An Informal Introduction to Multiplier Ideals

Summary

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.

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 Brian¸con–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. (In fact, the “multiplier” in the name refers to their analytic construction; see Section 2.4.) This machine was developed and applied with great success by Demailly, Siu and others. Algebro-geometrically, the foundations were laid in passing by Esnault and Viehweg in connection with their work involving the Kawamata–Viehweg vanishing theorem. More systematic developments of the geometric theory were subsequently undertaken by Ein, Kawamata and Lazarsfeld. We will take the geometric approach here.

The present notes follow closely a short course on multiplier ideals given by Lazarsfeld at the Introductory Workshop for the Commutative Algebra Program at the MSRI in September 2002. The three main lectures were supplemented with a presentation by Blicke on multiplier ideals associated to monomial ideals (which appears here in Section 3). We have tried to preserve in this write-up the informal tone of these talks: thus we emphasize simplicity over generality in statements of results, and we present very few proofs. Our primary hope is to give the reader a feeling for what multiplier ideals are and how they are used. For a detailed development of the theory from an algebro-geometric perspective we refer to Part Three of the forthcoming book [Lazarsfeld 2004]. The analytic picture is covered in Demailly’s lectures [2001].