L2 vanishing theorems for positive line bundles and adjunction theory

Abstract

The notes start with an elementary introduction to a few important analytic techniques of algebraic geometry: closed positive currents, $L^2$ estimates for the $\dbar$-operator on positive vector bundles, Nadel's vanishing theorem for multiplier ideal sheaves (a generalization of the well-known Kawamata-Viehweg vanishing theorem). Applications to adjoint line bundles are then discussed. T.~Fujita conjectured in 1987 that $K_X+(n+2)L$ is very ample for every ample line bundle $L$ on a non singular projective variety $X$ with $\dim X=n$. The answer is known only for $n\le 2$ (I.~Reider, 1988). In the last years, various bounds have been obtained for integers $m$ such that $2K_X+mL$ is very ample (by J.~Koll\'ar, L.~Ein-R.~Lazarsfeld, Y.T.~Siu and the author, among others). Two approaches are discussed: an analytic approach via Monge-Amp\`ere equations and current theory, and a more algebraic one (due to Siu) via multiplier ideal sheaves and Riemann-Roch. Finally, an effective version of the big Matsusaka theorem is derived, in the form of an explicit bound for an integer $m$ such that $mL$ is very ample, depending only on $L^n$ and $L^{n-1}\cdot K_X$; these bounds improve Siu's results (1993), and essentially contain the optimal bounds obtained by Fernandez del Busto for the surface case.