Marco Brunella

A POSITIVITY PROPERTY FOR FOLIATIONS ON COMPACT KÄHLER MANIFOLDS

We prove that the canonical bundle of a foliation by curves on a compact Kahler manifold is pseudoeffective, unless the foliation is a (special) foliation by rational curves.

Complete polynomial vector fields on the complex plane

Abstract We give a full classification, up to polynomial automorphisms, of complete polynomial vector fields in two complex variables.

A characterization of Inoue surfaces

We give a characterization of Inoue surfaces in terms of automorphic pluriharmonic functions on a cyclic covering. Together with results of Chiose and Toma, this completes the classification of compact complex surfaces of Kaehler rank one.

PLURISUBHARMONIC VARIATION OF THE LEAFWISE POINCARÉ METRIC

This paper is concerned with a higher dimensional generalization of the main result of our previous paper [2]: we shall prove that the Poincare metric on the leaves of a one-dimensional holomorphic foliation on a compact Kahler manifold has a plurisubharmonic variation.

Some remarks on meromorphic first integrals

We give a simple proof, with some complements, of a result of Cerveau and Lins Neto, concerning the existence of meromorphic first integrals for germs of codimension one foliations with an invariant real hypersurface.

Uniformisation of Foliations by Curves

These lecture notes provide a full discussion of certain analytic aspects of the uniformisation theory of foliations by curves on compact Kahler manifolds, with emphasis on convexity properties and their consequences on positivity properties of the corresponding canonical bundles.

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].

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.

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.

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].