1.1. From the Introduction.

These notes originated at a seminar that was held during July and August of 1987 at Salt Lake City. The original aim of the seminar was to get an overview of the following three topics:

1. Recent advances in the classification program of three (and higher) dimensional algebraic varieties.

2. Existence of rational curves and other special subvarieties.

3. Existence and nature of special metrics on varieties.

We also hoped to then go further and study the relationships between these three approaches. Time however proved to be insufficient to complete even the limited programme.

The first part of the program was considered in detail. In that part, the central theme is the investigation of varieties on which the canonical class is not numerically effective. For smooth threefolds this was done in [S Mori, Threefolds whose canonical bundles are not numerically effective (1982)] and later extended considerably. The original approach of [S Mori, Threefolds whose canonical bundles are not numerically effective (1982)] is geometrically very clear, therefore it is given in detail. Subsequent generalisations were also considered at length.

Considerable attention was paid also to the study of special curves on hypersurfaces and some related examples. There seems to be a lot of experimental evidence to indicate that there is a very close relationship between the Kodair a dimension of a threefold (a property of a threefold from classification theory) and the existence of rational curves. These problems are very interesting but they also seem quite hard. Our contribution in this direction is mostly limited to presenting some examples and conjectures.

In the second direction, one of the questions we were interested in was that of understanding rational curves on quintic hypersurfaces in $\mathbb{P}^{4}$. Later this was scaled down to understand lines on quintic hypersurfaces in $\mathbb{P}^{4}$, but even this seems a hard problem. We began to understand it more completely only after the seminar had ended.

Very little time was left to consider the third direction. W e were fortunate to have a series of lectures, but we could not pursue this interesting and important direction in any detail.

The style of the seminars was very informal. We tried to keep them discussion-and-problem oriented. Notes were taken by H Clemens who typed them up by the next day. These notes constituted the first version of the present text. During the seminar and afterwards, these notes were considerably revised, cut, expanded and edited. During this process we tried to keep the original informality of the talks alive.

1.2. Review by: Peter Nielsen.
Mathematical Reviews MR1004926 (90j:14046).

A considerable time passed between the completion of the Enriques-Zariski theory of minimal models of surfaces and the formulation of a conjectural analogue in higher dimensions. Mori's theory of extremal rays gave a framework, inter alia, for a theory of minimal models of higher-dimensional varieties. A series of interlocking theorems and conjectures form the so-called minimal model program, the strongest evidence for which is Mori's proof of the flip theorem, thereby completing the program for 3-folds. Significant contributions were made by Benveniste, Kawamata, Kollár, Reid and Shokurov. The first two-thirds of this book are devoted to the main results of this still developing theory. In the first two lectures a very clear exposition is given of Mori's theorem concerning the existence of rational curves on 3-folds with non-nef canonical bundle. In particular the role of the reduction to characteristic p in the deformation theory is clearly explained. The cone of curves is introduced early, providing the motivation for Mori's program. Numerous examples and diagrams are used to illustrate the theory of the Kleiman-Mori cone. Complete proofs are given in subsequent chapters of key results such as Shokurov's nonvanishing theorem and the rationality theorem, and a new proof of the cone theorem is also given. The role of the various singularities which arise in the theory of smooth 3-folds is outlined with particular emphasis placed on the notion of discrepancy. The sections on flips and extremal neighbourhoods provide a rather detailed introduction to Mori's proof of the flip theorem for 3-folds. This portion of the book concludes with a characterisation of 3-dimensional terminal Gorenstein singularities and their relation to flops.

Several subsequent lectures are devoted to results of Carlson and Toledo on Kähler structures on Riemannian locally symmetric spaces. These involve results of Sampson on harmonic maps which are closely related to certain ideas in Hodge theory. Finally, subvarieties of hypersurfaces, in particular rational curves on quintics in $\mathbb{P}^{4}$, are studied from the point of view of deformation theory. These results are due to Clemens. The difficult nature of the subject of the title is well served by the clear and cogent exposition, which makes the book accessible to the general algebraic geometer.

2.1. From the Publisher.

The aim of this book is to study various geometric properties and algebraic invariants of smooth projective varieties with infinite fundamental groups. This approach allows for much interplay between methods of algebraic geometry, complex analysis, the theory of harmonic maps, and topology. Making systematic use of Shafarevich maps, a concept previously introduced by the author, this work isolates those varieties where the fundamental group influences global properties of the canonical class.

The book is primarily geared toward researchers and graduate students in algebraic geometry who are interested in the structure and classification theory of algebraic varieties. There are, however, presentations of many other applications involving other topics as well - such as Abelian varieties, theta functions, and automorphic forms on bounded domains. The methods are drawn from diverse sources, including Atiyah's $L^{2}$-index theorem, Gromov's theory of Poincaré series, and recent generalisations of Kodaira's vanishing theorem.

2.2. From the Preface.

The theory of automorphic forms goes back to L Euler, but the main development of its function-theoretic aspects started with H Poincaré. The aim of the theory is to study the function theory and geometry of complex manifolds whose universal covering space is well understood, for instance $\mathbb{C}^{n}$ or the unit ball in $\mathbb{C}^{n}$.

Around 1970, I R Shafarevich conjectured that the universal cover of a smooth projective algebraic variety is holomorphically convex. This question is still completely open. The Shafarevich conjecture suggests that the most general version of the theory of automorphic forms is the study of projective varieties whose universal covering space is Stein. The function theory of Stein spaces is quite well understood; therefore, one hopes that there are many interesting connections between the meromorphic function theory of a variety and the holomorphic function theory of its universal cover.

The aim of these notes is to connect a weakened version of the Shafarevich conjecture with the theory of automorphic forms, thereby providing general methods to study varieties whose fundamental group is large. Instead of attempting to prove the Shafarevich conjecture, I try to concentrate on some ideas that connect the fundamental group directly to algebro-geometric properties of a variety. Some of the results are rather promising but much more remains to be done.

These notes are a considerably expanded version of the Milton Brockett Porter lectures I delivered at Rice University in March 1993. I would like to thank the Mathematics Department of Rice University for its warm hospitality and for giving me the first occasion to test my theories on a large audience.

I had the opportunity to give further series of lectures about these topics at the University of Bayreuth, the University of Utah, and at the Regional Geometry Institute at Park City, Utah. I am grateful to my audiences for pointing out several mistakes and suggesting improvements in the presentation.

2.3. Review by: Kang Kuo.
Mathematical Reviews MR1341589 (96i:14016).

Let X be a smooth complex projective variety and $\pi_{1}(X)$ be the fundamental group of $X$. The main interest of the author is to see how the presence of an infinite $\pi_{1}(X)$ (of a topological nature) influences other algebro-geometric properties of $X$. As we know, the basic algebro-geometric invariants are the Kodaira dimension $\kappa(X)$ and the pluricanonical maps. Following from general conjectures about the minimal model program of algebraic varieties, the author suggests that the following should be true: Conjecture (Kollár): If $X$ has generically large $\pi_{1}(X)$ (the precise definition will be given below), then $\kappa(X) ≥ 0$. However, the minimal model conjecture is still unsolved.
...
In Part I of this book the author makes some very interesting connections between the classification of algebraic varieties and algebraic varieties with infinite fundamental groups.
...
In Part II various aspects of classical theory of automorphic forms are treated by the author.
...
In Part III the Kodaira vanishing theorem and its generalisations are discussed by the author. These theorems provide the main technical tool in the algebro-geometric treatment for the Shafarevich map in this book; the author gives a uniform treatment of them.
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In Part IV Kollár presents several nonvanishing theorems, only assuming that $\pi_{1}(X)$ is generically large.
...
In Part V, as an application of his nonvanishing theorem in Part IV, the author makes a nice characterisation of varieties which are birational to abelian varieties.
...
At the end of the book the author puts forth several very interesting conjectures. They are definitely new directions for the further study of projective varieties with nontrivial fundamental groups.

3.1. From the Publisher.

The aim of this book is to provide an introduction to the structure theory of higher dimensional algebraic varieties by studying the geometry of curves, especially rational curves, on varieties. The main applications are in the study of Fano varieties and of related varieties with lots of rational curves on them. This Ergebnisse volume provides the first systematic introduction to this field of study. The book contains a large number of examples and exercises which serve to illustrate the range of the methods and also lead to many open questions of current research.

3.2. From the Introduction.

The aim of this monograph is to give a comprehensive introduction to some of the recent progress in higher dimensional algebraic geometry. One of the most profound developments in the last two decades consists of the evolution of a structure theory for higher dimensional varieties. Its roots can be traced back to the classical theory of minimal models of surfaces and to the works of Fano in the thirties.

The works of Iskovskikh on Fano threefolds, Reid on canonical models and Mori on the minimal model program provided the major impetus for the recent resurgence of this field.

Since 1982 the theory has developed in two distinct directions. One is the cohomological approach of Kawamata, Reid and Shokurov whose main achievement is the proof of the Cone Theorem and the Contraction Theorem for mildly singular varieties. This direction is written up in the first Utah Summer Seminar notes.

The other direction is harder to classify. It concentrates on trying to find a more geometric approach to the structure of varieties by studying the geometry of curves, especially rational curves, on them. This approach relies on classical geometric ideas and strives to understand the intrinsic geometry of varieties. The most significant single achievement of this direction is the proof of the existence of flips in dimension three. Unfortunately, the proof is rather long and complicated, and it is not typical of the ideas and techniques in this field of study.

Therefore I present the circle of ideas which originate around the theme "rational curves on algebraic varieties", without touching on the theory of minimal models. The minimal model program, while not directly considered, always lurks in the background as motivation for the results and source of the techniques.

The core of these techniques is the theory of deformation of curves on varieties. This in turn requires the understanding of universal families of curves on varieties and of morphisms of curves to varieties. The necessary foundational results about Hilbert and Chow schemes have not been treated in the literature in the generality that is needed for our purposes. Therefore Chap. I contains a detailed introduction to the theory of Hilbert schemes and Chow varieties in the relative setting.

Technically this is the most demanding part of the book. Later chapters use only the basic definitions and theorems, but very few of the techniques are employed. Thus it may be easier to start with Chap. II, and refer back to Chap. I only as necessary.

The general results about Hilbert schemes and Chow varieties are considered for curves in Chap. II. There are several simplifications which greatly enhance their usefulness. The Cone of Curves, one of the fundamental objects of higher dimensional geometry, is introduced in (II.4). The previous results are employed in (II.5) to develop the bend-and-break method. The basic principle of the method asserts that if a curve in a variety has lots of deformations then there are rational curves around. In applications it is important to get detailed information about the resulting rational curves; this accounts for the technicalities involved.

The geometric methods allow us to view several aspects of the theory of surfaces in a unified way. The arguments are rather short and clear; they are presented in Chap. III. Hopefully this can serve as an introduction to the higher dimensional cases where the technical details become rather formidable.

The most important part of this book is Chap. IV. Its aim is to present a theory of uniruled varieties, i.e. those varieties which are covered by rational curves. Our circle of ideas applies most successfully to these problems. For this class of varieties, Mori's program does not produce a minimal model and is not expected to give a definitive structure theory. The techniques of this book are birational in nature. Thus we get a good understanding of this class without reference to minimal models.

The results of the previous chapters are applied to the study of Fano varieties in Chap. V. The theory of Fano varieties developed considerably in the last five years, but many good open problems remain.

The last chapter covers two topics that did not fit anywhere else but are needed in order to get a complete picture.

The book contains a large number of exercises. Their role is to give further applications and examples. They are an integral part of the book.

There are two important recent approaches to the study of rational curves on varieties which are not covered in this book. The first is the study of rational curves on varieties with trivial canonical class. For such varieties one expects relatively few, but frequently infinitely many, rational curves. The study of this situation is intimately connected with the general theory of mirror symmetry. The second is the development of a quantum cohomology theory of varieties with some very deep applications to enumerative questions involving rational curves.

3.3. Review by: Yuri G Prokhorov.
Mathematical Reviews MR1440180 (98c:14001).

In the classification theory of algebraic varieties families of rational curves appear to be significant objects to study. Now, from the modern minimal model program point of view, it is clear that certain classes of algebraic varieties can be characterised by properties of these families. Many results in this direction were obtained in works of Mori, Miyaoka, the author and others. This book contains a good exposition of the theory of rational curves.

The book is divided into five chapters. The first two chapters are preliminary. Chapter 1 contains a detailed introduction to the theory of Hilbert schemes and Chow varieties. In the next chapter the author develops the necessary technique from the deformation theory of curves. Some applications (for example, some vanishing theorems for $H^{1}$ in positive characteristic) are given.

Chapter 3 contains Mori's cone theorem and application of the above results to the birational classification theory of surfaces. The geometric methods used are very short and clear.

Chapter 4 of the book is devoted to the study of uniruled varieties, i.e. those varieties which are covered by rational curves. The main topics of this chapter are the numerical criterion of uniruledness of Miyaoka and Mori and results about rationally connected varieties.

In Chapter 5 the previous results are applied to the study of Fano varieties. In particular, it contains the theorem about boundedness of Fano varieties. Another important topic of this chapter is the author's new approach to proofs of nonrationality of Fano varieties by using positive characteristic trick.

The book is a very good introduction to the theory of rational curves. It will be very useful to a wide audience.

3.4. Review by: Miles Reid.
Bulletin of the American Mathematical Society 38 (1) (2000), 109-115.

Two papers of Mori in 1979 and 1982 used deformations of curves to give magnificent new results in the classification of varieties. Kollár's book sets itself the task of working out these arguments and their subsequent developments at a reasonable level of detail and technicality. The book is an irreplaceable source of information for many recent topics in algebraic geometry. It contains in particular a reworking of Mori's treatment of the existence of rational curves on Fano varieties, of Mori's fundamental Theorem on the Cone, and an extended treatment of the notion of rational connectedness that has developed over the last 15 years into the modern replacement for the old question of rationality.

Some sections of this book certainly take perseverance on the part of the reader before yielding up their delights; the first two chapters in particular are certainly harder than anything else in the book, and the reader should skim through these briefly and lazily (thus emulating the experts), but work seriously at some of the exercises later in the book. After some philosophy and simple-minded introduction, this review works backwards through the book, starting with the new ideas and applications, and finishing with some remarks to encourage the reader through the more technical early chapters.
...

Chapter I and Section II.1-3 of Kollár's book confront head-on in heroic style many difficult technical issues for which no easy treatment should be expected. As Kollár says, it seems to be pretty well established that no treatment of deformation theory can be technically adequate while remaining comprehensible. Most algebraic geometers and singularity theorists know a little bit about deformation theory from experience of a number of examples, but this knowledge is fundamentally inadequate for many purposes. Kollár makes a brilliant job of bringing the most important ones out into the open. Many of us will sleep or wave our arms more comfortably in our beds, respectively seminars, for knowing that Kollár has written up this technical material.

4.1. From the Publisher.

One of the major discoveries of the last two decades of the twentieth century in algebraic geometry is the realisation that the theory of minimal models of surfaces can be generalised to higher dimensional varieties. This generalisation, called the minimal model program or Mori's program, has developed into a powerful tool with applications to diverse questions in algebraic geometry and beyond. This book provides the a comprehensive introduction to the circle of ideas developed around the program, the prerequisites being only a basic knowledge of algebraic geometry. It will be of great interest to graduate students and researchers working in algebraic geometry and related fields.

4.2. From the Preface.

One of the major discoveries of the last two decades in algebraic geometry is the realisation that the theory of minimal models of surfaces can be generalised to higher dimensional varieties. This generalisation is called the minimal model program or Mori's program. While originally the program was conceived with the sole aim of constructing higher dimensional analogues of minimal models of surfaces, by now it has developed into a powerful tool with applications to diverse questions in algebraic geometry and beyond.

So far the program is complete only in dimension 3, but large parts are known to work in all dimensions.

The aim of this book is to introduce the reader to the circle of ideas developed around the minimal model program, relying only on knowledge of basic algebraic geometry.

In order to achieve this goal, considerable effort was devoted to make the book as self-contained as possible. We managed to simplify many of the proofs, but in some cases a compromise seemed a better alternative. There are quite a few cases where a theorem which is local in nature is much easier to prove for projective varieties. For these, we state the general theorem and then prove the projective version, giving references for the general cases. Most of the applications of the minimal model program ultimately concern projective varieties, and for these the proofs in this book are complete.

4.3. From the Introduction.

From the beginnings of algebraic geometry it has been understood that birationally equivalent varieties have many properties in common. Thus it is natural to attempt to find in each birational equivalence class a variety which is simplest in some sense, and then study these varieties in detail.

Each irreducible curve is birational to a unique smooth projective curve, thus the investigation of smooth projective curves is equivalent to the study of all curves up to birational equivalence.

For surfaces the situation is more complicated. Each irreducible surface is birational to infinitely many smooth projective surfaces. The theory of minimal models of surfaces, developed by the Italian algebraic geometers at the beginning of the twentieth century, aims to choose a unique smooth projective surface from each birational equivalence class. The recipe is quite simple. If a smooth projective surface contains a smooth rational curve with self-intersection 1, then it can be contracted to a point and we obtain another smooth projective surface. Repeating this procedure as many times as possible, we usually obtain a unique 'minimal model'. In a few cases we obtain a model that is not unique, but these cases can be described very explicitly.

A search for a higher dimensional analogue of this method started quite late. One reason is that some examples indicated that a similar approach fails in higher dimensions.

The works of Reid and Mori in the early 1980s raised the possibility that a higher dimensional theory of minimal models may be possible if we allow not just smooth varieties but also varieties with certain mild singularities. This approach is called the Minimal Model Program or Mori's Program. After many contributions by Benveniste, Kawamata, Kollár, Reid, Shokurov, Tsunoda, Viehweg and others, the program was completed in dimension three by Mori in 1988.

Since then this program has grown into a method which can be applied successfully to many problems in algebraic geometry.

The aim of this book is to provide an introduction to the techniques and ideas of the minimal model program.

Chapter 1 gives an introduction to the whole program through a geometric approach. Most of these results are not used later, but they provide a useful conceptual foundation.

Chapter 2 is still introductory, discussing some aspects of singularities and the relevant generalisations of the Kodaira Vanishing Theorem.

The first major part of the program, the Cone Theorem, is proved in Chapter 3. These results work in all dimensions.

The rest of the book is essentially devoted to the study of 3-dimensional flips and flops. Flips and flops are new types of birational transformations which first appear in dimension 3. Most major differences between the theory of surfaces and 3-folds can be traced back to flips and flops.

Chapter 4 is devoted to the classification of certain surface singularities. These results are needed in further work on the 3-dimensional theory.

The singularities appearing in the course of the minimal model program are investigated in Chapter 5. The results are again rather complete in all dimensions.

Flops are studied in Chapter 6. Flops are easier to understand than flips, and, at least in dimension 3, their description is rather satisfactory.

Chapter 7 is devoted to 3-dimensional flips. The general theory is still too complicated and long to be included in a textbook, thus we restrict ourselves to the study of a special class, the so-called semi-stable flips. We have succeeded in simplifying the proofs in this case considerably. Semi-stable flips appear naturally in many contexts, and they are sufficient for several of the applications.

A more detailed description of the contents of each chapter is given at its beginning.

Sections 4.5 and 5.5 are each a side direction, rather than being part of the main line of arguments. In each case we felt that the available references do not adequately cover some results we need, and that our presentation may be of interest to the reader.

4.4. Review by: Mark Gross.
Mathematical Reviews MR1658959 (2000b:14018).

The book under review, written by two of the leaders in the field, is a comprehensive treatment of the minimal model program. The text strives to be self-contained and to give complete proofs. The book begins with a survey of the earlier techniques (the method of bend and break) used in the theory, which enables one to prove many of the important theorems in the non-singular case. In the second chapter, the many different flavours of singularities which have been introduced by minimal model theorists are defined. In the third chapter, the cone theorems are proved in complete generality. At this point, modulo flips, one is ready to run the minimal model program. The fourth and fifth chapters go into more detail on the types of singularities under discussion, with surface singularities and results about simultaneous resolutions treated in the fourth chapter and threefold singularities in the fifth chapter.

The sixth chapter deals with threefold flops. These are birational operations analogous to flips, and are associated to small contractions $f:X \rightarrow Y$ for which $K_{X}$ is relatively trivial. These operations get at the issue of non-uniqueness of minimal models of threefolds: two birational minimal models must differ by a sequence of flops. Existence of flops for threefolds with canonical singularities is proven, as well as the existence of $Q$-factorisations.

The last chapter deals with semistable flips, and proves their existence. This is a subclass of all flips, whose existence was proven before the general case, the latter being far too technical and lengthy for a textbook. Finally the chapter ends with a brief survey on other work involving threefolds and higher-dimensional birational geometry.

The beginning reader might find the going rather technical at times, and might want to start with the earlier text Higher-dimensional complex geometry by H Clemens, Kollár and Mori (1988), from which the first chapter of this text is drawn. Nevertheless, the text under review will prove invaluable for the more advanced student of the minimal model program, as well as researchers in the field.

4.5. Review by: Yujiro Kawamata.
Bulletin of the American Mathematical Society 38 (2) (2001), 267-272.

There has been a revolutionary change in the field of birational geometry in the last twenty plus years. This is based on the theory of extremal rays initiated by Mori and is central to the investigation of minimal models called the Minimal Model Program (MMP) or Mori program. But there has been no reasonably accessible textbook for this theory besides more professionally oriented surveys. In this sense, this book, written by two of the main players in this development, answers a demand for a long awaited introductory textbook for the beginners in this field. The exposition is sufficiently elementary, self-contained and comprehensive, and requires fewer prerequisites, so this book will become a standard reference. A caution is that the proof of the existence theorem of 3-dimensional flips is treated only in the easy semistable case. One has to refer to the original articles in order to study the general case.

5.1. From the Publisher.

The most basic algebraic varieties are the projective spaces, and rational varieties are their closest relatives. In many applications where algebraic varieties appear in mathematics and the sciences, we see rational ones emerging as the most interesting examples. The authors have given an elementary treatment of rationality questions using a mix of classical and modern methods. Arising from a summer school course taught by János Kollár, this book develops the modern theory of rational and nearly rational varieties at a level that will particularly suit graduate students. There are numerous examples and exercises, all of which are accompanied by fully worked out solutions, that will make this book ideal as the basis of a graduate course. It will act as a valuable reference for researchers whilst helping graduate students to reach the point where they can begin to tackle contemporary research problems.

5.2. From the Introduction.

The most basic algebraic varieties are the projective spaces, and rational varieties are their closest relatives. Rational varieties are those that are birationally equivalent to projective space. In many applications where algebraic varieties appear in mathematics, we see rational ones emerging as the most interesting examples. This happens in such diverse fields as the study of Lie groups and their representations, in the theory of Diophantine equations, and in computer-aided geometric design.

This book provides an introduction to the fascinating topic of rational, and "nearly rational." varieties. The subject has two very different aspects, and we treat them both. On the one hand, the internal geometry of rational and nearly rational varieties tends to be very rich. Their study is full of intricate constructions and surprising coincidences, many of which were thoroughly explored by the classical masters of the subject. On the other hand, to show that particular varieties are not rational can be a difficult problem: the classical literature is riddled with serious errors and gaps that require sophisticated general methods to repair. Indeed, only recently, with the advent of minimal model theory, have all the difficulties in classical approaches to proving nonrationality based on the study of linear systems and their singularities been ironed out.

While presenting some of the beautiful classical discoveries about the geometry of rational varieties, we pay careful attention to arithmetic issues. For example, we consider whether a variety defined over the rational numbers is rational over $\mathbb{Q}$, which is to say, whether there is a birational map to projective space given locally by polynomials with coefficients in $\mathbb{Q}$.

The hardest parts of the book focus on how to establish nonrationality of varieties, a difficult problem with many basic questions remaining open today. There are good general criteria, involving global differential forms, that can be used in many cases, but the situation becomes very difficult when these tests fail. For example, using simple numerical invariants called the plurigenera, it is easy to see that a smooth hypersurface in projective space whose degree exceeds its embedding dimension can not be rational. However, it is a very delicate problem to determine whether or not a lower degree hypersurface is rational.

Rationality of quadric and cubic surfaces was completely settled in the nineteenth century, but rationality for threefolds occupied the attention of algebraic geometers for most of the twentieth century. In the 1970s. Clemens and Griffith identified a new obstruction to rationality for a threefold inside its third topological (singular) cohomology group. This method of intermediate Jacobians provided the first proof that no smooth cubic threefold is rational. Because this approach fits better in a book about Hodge theory, we do not discuss it here. Instead, we prove that no smooth quartic threefold in projective four-space is rational, drawing on ideas from the minimal model program. Beyond this. very little is known: no one knows whether or not all smooth cubic fourfolds are rational, or indeed, whether there exists any nonrational smooth cubic hypersurface of any dimension greater than three.

On the other hand, in this book we do present a technique for proving non-rationality of "very general" hypersurfaces of any dimension greater than two whose degree is close to their dimension. Like other approaches to proving nonrationality, this technique uses differential forms; the novelty here is that the differential forms we use are defined on varieties of prime characteristic.

Our biggest omission is perhaps never to define precisely what we mean by a "nearly rational variety." Current research in birational algebraic geometry indicates that the most natural class of nearly rational varieties is formed by rationally connected varieties, introduced in Kollár et al. (1992). Although it is easy to state the definition, it is harder to appreciate why we claim that this is indeed the most natural class of nearly rational varieties to consider. Our aim in this book is more modest; we hope to inspire the reader to learn more about rationality questions. As a next step, we recommend the general introduction to rationally connected varieties given in Kollár (2001). Kollár (1996) contains a detailed treatment for the technically advanced.

5.3. Review by: Alexandr V Pukhlikov.
Mathematical Reviews MR2062787 (2005i:14063).

According to the Introduction, this book began with a series of lectures given by J. Kollár at the European Mathematical Society Summer School in Algebraic Geometry (Hungary, 1996). The notes were written up by K E Smith. Later some new material was added (evidently, Chapters 5 and 6, which cover the area of study of the third author, A Corti).

The book is an introduction to the theory of algebraic varieties, which have similar properties to rational varieties. In modern terminology, the main subject of the book is the class of rationally connected varieties. Typical examples of rationally connected varieties are given by the projective space $\mathbb{P}^{n}$ itself, by all rational varieties, also by Fano varieties and Fano fiber spaces over a rationally connected base.

Since the book is meant to be an introduction to the subject, the authors discuss some important results and examples on a more or less elementary level, thus avoiding more difficult questions. The book contains exercises with solutions to them. Let us describe briefly the contents of individual chapters.

In Chapter 1 examples of rational varieties are given (mainly the classical constructions). How non-rationality follows from the existence of global differential forms is explained.

In Chapter 2 cubic surfaces are discussed; the authors explain the classical results of Manin and Segre on cubic surfaces over non-closed fields.
...
Chapter 3, which is entitled "Rational surfaces", contains the Castelnuovo rationality criterion, Iskovskikh's classification of minimal surfaces over perfect fields and classification of del Pezzo surfaces over arbitrary fields. Section 5 of this chapter introduces the concept of a weighted projective space.

Chapter 4 is an elementary introduction to Kollár's method of proving non-rationality via reduction modulo p; Matsusaka's theorem, on which the method is based, is also discussed. Some explicit examples of non-rational Fano varieties are given. Prior to this book the material in Chapter 4 could be found only in papers which are not quite accessible to a beginner.

Chapters 5 and 6 present an attempt to give an introduction to one of the most powerful methods of modern birational geometry, that is, the method of maximal singularities.
...
Chapters 5 and 6 contain a lot of interesting material (proof of birational superrigidity of the quartics, the connectedness principle of Shokurov and Kollár, a theorem of Varchenko on singularities of curves, some quite nontrivial computations), however using these chapters as an introduction to the subject could hardly be recommended, since they could sometimes mislead the reader both in the sense of mathematical content and historical context.

6.1. From the Publisher.

Resolution of singularities is a powerful and frequently used tool in algebraic geometry. In this book, János Kollár provides a comprehensive treatment of the characteristic 0 case. He describes more than a dozen proofs for curves, many based on the original papers of Newton, Riemann, and Noether. Kollár goes back to the original sources and presents them in a modern context. He addresses three methods for surfaces, and gives a self-contained and entirely elementary proof of a strong and functorial resolution in all dimensions. Based on a series of lectures at Princeton University and written in an informal yet lucid style, this book is aimed at readers who are interested in both the historical roots of the modern methods and in a simple and transparent proof of this important theorem.

6.2. From the Introduction.

A complex algebraic variety $X$ is a subset of complex affine $n$-space $\mathbb{C}^{n}$ or of complex projective $n$-space $\mathbb{C}P^{n}$ defined by polynomial equations.

A point $x \in X$ is called a smooth point if, up to a complex analytic local coordinate change, $X$ looks like a linear subspace near $x$. Otherwise, a point is called singular. At singular points $X$ may have self-intersections, it may look like the vertex of a cone, or it can be much more complicated.

Roughly speaking, resolution of singularities asserts that an arbitrary singular variety $X$ can be parametrised by a smooth variety $X'$. That is, all the points of $X'$ are smooth, and there is a surjective and proper map $f: X' \rightarrow X$ defined by polynomials.

The whole history of algebraic geometry is intertwined with the development of resolution of singularities. One of the first deep results in algebraic geometry is Newton's proof of resolution for curves in the complex plane $\mathbb{C}^{2}$, that is, for zero sets of two variable polynomials $f(x, y) = 0$.

The next two centuries produced many more proofs of resolution for curves, and by the early part of the twentieth century the resolution of algebraic surfaces was also settled.

After much effort, Zariski proved resolution for 3-folds, and finally Hironaka settled the general case in a 218-page paper in 1964.

Since then resolution has had an unusual role in algebraic geometry. On the one hand, it is a basic result and almost everyone working in the field uses it; many of us use it regularly. On the other hand, the proof was generally viewed as too long and complicated to fathom in detail, and few people actually read all of it. (I must confess that I have not been among these few until recently.)

The lingering perception that the proof of resolution is very hard gradually diverged from reality. While Hironaka's original arguments are indeed very subtle and lengthy, during the last forty years a small group of initiates has been improving and simplifying the proof. In particular, all the technical machinery has been removed.

I gave a graduate course devoted to resolution of singularities at Princeton University during the 2004/05 academic year. The first semester gave ample time to explore many different approaches for curves and surfaces. It is quite interesting to see how the nice methods for curves become much harder for surfaces before a clear and simple proof emerges for the general case. Finally it is feasible to prove resolution in the last two weeks of a beginning algebraic geometry course.

Chapter 1 is devoted to resolution of curve singularities. It was very enjoyable to work through thirteen methods, read some old papers, and see the roots of many later techniques. This chapter is very elementary, and many of the proofs could be presented in a first course on algebraic geometry.

Chapter 2 studies resolution for surfaces. Several pretty ideas work for surfaces but seem to fall short in higher dimensions. In addition, the proofs need more technical background.

By contrast, the methods of the general case presented in Chapter 3 are again elementary. The actual proof occupies only thirty pages in Sections 7-13, and most of Chapter 3 is devoted to motivation and examples. The approach is based on the recent work of Włodarczyk, which in turn is built on the earlier results of Hironaka, Giraud and Villamayor.

Other important advances in the Hironaka method [have been given], whose connections with the present proof are more indirect.

It is worth noting that before the appearance of Hironaka's 1964 work it was not obvious that resolutions existed, much less that they could be achieved by repeatedly blowing up smooth subvarieties. The 1950s saw several wrong proofs, and even Zariski - the grandfather of modern resolution seems to have doubted at some point that the Hironaka method would ever work.

This short book is aimed at readers who are interested in the subject and would like to understand one proof of resolution in characteristic 0 but do not (yet) plan to get acquainted with every aspect of the theory. Any thorough treatment of resolutions should encompass many other topics that are barely mentioned here.

6.3. Review by: Dan Abramovich.
Mathematical Reviews MR2289519 (2008f:14026).

This is the second book published within about three years on the subject of resolution of singularities, including a treatment of H Hironaka's theorem on resolution of singularities in characteristic 0. This in itself demonstrates a fact previously known to a select few: resolution of singularities in characteristic 0 is well understood. One can now give a completely comprehensible semester course on the topic of resolution of singularities. One can also devote a few weeks in a first course on algebraic geometry to give just a complete proof of resolution of singularities in characteristic 0 (Chapter 3 of the present book, which is largely self-contained). Just a few years ago resolution of singularities in characteristic 0 was taken by most algebraic and complex geometers to be a black box. Even A Grothendieck in Actes du Congrès International des Mathématiciens (Nice, 1970) admitted openly that he did not completely understand Hironaka's proof.
...
The text is decidedly in the style of lecture notes. Kollár aims to demystify the main results and techniques for students and researchers possibly working on unrelated subjects. This style gives the reader a hint of Kollár's distinctive way of thinking (sometimes at the expense of clarity ...

People are already using this book. I am using this book now. I expect it will be used well into the future.

6.4. Review by: Dan Abramovich.
Bulletin of the American Mathematical Society 48 (1) (2011), 115-122.

Resolution of singularities in characteristic 0 - what a change of fortunes!

Hironaka's proof appeared 46 years ago. Yet in recent years there is a sense of excitement about the subject: young geometers are giddy about learning it; lectures about it are given in regular courses, summer schools, and conferences; and more people are seriously trying their hands at the notorious problem of resolution of singularities in positive characteristics. What is all the fuss about?

Of course the answer has several aspects. Efforts at good exposition in the last two decades and the resulting widening of the group of interested researchers is important. Some publicity certainly helped. The book under review is both an outcome of the process and a new driving force for the excitement. Also essential is new mathematical substance.

The main point, however, is different. Here is a wonderful result of fundamental importance in algebraic and complex geometry. Its uses are far and wide - an impressive list was included already in Grothendieck's address at the 1970 Nice International Congress of Mathematicians, where Hironaka received the Fields Medal. It has every right to be included in a basic graduate course on the subject - after all, our students get to use it right away. Yet for many years it was used as a black box. Grothendieck admitted in his address that he had not fully understood the proof, and the embarrassment is palpable. Much later still, Kollár and Mori referred to the possibility of using weaker results, which did not make them too happy.

The situation now is dramatically changed. Here is a book you can use to introduce your students to the subject. Self-contained proofs are given in Chapter 3 of Kollár's book. I used Kollár's treatment as one topic in a basic graduate course, and it is truly remarkable. After two weeks of introductory material, the students took over lecturing. The proof breaks nicely into manageable pieces, each taking about one hour of lecture. The whole thing took about six weeks, and left ample time for other topics. Others who lectured on their own covered this faster. This is finally the way things should be! If you are teaching algebraic geometry and considering topics for the second term, do consider this subject - you will enjoy it whether or not you are an algebraic geometer. You may choose what fits your style: Kollár's book is in the style of lecture notes. In short, there is no excuse now for using resolution of singularities as a black box - it is well understood and fully transparent. Certainly if you are an algebraic geometer you had better learn this stuff before the next time you meet the author.

7.1. From the Publisher.

This book gives a comprehensive treatment of the singularities that appear in the minimal model program and in the moduli problem for varieties. The study of these singularities and the development of Mori's program have been deeply intertwined. Early work on minimal models relied on detailed study of terminal and canonical singularities but many later results on log terminal singularities were obtained as consequences of the minimal model program. Recent work on the abundance conjecture and on moduli of varieties of general type relies on subtle properties of log canonical singularities and conversely, the sharpest theorems about these singularities use newly developed special cases of the abundance problem. This book untangles these interwoven threads, presenting a self-contained and complete theory of these singularities, including many previously unpublished results.

7.2. From the Preface.

In 1982 Shigefumi Mori outlined a plan now called Mori's program or the minimal model program - whose aim is to investigate geometric and cohomological questions on algebraic varieties by constructing a birational model especially suited to the study of the particular question at hand.

The theory of minimal models of surfaces, developed by Castelnuovo and Enriques around 1900, is a special case of the 2-dimensional version of this plan. One reason that the higher dimensional theory took so long in coming is that, while the minimal model of a smooth surface is another smooth surface, a minimal model of a smooth higher dimensional variety is usually a singular variety. It took about a decade for algebraic geometers to understand the singularities that appear and their basic properties. Rather complete descriptions were developed in dimension 3 by Mori and Reid and some fundamental questions were solved in all dimensions.

While studying the compactification of the moduli space of smooth surfaces, Kollár and Shepherd-Barron were also led to the same classes of singularities.

At the same time, Demailly and Siu were exploring the role of singular metrics in complex differential geometry, and identified essentially the same types of singularities as the optimal setting.

The aim of this book is to give a detailed treatment of the singularities that appear in these theories.

We started writing this book in 1993, during the 3rd Salt Lake City summer school on Higher Dimensional Birational Geometry. The school was devoted to moduli problems, but it soon became clear that the existing literature did not adequately cover many properties of these singularities that are necessary for a good theory of moduli for varieties of general type. A few sections were written and have been in limited circulation, but the project ended up in limbo.

The main results on terminal, canonical and log terminal singularities were treated in Kollár and Mori (1998) and for many purposes of Mori's original program these are the important ones.

There have been attempts to revive the project, most notably an AIM conference in 2004, but real progress did not restart until 2008. At that time several long-standing problems were solved and it also became evident that for many problems, including the abundance conjecture, a detailed understanding of log canonical and semi-log canonical singularities and pairs is necessary. In retrospect we see that many of the necessary techniques have not been developed until recently, so the earlier efforts were rather premature.

Although the study of these singularities started only 30 years ago, the theory has already outgrown the confines of a single monograph. Thus many of the important developments could not be covered in detail. Our aim is to focus on the topics that are important for moduli theory. Many other areas are developing rapidly and deserve a treatment of their own.

7.3. Review by: Tommaso De Fernex.
Mathematical Reviews MR3057950.

The book gives a detailed treatment of the theory of singularities appearing in the minimal model program.

Terminal and canonical singularities are the only classes of singularities that are needed, strictly speaking, to run a minimal model program starting from a smooth manifold, the former being the class of singularities appearing in the terminal models, and the latter being the singularities of the canonical models. Inductive approaches to building the steps of the program have led to work in the larger framework of log pairs, and in a larger class of singularities such as log terminal and log canonical singularities, and variants of these notions such as divisorial log terminal (dlt) singularities. Semi-log canonical pairs naturally arise in the context of moduli theory. Their support is not a variety anymore, but a reduced equidimensional scheme whose components have mild singularities and intersect nicely. The model to keep in mind is that of a stable degeneration of a one-parameter family of smooth varieties of general type. The minimal model program is a powerful tool to study compactifications of moduli spaces, and semi-log canonical pairs come up as the limit objects.

These various classes of singularities have been introduced and studied throughout the last 30 years, roughly following the order above. The book by J Kollár and S Mori [Birational geometry of algebraic varieties (1998)] provides an excellent introduction to the minimal model program and a good treatment of terminal, canonical and log terminal singularities. The book under review can be regarded, in a way, as a natural continuation of that book. Rather technical in nature, it aims to solidify the foundations of the theory of singularities, and it is safe to expect that it will become a point of reference for people working in birational geometry and on moduli problems.

8.1. From the Publisher.

This book establishes the moduli theory of stable varieties, giving the optimal approach to understanding families of varieties of general type. Starting from the Deligne-Mumford theory of the moduli of curves and using Mori's program as a main tool, the book develops the techniques necessary for a theory in all dimensions. The main results give all the expected general properties, including a projective coarse moduli space. A wealth of previously unpublished material is also featured, including Chapter 5 on numerical flatness criteria, Chapter 7 on $K$-flatness, and Chapter 9 on hulls and husks.

8.2. From the Preface.

The aim of this book is to generalise the moduli theory of algebraic curves - developed by Riemann, Cayley, Klein, Teichmüller, Deligne, and Mumford - to higher dimensional algebraic varieties.

Starting with the theory of algebraic surfaces worked out by Castelnuovo, Enriques, Severi, Kodaira, and ending with Mori's program, it became clear that the correct higher dimensional analogue of a smooth projective curve of genus ≥ 2 is a smooth projective variety with ample canonical class. We establish a moduli theory for these objects, their limits, and generalisations.

The first attempt to write a book on higher dimensional moduli theory was the 1993 Summer School in Salt Lake City, Utah. Some notes were written, but it soon became evident that, while the general aims of a theory were clear, most of the theorems were open and even many of the basic definitions unsettled.

The project was taken up again at an AIM conference in 2004, which eventually resulted in solving the moduli-theoretic problems related to singularities; these were written up in Kollár. After 30 years, we now have a complete theory, the result of the work of numerous people.

While much of the early work focused on the construction of moduli spaces, later developments in the theory of stacks emphasised families. We also follow this approach and spend most of the time understanding families. Once this is done at the right level, the existence of moduli spaces becomes a natural consequence.

8.3. Review by: Chenyang Xu.
Mathematical Reviews MR4566297.

In April 2007, after the seminal work of C Birkar et al. appeared online, in a conference at the Simons Laufer Mathematical Sciences Institute (then called MSRI), Kollár announced that he would begin to advance the program of constructing moduli spaces parametrising canonical models of pairs. Fast forward to 2023, when Kollár published the volume under review, which summarises the progress of the last one and a half decades. In particular, a complete solution to the moduli problem of canonical models is included.
...
Written by a leader of the field, the book sets a milestone in the moduli theory of high-dimensional pairs. It presents the evolution of the topic, as well as Kollár's distinct way of thinking about it.

9. What determines an algebraic variety? (2023), by János Kollár, Max Lieblich, Martin Olsson and Will Sawin.

9.1. From the Publisher.

A pioneering new nonlinear approach to a fundamental question in algebraic geometry.

One of the crowning achievements of nineteenth-century mathematics was the proof that the geometry of lines in space uniquely determines the Cartesian coordinates, up to a linear ambiguity. What Determines an Algebraic Variety? develops a nonlinear version of this theory, offering the first nonlinear generalisation of the seminal work of Veblen and Young in a century. While the book uses cutting-edge techniques, the statements of its theorems would have been understandable a century ago; despite this, the results are totally unexpected. Putting geometry first in algebraic geometry, the book provides a new perspective on a classical theorem of fundamental importance to a wide range of fields in mathematics.

Starting with basic observations, the book shows how to read off various properties of a variety from its geometry. The results get stronger as the dimension increases. The main result then says that a normal projective variety of dimension at least 4 over a field of characteristic 0 is completely determined by its Zariski topological space. There are many open questions in dimensions 2 and 3, and in positive characteristic.

9.2. From the Preface.

Geometry starts with the study of points and lines in the plane. The simplest objects are the points in the plane (as Euclid says, they have no parts) and lines are certain sets of points.

Descartes put coordinates on the plane, so now we usually think of the plane as a 2-dimensional vector space $\mathbb{R}^{2}$, and the lines as solution sets of linear equations. This is the starting point of algebraic geometry, where one can use algebraic methods to solve geometric problems.

The reworking of the foundations of algebraic geometry, started in the 1930s by van der Waerden, Weil, and Zariski, culminated in Grothendieck's theory of schemes around 1960. This turned 'algebraic geometry' into 'algebraic geometry.' In these treatments the primary object is no longer the $n$-space $\mathbb{R}^{n}$, but the polynomial ring $\mathbb{C}[x_{1}, ..., x_{n}]$.

Thus in contemporary algebraic geometry we think of $\mathbb{C}^{n}$ as consisting of two parts:

- (geometry) the set of points $\mathbb{C}^{n}$ together with 'closed subsets' given by solution sets of systems of polynomial equations, and

- (algebra) the ring of all polynomial functions $\mathbb{C}^{n} \rightarrow \mathbb{C}$.

Rather sloppily, standard usage does not distinguish between $\mathbb{C}^{n}$as a point set, vector space, Zariski topological space, variety or scheme, but for us these distinctions are crucial. We call the geometric object the Zariski topological space, and denote it by $|\mathbb{C}^{n}|$.

By Hilbert's Nullstellensatz, the points of $|\mathbb{C}^{n}|$ correspond to maximal ideals of $\mathbb{C}[x_{1}, ..., x_{n}]$, and the closed subsets to other radical ideals. Thus, the ring $\mathbb{C}[x_{1}, ..., x_{n}]$ uniquely determines the geometry $|\mathbb{C}^{n}|$.

The main question we aim to study in these notes is a converse: Does geometry determine algebra and function theory?

We compactify $\mathbb{C}^{n}$ to projective space $\mathbb{C}P^{n}$ and, for full generality, we work with projective $n$-space $P_{K}^{n}$ over any commutative field $K$. Any Zariski closed subset $X \subset P_{K}^{n}$ has an underlying geometry, denoted by $|X|$, and a function theory or structure sheaf, denoted by $\mathcal{O}_X$.

Thus the general question that we address is: Given a projective algebraic set $X \subset P_{K}^{n}$, does the geometry $|X|$ determine the algebra and function theory of $\mathcal{O}_X$?

For algebraic curves the geometry is not rich enough to give any information, and there are a few other, mostly obvious, exceptions. The main theorem - stated in Section 1.2 - says that, for projective varieties of dimension ≥ 4, the geometry does determine the algebra and function theory, at least over fields of characteristic 0. Our answers are less complete in dimensions 2 and 3.