##
**Higher-dimensional algebraic geometry.**
*(English)*
Zbl 0978.14001

Universitext. New York, NY: Springer. xiii, 233 p. (2001).

Higher-dimensional algebraic geometry, that is the study of algebraic varieties of dimension greater than two, is a branch of algebraic geometry which is largely still in progress.

Opposed to the theories of algebraic curves and surfaces, which are basically classical and almost perfectly understood, in the meantime, the classification theory of higher-dimensional varieties is a comparatively young subdiscipline of algebraic geometry and, in its systematic framework, barely older than twenty-five years. However, an amazing quantity of knowledge in this new area has accumulated, in the course of the past twenty years, together with a vast spectrum of new, specific concepts, methods, techniques, and deep-going results. The whole subject has now reached a stage where it is rather hard for the beginner to enter this magic world built up by S. Mori, F. Campana, M. Reid, Y. Kawamata, J. Kollár, Y. Miyaoka, V. Shokurov, V. Iskovskikh, E. Viehweg, and many others.

As a result, higher-dimensional algebraic geometry, in its present state of art, somehow makes the impression of a private club of ultimate experts in the field, and has remained much more confidential than it should have. Certainly, this is particularly due to the fact that there has been no beginner-friendly textbook on this topic, so far, guiding the non-expert through the specific terminology, the considerable amount of technicalities, and the systematic strategy underlying the classification theory of higher-dimensional algebraic varieties.

The book under review represents the first attempt to provide an easily accessible introduction to this subject, and that in the form of a systematic textbook. Of course, there are already several excellent survey articles on topics in higher-dimensional algebraic geometry, and there are also a few special research monographs and high-level reference books, for example J. Kollár’s book: “Rational curves on algebraic varieties” (1996; Zbl 0877.14012) but those can scarely be regarded as comprehensive or easy introductions to the field.

The author’s textbook is based on notes from a class taught at Harvard University during the spring term 1999. Due to the introductory character of this text, the author has not tried to be exhaustive, nor to write another reference book. He has, on the contrary, selected suitable parts of the theory and tried to give basic definitions, essential proofs, and important examples with as many details as possible. In this vein, the material covered in this book falls roughly into three parts:

(1) preparatory and standard facts on Cartier divisors and parameter spaces for morphisms from curves to varieties;

(2) various aspects of the geometry of smooth projective varieties with many rational curves;

(3) first steps toward Mori’s minimal program of birational classification of higher-dimensional algebraic varieties.

More precisely, the entire text consists of seven chapters. Chapter 1, entitled “Curves and divisors on algebraic varieties”, deals with the intersection theory of Cartier divisors, cones of curves, ampleness of divisors, nef and big divisors, and rational curves on exceptional loci of birational morphisms.

Chapter 2, still belonging to the preparatory first part of the text, discusses the space parametrizing curves on a given variety, or more precisely morphisms from a given smooth projective curve to a given smooth quasi-projective variety. The author does not reproduce Grothendieck’s original and very general construction of these spaces, but he explains in some detail their classifying character and their local geometry. In order to keep the discussion as concise as possible, and to avoid too many technical intricacies, complete proofs are given only for the simplest version of these spaces, whilst additional facts are then simply sketched or listed.

Chapter 3 comes with the title “Bend-and-break lemmas”. Here the author introduces Mori’s ingenious techniques for studying deformations of curves on a projective variety passing through a fixed point. Mori’s so-called bend-and-break lemmas, in their various incarnations, provide the basic toolkit for the study of projective varieties with many rational curves, and the author explains them here in a detailed manner.

Chapter 4 turns to the special case of uniruled and rationally connected varieties, including the basic facts on so-called free rational curves, very free rational curves, rationally chain-connected varieties, and smoothing trees of rational curves.

This is used, in the subsequent chapter 5, to study the rational quotient of a variety. This concept, established by F. Campana some years ago, is a very good algebro-geometric substitute for the (generally non-algebraic) quotient of a variety modulo rational chain equivalence of points. The author presents here a construction of this rational quotient, which follows the work of J. Kollár, Y. Miyaoka and S. Mori (1992) in the more special case of Fano varieties rather than Campana’s general method. At the end, this chapter also provides some new results on Fano varieties with high degree \((-K_X)^{\text{dim} X}\).

Whereas, chapters 3, 4, and 5 form the second main part of the book, the following chapters 6 and 7 represent the third part of it, that is an introduction to Mori’s minimal model program of the birational classification of higher-dimensional algebraic varieties.

Chapter 6 studies the cone \(NE(X)\) of effective curves of a smooth projective variety \(X\). The author focuses here on proving and illustrating Mori’s famous cone theorem, which gives a geometric description of the closure of \(NE(X)\) in the smooth case, and at the end of this chapter a closer study of contractions of extremal rays is carried out.

While in the smooth case the proof of the cone theorem can be obtained as an application of Mori’s bend-and-break techniques (explained in chapter 3), the contraction theorem for extremal rays and the construction of a suitable “minimal model” for a given variety are unattainable by means of these ideas. It turns out that one has to allow some kind of singular varieties for that purpose, but Mori’s techniques do not work for singular varieties. Therefore, other methods are required for tackling the minimal model program, and those are the subject of study in the concluding chapter 7.

This chapter is entitled “Cohomological methods”, because the material presented here is mainly based on the whole machinery of cohomological vanishing theorems. Being much less geometrical than the foregoing chapters, this final chapter is by far the most involved and difficult part of the book. The author presents an approach (initiated by Y. Kawamata) which culminates in proving the cone and contraction theorems, in a unified way, as well as in doing the first steps toward the general minimal model program. This includes sections on canonical models, their singularities, singularities of pairs, the Kawamata-Viehweg vanishing theorem, the base-point-free theorem, the rationality theorem, and the length of extremal rays. Methodologically, the author uses the so-called “logarithmic” framework, which makes the entire treatment look more natural and conceivable.

Each chapter comes with a set of exercises, many of which point to additional, further-reaching theorems, and the more difficult ones of them are equipped with brief hints for solution. A particular feature of the text is provided by the numerous instructive examples and comments illustrating the undoubtedly demanding material.

The author has tried to keep the text as self-contained as possible, which has been a fairly difficult task, and all in all he has succeeded in writing the first introductury textbook on higher-dimensional varieties which should be accessible to any motivated reader familiar with the basics of modern algebraic geometry, about at the level of R. Hartshorne’s celebrated standard text. In this regard, the book provides an excellent source for graduate students in algebraic geometry, seasoned mathematicians in general, and theoretical physicists using algebro-geometric methods. The exposition of the material is characterized by a very lucid, refined, and user-friendly style of writing.

Without any doubt, this book fills a gap in the existing textbook literature on algebraic geometry.

Opposed to the theories of algebraic curves and surfaces, which are basically classical and almost perfectly understood, in the meantime, the classification theory of higher-dimensional varieties is a comparatively young subdiscipline of algebraic geometry and, in its systematic framework, barely older than twenty-five years. However, an amazing quantity of knowledge in this new area has accumulated, in the course of the past twenty years, together with a vast spectrum of new, specific concepts, methods, techniques, and deep-going results. The whole subject has now reached a stage where it is rather hard for the beginner to enter this magic world built up by S. Mori, F. Campana, M. Reid, Y. Kawamata, J. Kollár, Y. Miyaoka, V. Shokurov, V. Iskovskikh, E. Viehweg, and many others.

As a result, higher-dimensional algebraic geometry, in its present state of art, somehow makes the impression of a private club of ultimate experts in the field, and has remained much more confidential than it should have. Certainly, this is particularly due to the fact that there has been no beginner-friendly textbook on this topic, so far, guiding the non-expert through the specific terminology, the considerable amount of technicalities, and the systematic strategy underlying the classification theory of higher-dimensional algebraic varieties.

The book under review represents the first attempt to provide an easily accessible introduction to this subject, and that in the form of a systematic textbook. Of course, there are already several excellent survey articles on topics in higher-dimensional algebraic geometry, and there are also a few special research monographs and high-level reference books, for example J. Kollár’s book: “Rational curves on algebraic varieties” (1996; Zbl 0877.14012) but those can scarely be regarded as comprehensive or easy introductions to the field.

The author’s textbook is based on notes from a class taught at Harvard University during the spring term 1999. Due to the introductory character of this text, the author has not tried to be exhaustive, nor to write another reference book. He has, on the contrary, selected suitable parts of the theory and tried to give basic definitions, essential proofs, and important examples with as many details as possible. In this vein, the material covered in this book falls roughly into three parts:

(1) preparatory and standard facts on Cartier divisors and parameter spaces for morphisms from curves to varieties;

(2) various aspects of the geometry of smooth projective varieties with many rational curves;

(3) first steps toward Mori’s minimal program of birational classification of higher-dimensional algebraic varieties.

More precisely, the entire text consists of seven chapters. Chapter 1, entitled “Curves and divisors on algebraic varieties”, deals with the intersection theory of Cartier divisors, cones of curves, ampleness of divisors, nef and big divisors, and rational curves on exceptional loci of birational morphisms.

Chapter 2, still belonging to the preparatory first part of the text, discusses the space parametrizing curves on a given variety, or more precisely morphisms from a given smooth projective curve to a given smooth quasi-projective variety. The author does not reproduce Grothendieck’s original and very general construction of these spaces, but he explains in some detail their classifying character and their local geometry. In order to keep the discussion as concise as possible, and to avoid too many technical intricacies, complete proofs are given only for the simplest version of these spaces, whilst additional facts are then simply sketched or listed.

Chapter 3 comes with the title “Bend-and-break lemmas”. Here the author introduces Mori’s ingenious techniques for studying deformations of curves on a projective variety passing through a fixed point. Mori’s so-called bend-and-break lemmas, in their various incarnations, provide the basic toolkit for the study of projective varieties with many rational curves, and the author explains them here in a detailed manner.

Chapter 4 turns to the special case of uniruled and rationally connected varieties, including the basic facts on so-called free rational curves, very free rational curves, rationally chain-connected varieties, and smoothing trees of rational curves.

This is used, in the subsequent chapter 5, to study the rational quotient of a variety. This concept, established by F. Campana some years ago, is a very good algebro-geometric substitute for the (generally non-algebraic) quotient of a variety modulo rational chain equivalence of points. The author presents here a construction of this rational quotient, which follows the work of J. Kollár, Y. Miyaoka and S. Mori (1992) in the more special case of Fano varieties rather than Campana’s general method. At the end, this chapter also provides some new results on Fano varieties with high degree \((-K_X)^{\text{dim} X}\).

Whereas, chapters 3, 4, and 5 form the second main part of the book, the following chapters 6 and 7 represent the third part of it, that is an introduction to Mori’s minimal model program of the birational classification of higher-dimensional algebraic varieties.

Chapter 6 studies the cone \(NE(X)\) of effective curves of a smooth projective variety \(X\). The author focuses here on proving and illustrating Mori’s famous cone theorem, which gives a geometric description of the closure of \(NE(X)\) in the smooth case, and at the end of this chapter a closer study of contractions of extremal rays is carried out.

While in the smooth case the proof of the cone theorem can be obtained as an application of Mori’s bend-and-break techniques (explained in chapter 3), the contraction theorem for extremal rays and the construction of a suitable “minimal model” for a given variety are unattainable by means of these ideas. It turns out that one has to allow some kind of singular varieties for that purpose, but Mori’s techniques do not work for singular varieties. Therefore, other methods are required for tackling the minimal model program, and those are the subject of study in the concluding chapter 7.

This chapter is entitled “Cohomological methods”, because the material presented here is mainly based on the whole machinery of cohomological vanishing theorems. Being much less geometrical than the foregoing chapters, this final chapter is by far the most involved and difficult part of the book. The author presents an approach (initiated by Y. Kawamata) which culminates in proving the cone and contraction theorems, in a unified way, as well as in doing the first steps toward the general minimal model program. This includes sections on canonical models, their singularities, singularities of pairs, the Kawamata-Viehweg vanishing theorem, the base-point-free theorem, the rationality theorem, and the length of extremal rays. Methodologically, the author uses the so-called “logarithmic” framework, which makes the entire treatment look more natural and conceivable.

Each chapter comes with a set of exercises, many of which point to additional, further-reaching theorems, and the more difficult ones of them are equipped with brief hints for solution. A particular feature of the text is provided by the numerous instructive examples and comments illustrating the undoubtedly demanding material.

The author has tried to keep the text as self-contained as possible, which has been a fairly difficult task, and all in all he has succeeded in writing the first introductury textbook on higher-dimensional varieties which should be accessible to any motivated reader familiar with the basics of modern algebraic geometry, about at the level of R. Hartshorne’s celebrated standard text. In this regard, the book provides an excellent source for graduate students in algebraic geometry, seasoned mathematicians in general, and theoretical physicists using algebro-geometric methods. The exposition of the material is characterized by a very lucid, refined, and user-friendly style of writing.

Without any doubt, this book fills a gap in the existing textbook literature on algebraic geometry.

Reviewer: Werner Kleinert (Berlin)

### MathOverflow Questions:

Mori’s cone theoremReference for Mori program

References about pseudoeffective cone

### MSC:

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |

14Jxx | Surfaces and higher-dimensional varieties |

14E30 | Minimal model program (Mori theory, extremal rays) |

14C20 | Divisors, linear systems, invertible sheaves |