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**Birational geometry of foliations.**
*(English)*
Zbl 1073.14022

Monografías de Matemática (Rio de Janeiro). Rio de Janeiro: Instituto de Matemática Pura e Aplicada (IMPA). 138 p. (2000).

Based upon a series of lectures delivered by the author at the First Latin-American Congress of Mathematicians (Rio de Janeiro, 2000), the book under review provides a detailed account of the classification of holomorphic foliations on complex algebraic surfaces. Apart from a comprehensive introduction to singular foliations on surfaces, the author’s main goal was to systematically describe some recent developments toward their classification theory, as they were achieved by the works of M. McQuillan, L. G. Mendes, the author himself, and others in the past few years.

As it turned out, a birational classification theory of (singular) foliations of complex surfaces could be built up along the lines of both the original Enriques classification of complex surfaces themselves and the more recent Mori approach, with many subtle and refined modifications, and the first coherent exposition of this novel program represents indeed the core of these notes under review.

The entire text consists of nine chapters. Chapters 1 and 2 review some basic material on foliations and their singularities, including Seidenberg’s reduction theorem. Chapter 3 explains the crucial index theorems of Baum-Bott and Camacho-Sad for foliations, together with some generalizations and applications for later use.

Chapter 4 introduces and studies two important classes of foliations, namely Ricci foliations and turbulent foliations, which are to play a special role in the birational classification of foliations on surfaces. Chapter 5 develops the author’s version of Zariski’s work on minimal models for foliations, whereas Chapter 6 discusses two other particular classes of foliations, including those generated by a global holomorphic 1-form and those generated by a global holomorphic vector field. Chapter 7 deals with a remarkable theorem of Y. Miyaoka, first proved in 1985 and asserting that a foliation with non-pseudo-effective cotangent bundle is necessarily a foliation by rational curves on the surface. Miyaoka’s rationality theorem for foliations is here proved in two different ways, following appoaches by N. Shepherd-Barron and by F. Bogomolov–M. McQuillan, together with an outline of yet another, however still incomplete approach of the author’s. The variety of these different viewpoints sheds some new light on the fundamental role of Miyaoka’s rationality criterion, both algebraically and analytically.

In Chapter 8, following M. McQuillan’s work, the author studies the first properties of the Zariski decomposition of the cotangent bundle of a non-rational foliation, defines the so-called numerical Kodaira dimension as a crucial invariant for foliations, and finally obtains a complete classification of foliations with vanishing numerical Kodaira dimension. In Chapter 9, the (ordinary) Kodaira dimension of a foliation is defined, and the relation between the ordinary and the numerical Kodaira dimension is studied in great detail. This leads, among other results, to a complete classification of foliations with Kodaira dimension 0 or 1. This concluding chapter ends with concrete examples of foliations for which the Kodaira dimension and the numerical Kodaira dimension do not coincide.

The entire book is very well written and, in the meantime has already become a standard reference on the subject. The first six chapters provide a comprehensive and systematic introduction to the theory of foliations on complex surfaces, whereas the remaining three chapters lead the reader to the forefront of current research in the classification theory of foliations. The exposition is utmost detailed, lucid, inspiring, and reader-friendly. As the prerequisites from algebraic geometry and complex-analytic geometry are kept to a minimum, i.e., not exceeding the (advanced) standard basics, these very useful notes should be accessible to a wide audience of complex geometers and analysts.

As it turned out, a birational classification theory of (singular) foliations of complex surfaces could be built up along the lines of both the original Enriques classification of complex surfaces themselves and the more recent Mori approach, with many subtle and refined modifications, and the first coherent exposition of this novel program represents indeed the core of these notes under review.

The entire text consists of nine chapters. Chapters 1 and 2 review some basic material on foliations and their singularities, including Seidenberg’s reduction theorem. Chapter 3 explains the crucial index theorems of Baum-Bott and Camacho-Sad for foliations, together with some generalizations and applications for later use.

Chapter 4 introduces and studies two important classes of foliations, namely Ricci foliations and turbulent foliations, which are to play a special role in the birational classification of foliations on surfaces. Chapter 5 develops the author’s version of Zariski’s work on minimal models for foliations, whereas Chapter 6 discusses two other particular classes of foliations, including those generated by a global holomorphic 1-form and those generated by a global holomorphic vector field. Chapter 7 deals with a remarkable theorem of Y. Miyaoka, first proved in 1985 and asserting that a foliation with non-pseudo-effective cotangent bundle is necessarily a foliation by rational curves on the surface. Miyaoka’s rationality theorem for foliations is here proved in two different ways, following appoaches by N. Shepherd-Barron and by F. Bogomolov–M. McQuillan, together with an outline of yet another, however still incomplete approach of the author’s. The variety of these different viewpoints sheds some new light on the fundamental role of Miyaoka’s rationality criterion, both algebraically and analytically.

In Chapter 8, following M. McQuillan’s work, the author studies the first properties of the Zariski decomposition of the cotangent bundle of a non-rational foliation, defines the so-called numerical Kodaira dimension as a crucial invariant for foliations, and finally obtains a complete classification of foliations with vanishing numerical Kodaira dimension. In Chapter 9, the (ordinary) Kodaira dimension of a foliation is defined, and the relation between the ordinary and the numerical Kodaira dimension is studied in great detail. This leads, among other results, to a complete classification of foliations with Kodaira dimension 0 or 1. This concluding chapter ends with concrete examples of foliations for which the Kodaira dimension and the numerical Kodaira dimension do not coincide.

The entire book is very well written and, in the meantime has already become a standard reference on the subject. The first six chapters provide a comprehensive and systematic introduction to the theory of foliations on complex surfaces, whereas the remaining three chapters lead the reader to the forefront of current research in the classification theory of foliations. The exposition is utmost detailed, lucid, inspiring, and reader-friendly. As the prerequisites from algebraic geometry and complex-analytic geometry are kept to a minimum, i.e., not exceeding the (advanced) standard basics, these very useful notes should be accessible to a wide audience of complex geometers and analysts.

Reviewer: Werner Kleinert (Berlin)

### MSC:

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

32S65 | Singularities of holomorphic vector fields and foliations |

37F75 | Dynamical aspects of holomorphic foliations and vector fields |

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

32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |