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 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 $$\Phi_{mL}\colon X\to {\bf P}^N,\quad x\mapsto(\sigma_0(x)\colon\sigma_1(x):\cdots:\sigma_)N(x))$$ 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 ${\bf P}^N$ 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.
Conjecture (([Fuj88]). -- 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.
Her e$K_X=\Lambda^nT^*_X$ 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. Redier [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´r ([Kol93]), and L. Ein-R. Lazarsfeld [EL93], [EL94]. Especially, J. Koll´r 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:
Main Theorem ([Dem94]). -- For any ample line bundle $L$ on a $n$-dimensional projective manifold $X$, the line bundle $2K_X+12n^nL$ is very ample.
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 numberical conditions satisfied by $L$ (see [De94], [EL94]). The basic tool for the analytic method is the following fundamental result due to A. Nadel, which is derived from Hörmander'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:
Nadel's vanishing theorem ([Nad89], [Dem89]). -- 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 $e^{-\phi}$ be the local weight representing the Hermitian metric of $L$, and let ${\scr I}(\phi)$ be the sheaf of ideal of germs of holomorphic functions $f$ such that $|f|^2e^{-\phi}$ is locally integrable. Then $H^q(X,{\scr O{(K_X+L)\otimes{\scr I}(\phi))=0$ for all $q\geq 1$.
Using vanishing of the $H^1$ group, one obtains a sufficient condition for the existence of sections generating given jets in $H^0(X,{\scr O}(K_X+L)\otimes {\scr O}/{\scr I}(\phi))$. The main point in this approach is thus to construct suitable plurisubharmonic weights $\phi$ achieving the desired ideals ${\scr I}(\phi)$ for the jets. This is done by solving a complex Monge-Ampère equation $(\omega+i\partial\overline\opartial\phi)=f$, where $f$ is taken to be a linear combination of Dirac measures and of a uniform density with respect to a given Kähler metric $\omega$. The existence of a solution is guaranteed by the Aubin-Calabi-Yau theorem. The computation of the ideals ${\scr IO}(\phi)$ relies heavily on the theory of positive currents and Lelong numbers, and on suitable
As an application of the above Main Theorem, Y. T. Siu [Siu94] obtained recently an effective vresion of the big Matsusaka theorem: if $L$ is ample, then $mL$ is very ample for $$m\geq m_0\coloneq {(2^{3^{n-1}}5n)^{4^{n-1}}(3(3n-2)^nL^n+K_X\cdot L^{n-1})^{4^{n-1}3n}\over (6(3n-2)^n-2n-2)^{4^{n-1}n-2/3}(L^n)^{4^{n-1}3(n-1)}}.$$
Bibliography: