##
**Introduction to the Mori program.**
*(English)*
Zbl 0988.14007

Universitext. New York, NY: Springer (ISBN 0-387-98465-8/hbk). xxiii, 478 p. (2002).

The so-called Mori program in projective algebraic geometry emerged in the last two decades as an effective approach toward the biregular and/or birational classification theory of higher-dimensional algebraic varieties. In our current understanding, the Mori program incorporates a broad spectrum of strategies, methods, techniques and results in higher-dimensional algebraic geometry, which grew out of the fundamental ideas and philosophies developed by S. Iitaka, S. Mori, Y. Kawamata, M. Reid, J. Kollár, V. Sarkisov, V. Shokurov, and many others. The entire theory is largely still in rapid progress, although an amazing quantity of knowledge has accumulated in the past twenty years, and we are now at a stage where understanding the recent developments requires mastering an immense multitude of subtle and highly involved concepts. In fact, up to now this topic of higher-dimensional classification theory in algebraic geometry has remained much more confidential and, unfortunately, too discouraging to young researchers than it should have.

This might be also due to the particular circumstance that, until very recently, no introductory and really comprehensible textbook on the subject had been written. Now, shortly after the appearance of O. Debarre’s textbook “Higher-dimensional algebraic geometry” (Universitext, 2000; Zbl 0978.14001), the book under review provides another available introduction to the classification of higher-dimensional algebraic varieties, this time with the Mori theory as the central topic. – The present book started as a collection of the author’s personal notes that he made to help himself to understand what the Mori program is all about. These notes grew into the present book form, after steady improvements due to various teaching experiences and discussions with leading experts in the field, and the outcome is the first systematic comprehensible, and really introductory textbook on Mori theory.

As the author emphasizes in the preface to this textbook, his main goal was to present the Mori program in as easy and digested a form as possible, with the degree of motivational background that many have wished for but has not hitherto been sufficiently revealed. In this vein, this book consists of what should roughly be the basics for a solid understanding of the global picture of the Mori program, leaving some of the highly special and technically involved topics to the original research literature.

The text consists in fourteen chapters, each of which is subdivided into several sections. The preceding introductory section is entitled “Introduction: The tale of the Mori program” and provides a first historical and strategical overview of what is to follow, and of what will be missing in the book. Then the fourteen chapters are organized as follows.

Chapter 1 (“Birational geometry of surfaces”): This chapter presents the main features of the classical birational geometry of surfaces in the framework of the Mori program, including Castelnuovo’s contractibility criterion, surfaces with non-nef canonical bundles, Mori fiber spaces in dimension two, minimal models in dimension two, canonical models in dimension two, and the Enriques classification of surfaces (in characteristic zero).

Chapter 2 (“Logarithmic category”): This chapter gives an introductory account of the so-called logarithmic category introduced by S. Iitaka. The main objects of study are here the “log-pairs”, \((X,D)\), associated with open subvarieties \(U=X\setminus D\) of compact varieties \(X\), their categorical properties, and Iitaka’s philosophy of studying the according “log-birational geometry” of complex algebraic varieties. This principle is then illustrated by discussing the log-birational geometry of surfaces in some more detail.

Chapter 3 (“Overview of the Mori program”): This chapter outlines the essentials of the Mori program in dimension three or higher. This is done by extending the analogy from dimension two to higher dimensions as much as possible, but also by pointing out those remarkable features (such as flips) that occur only in higher dimensions. The discussion includes the minimal model program in higher dimensions, the basic properties of Mori fiber spaces and models, and a few comments on recent variations of the Mori program such as the logarithmic Mori program and the relative Mori program.

Chapter 4 (“Singularities”): This chapter provides a unified and general account of those special singularities that are crucial in the study of the (logarithmic) Mori program in higher dimensions, namely terminal singularities, canonical singularities, log-terminal singularities, and log-canonical singularities. The unifying character of this analysis is achieved by using the concept of discrepancy of a logarithmic pair \((X,D)\). Moreover, in this chapter the author introduces the fundamental notion of the canonical cover of a variety \(X\), due to M. Reid and J. Wahl, and concludes the discussion of singularities by giving the analytical classification in dimension two of the various types of them. The latter part is particularly comprehensive.

Chapter 5 (“Vanishing theorems”): This chapter is devoted to the presentation of several vanishing theorems of certain cohomology groups that form the technical backbone of the approach to the Mori program in higher dimensions. The main theorems discussed here are the Kodaira vanishing theorem and the Kawamata-Viehweg vanishing theorem.

Chapter 6 (“Base point freeness of adjoint linear systems”): In this chapter, the author discusses the “base point freeness” of certain linear systems in the framework of the logarithmic category described in chapter 2. The arguments used here touch upon several important methods and results, among which are Fujita’s conjecture on base-point-free ample divisors, the method of Kawamata-Reid-Shokurov to solve Fujita’s conjecture in dimension three, and Shokurov’s non-vanishing theorem.

Chapter 7 (“The cone theorem”): This chapter deals with the convex cone \(NE(X)\) generated by the effective 1-cycles on a variety \(X\) and with its closure \(\overline {NE}(X)\) in the Néron-Severi group \(N_1(X)\). The author presents a proof of the famous cone theorem following the philosophy of Kawamata-Reid-Shokurov-Kollár. The basic ingredients are here the fundamental “theorem of rationality and boundedness of the denominator” and its logarithmic version.

Chapter 8 (“The contraction theorem”): This chapter turns to the key geometric operations within the minimal program, namely the so-called extremal contractions and contractions of extremal rays in \(\overline {NE}(X)\). The according structure theorems are enhanced by the discussion of the basic properties of the contracting morphism of an extremal ray (according to whether the latter one is of divisorial type, flipping type, or fibering type) as well as by some instructive examples in the case of threefolds.

Chapter 9 (“Flip”): The author explains here the main features of the so-called flips, the key operation in the center of the minimal model program, surrounding the two major conjectures concerning this concept (existence of flips and termination of flips). This chapter, although being comparatively short and sketchy, illustrates the link between the flip conjectures to other conjectures and gives some outlook to the affirmative answers in dimension three. As for further details and recent developments, the reader is referred to the original research papers.

Chapter 10 (“The cone theorem revisited”): The purpose of this chapter is to present Mori’s original idea to prove the cone theorem (in the smooth case) by means of his ingenious method of “bend and break” to produce rational curves of suitably bounded degree with respect to a given divisor.

Chapter 11 (“Logarithmic Mori program”): After a review of methods and facts developed so far for dealing with the Mori program in the framework of the logarithmic category, the author discusses some of the crucial subtleties that inevitably arise when going from the usual category (of varieties) over to the logarithmic category. This incorporates various conjectures and open problems, the statements of which the author analyzes thoroughly.

Chapter 12 (“Birational relation among minimal models”): Within the strategic scheme of the Mori program, a crucial problem is to find the birational relation among the minimal models of varieties. The purpose of this chapter is to investigate these birational relations along the “theory of flops” and the theorem on the “chamber structure” of ample cones of minimal models. The author concludes this chapter by discussing the question of how to count the number of minimal models in a fixed birational equivalence class.

Chapter 13 (“Birational relation among Mori fiber spaces”): This chapter focuses on the most important subject in the study of the Mori fiber spaces, the so-called Sarkisov program. This research strategy in higher dimensional algebraic geometry, developed by V. G. Sarkisov (1989), M. Reid (1991) and A. Corti (1995), aims at an algorithm for factoring a given birational map between Mori fiber spaces into a sequence of certain elementary transformations called “links”. Being a higher-dimensional analogue of the classical Castelnuovo-Noether theorem for surfaces, the Sarkisov program becomes clearer in the framework of the logarithmic category, and that is exactly what the author discusses in this chapter, together with some recent applications.

Chapter 14 (“Birational geometry of toric varieties”): While chapter 13 of this book can be regarded as the highlight of the entire text, and as the most advanced part of it, the concluding chapter 14 is intended as an exhilarating epilogue. The author illustrates how all the basic ingredients of the Mori program work in the concrete geometry of toric varieties, thereby presenting his personal reflections on M. Reid’s beautiful paper “Decomposition of toric morphisms” [in: Arithmetic and Geometry, Vol. II: Geometry, Prog. Math. 36, 395-418 (1983; Zbl 0571.14020)]. This includes discussions on the cone theorem and the contraction theorem for toric varieties, toric extremal contractions and flips, toric canonical and log-canonical divisors, the toric minimal program, and the toric Sarkisov program.

Altogether, this textbook is a highly welcome novelty in the literature on algebraic geometry. Written in a very lucid, rigorous and comprehensive style, this book helps to make the advanced topic of Mori theory a lot more attractive and accessible to non-specialists in the field, graduate students of algebraic geometry, and courageous physicists. The book is fairly self-contained in the sense that it requires “only” the basic facts from modern algebraic geometry as covered, for example, in R. Hartshorne’s standard text “Algebraic geometry” (1977; Zbl 0367.14001). The user-friendliness of K. Matsuki’s textbook on this important, advanced and utmost active area of research is bolstered up by detailed references to the literature for each single chapter, a complete list of notations used in the course of the text, and an extensive bibliography.

This might be also due to the particular circumstance that, until very recently, no introductory and really comprehensible textbook on the subject had been written. Now, shortly after the appearance of O. Debarre’s textbook “Higher-dimensional algebraic geometry” (Universitext, 2000; Zbl 0978.14001), the book under review provides another available introduction to the classification of higher-dimensional algebraic varieties, this time with the Mori theory as the central topic. – The present book started as a collection of the author’s personal notes that he made to help himself to understand what the Mori program is all about. These notes grew into the present book form, after steady improvements due to various teaching experiences and discussions with leading experts in the field, and the outcome is the first systematic comprehensible, and really introductory textbook on Mori theory.

As the author emphasizes in the preface to this textbook, his main goal was to present the Mori program in as easy and digested a form as possible, with the degree of motivational background that many have wished for but has not hitherto been sufficiently revealed. In this vein, this book consists of what should roughly be the basics for a solid understanding of the global picture of the Mori program, leaving some of the highly special and technically involved topics to the original research literature.

The text consists in fourteen chapters, each of which is subdivided into several sections. The preceding introductory section is entitled “Introduction: The tale of the Mori program” and provides a first historical and strategical overview of what is to follow, and of what will be missing in the book. Then the fourteen chapters are organized as follows.

Chapter 1 (“Birational geometry of surfaces”): This chapter presents the main features of the classical birational geometry of surfaces in the framework of the Mori program, including Castelnuovo’s contractibility criterion, surfaces with non-nef canonical bundles, Mori fiber spaces in dimension two, minimal models in dimension two, canonical models in dimension two, and the Enriques classification of surfaces (in characteristic zero).

Chapter 2 (“Logarithmic category”): This chapter gives an introductory account of the so-called logarithmic category introduced by S. Iitaka. The main objects of study are here the “log-pairs”, \((X,D)\), associated with open subvarieties \(U=X\setminus D\) of compact varieties \(X\), their categorical properties, and Iitaka’s philosophy of studying the according “log-birational geometry” of complex algebraic varieties. This principle is then illustrated by discussing the log-birational geometry of surfaces in some more detail.

Chapter 3 (“Overview of the Mori program”): This chapter outlines the essentials of the Mori program in dimension three or higher. This is done by extending the analogy from dimension two to higher dimensions as much as possible, but also by pointing out those remarkable features (such as flips) that occur only in higher dimensions. The discussion includes the minimal model program in higher dimensions, the basic properties of Mori fiber spaces and models, and a few comments on recent variations of the Mori program such as the logarithmic Mori program and the relative Mori program.

Chapter 4 (“Singularities”): This chapter provides a unified and general account of those special singularities that are crucial in the study of the (logarithmic) Mori program in higher dimensions, namely terminal singularities, canonical singularities, log-terminal singularities, and log-canonical singularities. The unifying character of this analysis is achieved by using the concept of discrepancy of a logarithmic pair \((X,D)\). Moreover, in this chapter the author introduces the fundamental notion of the canonical cover of a variety \(X\), due to M. Reid and J. Wahl, and concludes the discussion of singularities by giving the analytical classification in dimension two of the various types of them. The latter part is particularly comprehensive.

Chapter 5 (“Vanishing theorems”): This chapter is devoted to the presentation of several vanishing theorems of certain cohomology groups that form the technical backbone of the approach to the Mori program in higher dimensions. The main theorems discussed here are the Kodaira vanishing theorem and the Kawamata-Viehweg vanishing theorem.

Chapter 6 (“Base point freeness of adjoint linear systems”): In this chapter, the author discusses the “base point freeness” of certain linear systems in the framework of the logarithmic category described in chapter 2. The arguments used here touch upon several important methods and results, among which are Fujita’s conjecture on base-point-free ample divisors, the method of Kawamata-Reid-Shokurov to solve Fujita’s conjecture in dimension three, and Shokurov’s non-vanishing theorem.

Chapter 7 (“The cone theorem”): This chapter deals with the convex cone \(NE(X)\) generated by the effective 1-cycles on a variety \(X\) and with its closure \(\overline {NE}(X)\) in the Néron-Severi group \(N_1(X)\). The author presents a proof of the famous cone theorem following the philosophy of Kawamata-Reid-Shokurov-Kollár. The basic ingredients are here the fundamental “theorem of rationality and boundedness of the denominator” and its logarithmic version.

Chapter 8 (“The contraction theorem”): This chapter turns to the key geometric operations within the minimal program, namely the so-called extremal contractions and contractions of extremal rays in \(\overline {NE}(X)\). The according structure theorems are enhanced by the discussion of the basic properties of the contracting morphism of an extremal ray (according to whether the latter one is of divisorial type, flipping type, or fibering type) as well as by some instructive examples in the case of threefolds.

Chapter 9 (“Flip”): The author explains here the main features of the so-called flips, the key operation in the center of the minimal model program, surrounding the two major conjectures concerning this concept (existence of flips and termination of flips). This chapter, although being comparatively short and sketchy, illustrates the link between the flip conjectures to other conjectures and gives some outlook to the affirmative answers in dimension three. As for further details and recent developments, the reader is referred to the original research papers.

Chapter 10 (“The cone theorem revisited”): The purpose of this chapter is to present Mori’s original idea to prove the cone theorem (in the smooth case) by means of his ingenious method of “bend and break” to produce rational curves of suitably bounded degree with respect to a given divisor.

Chapter 11 (“Logarithmic Mori program”): After a review of methods and facts developed so far for dealing with the Mori program in the framework of the logarithmic category, the author discusses some of the crucial subtleties that inevitably arise when going from the usual category (of varieties) over to the logarithmic category. This incorporates various conjectures and open problems, the statements of which the author analyzes thoroughly.

Chapter 12 (“Birational relation among minimal models”): Within the strategic scheme of the Mori program, a crucial problem is to find the birational relation among the minimal models of varieties. The purpose of this chapter is to investigate these birational relations along the “theory of flops” and the theorem on the “chamber structure” of ample cones of minimal models. The author concludes this chapter by discussing the question of how to count the number of minimal models in a fixed birational equivalence class.

Chapter 13 (“Birational relation among Mori fiber spaces”): This chapter focuses on the most important subject in the study of the Mori fiber spaces, the so-called Sarkisov program. This research strategy in higher dimensional algebraic geometry, developed by V. G. Sarkisov (1989), M. Reid (1991) and A. Corti (1995), aims at an algorithm for factoring a given birational map between Mori fiber spaces into a sequence of certain elementary transformations called “links”. Being a higher-dimensional analogue of the classical Castelnuovo-Noether theorem for surfaces, the Sarkisov program becomes clearer in the framework of the logarithmic category, and that is exactly what the author discusses in this chapter, together with some recent applications.

Chapter 14 (“Birational geometry of toric varieties”): While chapter 13 of this book can be regarded as the highlight of the entire text, and as the most advanced part of it, the concluding chapter 14 is intended as an exhilarating epilogue. The author illustrates how all the basic ingredients of the Mori program work in the concrete geometry of toric varieties, thereby presenting his personal reflections on M. Reid’s beautiful paper “Decomposition of toric morphisms” [in: Arithmetic and Geometry, Vol. II: Geometry, Prog. Math. 36, 395-418 (1983; Zbl 0571.14020)]. This includes discussions on the cone theorem and the contraction theorem for toric varieties, toric extremal contractions and flips, toric canonical and log-canonical divisors, the toric minimal program, and the toric Sarkisov program.

Altogether, this textbook is a highly welcome novelty in the literature on algebraic geometry. Written in a very lucid, rigorous and comprehensive style, this book helps to make the advanced topic of Mori theory a lot more attractive and accessible to non-specialists in the field, graduate students of algebraic geometry, and courageous physicists. The book is fairly self-contained in the sense that it requires “only” the basic facts from modern algebraic geometry as covered, for example, in R. Hartshorne’s standard text “Algebraic geometry” (1977; Zbl 0367.14001). The user-friendliness of K. Matsuki’s textbook on this important, advanced and utmost active area of research is bolstered up by detailed references to the literature for each single chapter, a complete list of notations used in the course of the text, and an extensive bibliography.

Reviewer: Werner Kleinert (Berlin)

### MathOverflow Questions:

Example of a projective variety over a field of characteristic zero which is uniruled but not ruledReference for Mori program

### MSC:

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

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14J10 | Families, moduli, classification: algebraic theory |

14J30 | \(3\)-folds |