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Birational geometry of foliations. (English) Zbl 1082.32022
Publicações Matemáticas do IMPA. Rio de Janeiro: Instituto Nacional de Matemática Pura e Aplicada (IMPA) (ISBN 85-244-0219-9/pbk ). iv, 138 p. (2004).
The book under review is the expanded version of a series of lectures given by the author at IMPA in August 2000 (see Zbl 1073.14022). The aim of these notes is to provide a classification of (singular) holomorphic foliations on complex projective surfaces in the spirit of the classical Enriques classification of complex algebraic surfaces.
The classical Enriques classification of complex algebraic surfaces up to birational morphisms, is based on the Kodaira dimension which can be $$-\infty, 0, 1, 2$$. For surfaces of non general type (that is with Kodaira dimension $$<2$$) the classification is very deep and precise. The more recent Mori’s program, developed to classify higher dimensional complex varieties, based on the study of numerical properties of the canonical bundle – the so-called numerical Kodaira dimension – has been proved to be useful in the surfaces case as well.
With Enriques classification in mind and starting from previous results of Mendes and McQuillan (the relative papers being available only in a preprint form at the time the reviewer is writing this review) the author classifies a holomorphic foliation $$\mathcal F$$ on a complex projective surface by means of its Kodaira’s dimension $$\text{kod}(\mathcal F)$$ (defined as the “Kodaira dimension” of the dual of the tangent bundle of $$\mathcal F$$) and the related numerical Kodaira dimension $$\nu(\mathcal F)$$. Both numbers are birational invariants if one deals with foliations with only reduced singularities, which is always the case (up to monoidal transformations) by means of Seidenberg’s reduction theorem. As in Enriques’ classification, Kodaira’s and numerical Kodaira’s dimension of $$\mathcal F$$ can be $$-\infty, 0,1,2$$, and the latter case is the so called general type case (for which no complete refined classification is available).
For foliations of non general type a fine classification is provided. If $$\nu(\mathcal F)=-\infty$$ a deep result by Miyaoka allows to state that the foliation is a trivial $$\mathbb C\mathbb P^1$$-bundle over a curve. If $$\nu(\mathcal F)=0$$ McQuillan and Mendes proved that the foliation is given by a $$\mathbb C$$-action or by a very special ramified covering. The case $$\nu(\mathcal F)=1$$ and $$\text{kod}(\mathcal F)\geq 0$$ is solved again by McQuillan who proved that necessarily then $$\text{kod}(\mathcal F)=1$$ and the foliation is either Riccati, or turbolent or another very special foliation. The case $$\nu(\mathcal F)=1$$ and $$\text{kod}(\mathcal F)=-\infty$$, still unsolved at the time the book under review was written, has been solved recently by the author himself [see M. Brunella, Invent. Math. 152, No. 1, 119–148 (2003; Zbl 1029.32014)] and McQuillan, proving that in this case the foliation is a so called Hilbert modular foliation (and, contrarily to the surfaces classification, there exist many of such foliations).
The main ingredients for the classification are the Baum-Bott index formula [P. Baum and R. Bott, J. Differ. Geom. 7, 279–342 (1972; Zbl 0268.57011)], the Camacho-Sad index theorem [C. Camacho and P. Sad, Ann. Math. (2) 115, 579–595 (1982; Zbl 0503.32007)], the Brunella tangential index theorem [M. Brunella, Ann. Sci. Éc. Norm. Supér. (4) 30, 569–594 (1997; Zbl 0893.32019)], together with tools such as the Castelnuovo-de Franchis lemma and Bogomolov’s lemma and the already mentioned Miyaoka’s rationality criterion.
The book is really well written and the first chapters can also be used as a primary on holomorphic foliations theory on complex surfaces. Often the author provides simple proofs of well known results and essentially all the main tools used come with at least a sketch of a proof.

##### MSC:
 32S65 Singularities of holomorphic vector fields and foliations 37F75 Dynamical aspects of holomorphic foliations and vector fields 14E05 Rational and birational maps 32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces