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**Positivity in algebraic geometry I: Classical setting: Line bundles and linear series. II: Positivity for vector bundles, and multiplier ideals.**
*(English)*
Zbl 1066.14021

Berlin: Springer (ISBN 3-540-22528-5/vol.1; 3-540-22531-5/vol.2). xviii, 387 p./vol.1; xvii, 385 p./vol.2. (2004).

This two-volume book offers a comprehensive, up-to-date account on ampleness and positivity in complex algebraic geometry. The book has three parts: Part one, which occupies the first volume, presents the theory of ample, nef and big line bundles and its applications. Part two deals with the corresponding theory for vector bundles. Part three develops the theory of multiplier ideals. Throughout, the exposition is clear and polished, and puts much emphasis on examples, remarks, and constructions. There are no exercises. The bibliography contains over 600 entries.

The content of part one: The first chapter recalls the classical theory of ample and nef line bundles, treating their geometrical, cohomological, and numerical significance. There is also a discussion of the cone theorem and Castelnuovo–Mumford regularity. Chapter 2 deals with line bundles that may not be ample, and in particular with big line bundles. Chapter 3 starts with the various Lefschetz hyperplane theorems, goes on to Barth’s Theorem on varieties of small codimension in projective space, and continues with the connectedness theorems of Fulton-Hansen and Grothendieck. Chapter 4 is devoted to vanishing theorems. Here we have the results of Kodaira and Nakano for ample line bundles, and of Kawamata-Viehweg for nef and big line bundles, as well as the generic vanishing theorems of M. Green and R. Lazarsfeld [Invent. Math. 90, 389–407 (1987; Zbl 0659.14007)]. The last chapter discusses the theory of Seshadri constants.

The second part treats positivity for vector bundles. Chapter 6 recalls the definitions and basic properties, including a discussion of twisting vector bundles with \(\mathbb{Q}\)-line bundles, and the relation of ampleness with stability on curves. Chapter 7 treats analogs of Lefschetz theorems for ample vector bundles, together with results on degeneracy loci and vanishing theorems. Chapter 8 discusses positivity of Chern classes, in particular the characterization of positive polynomials by W. Fulton and R. Lazarsfeld [Ann. Math. (2) 118, 35–60 (1983; Zbl 0537.14009)]. Part three gives a systematic development of multiplier ideals, from an algebraic viewpoint. Chapter 9 starts with the Kawamata–Viehweg vanishing theorem for \(\mathbb{Q}\)-divisors, moves on to the definition of multiplier ideals, and then gives vanishing theorems for multiplier ideals, as well as Skoda’s theorem on multiplier ideals for ideal powers. Chapter 10 contains many applications of multiplier ideals. Chapter 11 deals with asymptotic multiplier ideals and several applications, the highlight being Y.-T. Siu’s [Invent. Math. 134, No. 3, 661–673 (1998; Zbl 0955.32017)] proof for the invariance of plurigenera.

The book contains a wealth of material and aims at readers with a certain overview in complex algebraic geometry. However, the text never gets bogged down in technicalities, and is a pleasure to read. The presentation nicely reveals historical developments and mathematical interplay between various results. Lazarsfeld has a knack of explaining important results through their applications. Much of the material in part three appear here for the first time in book form. A fine book indeed.

The content of part one: The first chapter recalls the classical theory of ample and nef line bundles, treating their geometrical, cohomological, and numerical significance. There is also a discussion of the cone theorem and Castelnuovo–Mumford regularity. Chapter 2 deals with line bundles that may not be ample, and in particular with big line bundles. Chapter 3 starts with the various Lefschetz hyperplane theorems, goes on to Barth’s Theorem on varieties of small codimension in projective space, and continues with the connectedness theorems of Fulton-Hansen and Grothendieck. Chapter 4 is devoted to vanishing theorems. Here we have the results of Kodaira and Nakano for ample line bundles, and of Kawamata-Viehweg for nef and big line bundles, as well as the generic vanishing theorems of M. Green and R. Lazarsfeld [Invent. Math. 90, 389–407 (1987; Zbl 0659.14007)]. The last chapter discusses the theory of Seshadri constants.

The second part treats positivity for vector bundles. Chapter 6 recalls the definitions and basic properties, including a discussion of twisting vector bundles with \(\mathbb{Q}\)-line bundles, and the relation of ampleness with stability on curves. Chapter 7 treats analogs of Lefschetz theorems for ample vector bundles, together with results on degeneracy loci and vanishing theorems. Chapter 8 discusses positivity of Chern classes, in particular the characterization of positive polynomials by W. Fulton and R. Lazarsfeld [Ann. Math. (2) 118, 35–60 (1983; Zbl 0537.14009)]. Part three gives a systematic development of multiplier ideals, from an algebraic viewpoint. Chapter 9 starts with the Kawamata–Viehweg vanishing theorem for \(\mathbb{Q}\)-divisors, moves on to the definition of multiplier ideals, and then gives vanishing theorems for multiplier ideals, as well as Skoda’s theorem on multiplier ideals for ideal powers. Chapter 10 contains many applications of multiplier ideals. Chapter 11 deals with asymptotic multiplier ideals and several applications, the highlight being Y.-T. Siu’s [Invent. Math. 134, No. 3, 661–673 (1998; Zbl 0955.32017)] proof for the invariance of plurigenera.

The book contains a wealth of material and aims at readers with a certain overview in complex algebraic geometry. However, the text never gets bogged down in technicalities, and is a pleasure to read. The presentation nicely reveals historical developments and mathematical interplay between various results. Lazarsfeld has a knack of explaining important results through their applications. Much of the material in part three appear here for the first time in book form. A fine book indeed.

Reviewer: Stefan Schröer (Düsseldorf)