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The text presents the birational classification of holomorphic foliations of surfaces. It discusses at length the theory developed by L.G. Mendes, M. McQuillan and the author to study foliations of surfaces in the spirit of the classification of complex algebraic surfaces.



Chapter 1. Local Theory

In this chapter we recall some basic facts concerning singular points of holomorphic foliations on surfaces, and in particular Seidenberg’s Theorem [39] which will play a fundamental role also in the global theory. A good reference for this material is [11].
Marco Brunella

Chapter 2. Foliations and Line Bundles

In this chapter we start the global study of foliations on complex surfaces. The most basic global invariants which may be associated with such a foliation are its normal and tangent bundles, and here we shall prove several formulae and study several examples concerning the calculation of these bundles. We shall mainly follow the presentation given in [5]; the book [20] may also be of valuable help.
Marco Brunella

Chapter 3. Index Theorems

In this chapter we prove two important results of Baum–Bott [3] and Camacho–Sad [10, 11], the first one concerning the computation of \(N_{\mathcal{F}}\cdot N_{\mathcal{F}}\) for a foliation \(\mathcal{F}\) on a compact surface, the second one concerning the computation of \(C \cdot C\) for a compact curve C invariant by a foliation. These two results, which are in fact two manifestations of the same “vanishing principle”, have a quite different nature than the easier formulae of the previous chapter: here the integrability of \(\mathcal{F}\) , that is the existence of leaves, plays a fundamental role. We shall also give several applications of these formulae. A comprehensive reference for these index theorems, and much beyond, is [45], especially Chapter V; see also [6, 7].
Marco Brunella

Chapter 4. Some Special Foliations

In this chapter we study two classes of ubiquitous foliations: Riccati foliations and turbulent foliations. A section will also be devoted to a very special foliation, which will play an important role in the minimal model theory.
Marco Brunella

Chapter 5. Minimal Models

In this chapter we introduce, following [8], a notion of minimal model adapted to the bimeromorphic study of foliations, as well as a notion of relatively minimal model (implicit in [8]). We show that any foliation outside a list of notable exceptions has a minimal model, or equivalently a unique relatively minimal model. Be careful that in [30] there is a slightly different notion of minimal model (which could be called “in Mori sense”, whereas our definition is “in Zariski sense”).
Marco Brunella

Chapter 6. Global 1-Forms and Vector Fields

In this chapter we recall some fundamental facts concerning holomorphic 1-forms on compact surfaces: Albanese morphism, Castelnuovo–de Franchis Lemma, Bogomolov Lemma. We also discuss the logarithmic case, which is extremely useful in the study of foliations with an invariant curve. Finally we recall the classification of holomorphic vector fields on compact surfaces. All of this is very classical and can be found, for instance, in [2, Chapter IV] and 24, 35].
Marco Brunella

Chapter 7. The Rationality Criterion

In this chapter we explain a remarkable theorem of Miyaoka [32] which asserts that a foliation whose cotangent bundle is not pseudoeffective is a foliation by rational curves. The original Miyaoka’s proof can be thought as a foliated version of Mori’s technique of construction of rational curves by deformations of morphisms in positive characteristic [33].
Marco Brunella

Chapter 8. Numerical Kodaira Dimension

In this chapter we study, following [30] , the first properties of the Zariski decomposition of the cotangent bundle of a nonrational foliation. In particular, we shall give a detailed description of the negative part of that Zariski decomposition, and we shall obtain a detailed classification of foliations whose Zariski decomposition is reduced to its negative part (i.e. foliations of numerical Kodaira dimension 0). We shall also discuss the “singular” point of view adopted in [30].
Marco Brunella

Chapter 9. Kodaira Dimension

In this chapter we classify, following [30] , foliations of Kodaira dimension 0 or 1. The main step is to prove that Kodaira dimension 0 actually implies that also the numerical Kodaira dimension is 0: this is a particular case of the so-called “abundance”. We also discuss foliations of Kodaira dimension −∞, but here the classification is still incomplete; the problem is that in this case the abundance does not hold, because there exist foliations of Kodaira dimension −∞ and numerical Kodaira dimension 1. As an application of all of this, we give a complete description of foliations with an entire transcendental leaf [7, 29].
Marco Brunella


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