One important question arising in Algebraic Geometry is the
computation of effective bounds for the degree of embeddings in
projective space of given algebraic varieties. If $L$ is a
holomorphic line bundle on a projective algebraic manifold $X$, it is
well known since Kodaira's fundamental work that $L$ is ample if and
only if $L$ carries a smooth Hermitian metric with positive curvature.
Recall (using an additive notation for the Picard group) that $L$ is
said to be ample if some multiple $mL$ is very ample, i.e. if $mL$ is
spanned by global sections and the map $$\Phi_{mL}\colon X\to \bP^N,\quad
x\mapsto(\sigma_0(x)\colon\sigma_1(x):\cdots:\sigma_N(x))$$ defined
by a basis of sections $(\sigma_j)_{0\leq j\leq N}$ in $H^0(X,mL)$ is
an embedding. The basic question is to compute an effective bound
$m_0>0$ such that $mL$ is very ample for $m\geq m_0$; the variety $X$
can then be embedded as a subvariety of $\bP^N$ of degree
$m^n_0L^n$ where $L^n\coloneq  \int_Xc_1(L)^n$, $n=\dim X$. In relation
with this question, T.~Fujita has raised the following important.
\bigskip

\noindent {\bf Conjecture} (([Fuj88]). --- {\it For any ample line
  bundle $L$ on a $n$-dimensional projective manifold $X$, then
  $K_X+(n+1)L$ is spanned by sections, and $K_X+(n+2)L$ is very ample.}
\bigskip

Here $K_X=\Lambda^nT^*_X$ is the canonical bundle. The examples of
curves and projective spaces already show that Fujita's bounds would
be optimal. Such questions have attracted a lot of attention in recent
years. The first breakthrough is due to I.~Reider [Rdr88], who proved
the conjecture in the case of surfaces, using Bogomolov's instability
criterion for rank 2 bundles. An insight on the hgiher dimenaional
case has been obtained through various algebraic methods in subsequent
works by J. Koll\'ar ([Kol93]), and L.~Ein-R.~Lazarsfeld [EL93],
[EL94]. Especially, J.~Koll\'ar has obtained the following effective
version of the Kawamata base point free theorem: if $L$ is nef
(numerically effective) with $L^n>0$, and if $K_X+L$ is nef, then
$m(K_X+L)$ is spanned for $m>2(n+2)!(n+1)$. In dimension 3, the
spannedness part of Fujita's conjecture has been solved by [EL93], and
their result was improved shortly afterwards by T. Fujita ([Fuj94]).
Nevertheless, as far as very ampleness is concerned, the best known
result for $n\geq 4$ is the following one, which has been obtained by
an analytic method:
\bigskip

{\bf Main Theorem}. --- {\it For any ample line bundle $L$
  on a $n$-dimensional projective manifold $X$, the line bundle
  $2K_X+mL$ is very ample for $m\ge m_n$ where $m_n$ depends only on
dimension. The best known bound $m_n$ has been successively evaluated
as 
$$\eqalign{
m_n&=12\,n^n\kern103pt\hbox{\rm([Dem93])},\cr
m_n&=2\Big(n+1+n{3n+1 \choose n}\Big)\kern25pt\hbox{\rm([Siu94])},\cr 
m_n&=2+{3n+1\choose n}\kern65pt\hbox{\rm([Dem95])}.\cr}
$$}
\medskip

All the above mentioned results come from more technical criteria for
the existence of global sections in the adjoint linear system
$|K_X+L|$ assuming given 1-jets or higher order jets, under suitable
numerical conditions satisfied by $L$ (see [Dem94], [EL94]). The basic
tool for the analytic method is the following fundamental result due
to A. Nadel, which is derived from H\"ormander's $L^2$ estimates for
the solutions of the $\overline\partial$ operator. This result can be
seen as an analytic version of the Kawamata-Viehweg vanishing theorem:
\bigskip

\noindent
{\bf Nadel's vanishing theorem} ([Nad89], [Dem89]). --- {\it Let $X$ be a
holomorphic line bundle on a projective algebvraic manifold $X$ and
suppose that $L$ can be equipped with a possibly singular Hermitian
metric such that the curvature in the sense of currents is positive
definite everywhere. Let $e^{-\varphi}$ be the local weight
representing the Hermitian metric of $L$, and let $\cI(\varphi)$
be the sheaf of ideal of germs of holomorphic functions $f$ such that
$|f|^2e^{-\varphi}$ is locally integrable. Then $H^q(X,\cO(K_X+L)
\otimes\cI(\varphi))=0$ for all $q\geq 1$.}
\bigskip

Using vanishing of the $H^1$ group, one obtains a sufficient condition
for the existence of sections generating given jets in $H^0(X,\cO
(K_X+L)\otimes \cO/\cI(\varphi))$. The main point in
this approach is thus to construct suitable plurisubharmonic weights
$\varphi$ achieving the desired ideals $\cI(\varphi)$ for the
jets. This is done by solving a complex Monge-Amp\`ere equation
$(\omega+i\partial\overline\partial\varphi)=f$, where $f$ is taken to
be a linear combination of Dirac measures and of a uniform density
with respect to a given K\"ahler metric $\omega$. The existence of a
solution is guaranteed by the Aubin-Calabi-Yau theorem. The
computation of the ideals $\cI(\varphi)$ relies heavily on the
theory of positive currents and Lelong numbers.

As an application of the above Main Theorem, Y.T.~Siu [Siu94]
obtained recently an effective version of the big Matsusaka theorem.
Siu's bound has been refined further in [Dem95]~:
{\it if $L$ is ample, then $mL$ is very ample for 
$$
m\ge m_0\coloneq C_n~(L^n)^{3^{n-2}}\left(n+2+{L^{n-1}\cdot K_X\over L^n}
\right)^{3^{n-2}(n/2+3/4)+1/4}
$$
with $C_n=(2n)^{(3^{n-1}-1)/2}\big({3n+1\choose n}-2n
\big)^{3^{n-2}(n/2+3/4)+1/4}$.
\vfill\eject

\noindent
{\twelvebf References}
\bigskip

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