$L^2$ vanishing theorems for positive line bundles and adjunction theory, Lecture Notes of a CIME course on "Transcendental Methods of Algebraic Geometry" (Cetraro, Italy, July 1994)

TL;DR

This paper discusses $L^2$ vanishing theorems for positive line bundles, their applications to adjoint line bundles, and compares analytic and algebraic approaches, providing effective bounds for very ampleness.

Contribution

It introduces new $L^2$ vanishing theorems, compares analytic and algebraic methods for positivity, and derives explicit bounds for very ampleness of line bundles.

Findings

Established new $L^2$ vanishing theorems for positive line bundles.

Compared analytic and algebraic approaches to adjoint bundle problems.

Provided explicit bounds for very ampleness depending on intersection numbers.

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.