An Informal Introduction to Multiplier Ideals

Summary

Abstract. Multiplier ideals are associated with a complex variety and an ideal or ideal sheaf thereon, and satisfy certain vanishing theorems that have proved rich in applications, for example in local algebra. This article offers an introduction to the study of multiplier ideals, mainly adopting the geometric viewpoint.

Introduction

Given a smooth complex variety X and an ideal (or ideal sheaf) a on X, one can attach to a a collection of multiplier ideals J(a^c) depending on a rational weighting parameter c > 0. These ideals, and the vanishing theorems they satisfy, have found many applications in recent years. In the global setting they have been used to study pluricanonical and other linear series on a projective variety [Demailly 1993; Angehrn and Siu 1995; Siu 1998; Ein and Lazarsfeld 1997; 1999; Demailly 1999]. More recently they have led to the discovery of some surprising uniform results in local algebra [Ein et al. 2001; 2003; 2004]. The purpose of these lectures is to give an easy-going and gentle introduction to the algebraically-oriented local side of the theory.

Multiplier ideals can be approached (and historically emerged) from three different viewpoints. In commutative algebra they were introduced and studied by Lipman [1993] (under the name “adjoint ideals”, which now means something else), in connection with the Briançon–Skoda theorem. On the analytic side of the field, Nadel [1990] attached a multiplier ideal to any plurisubharmonic function, and proved a Kodaira-type vanishing theorem for them.