: École d’Été 2019 - Feuilletages et géométrie algébrique

The aim of this mini course is to highlight some links between the study of the Kobayashi hyperbolicity properties of complex projective manifolds and holomorphic foliations.A compact complex space is Kobayashi hyperbolic if and only if every holomorphic map from the complex plane to it is constant. Projective (or more generally compact Kähler) Kobayashi hyperbolic manifolds share many features with projective manifolds of general type, and it is nowadays a classical and important conjecture (due to S. Lang) that a complex projective manifold should be hyperbolic if and only if it is of general type together with all of its subvarieties.One essential tool in this business (introduced by Green-Griffiths, and later refined by Demailly) are the so-called (invariant) jet differentials: they are algebraic differential equations which every holomorphic image of the complex plane must satisfy, provided they are with values in an anti-ample divisor. The abundance of such jet differentials provide then a strong constraint to the existence of non constant holomorphic map from the complex plane.In this series of lectures we shall first of all introduce the basic notions and facts about Kobayashi hyperbolicity, explain what jet differentials are, and how to use them. Next, we shall describe a series of counterexamples built using holomorphic foliations in an essential way, which explain why jet differentials are not enough to obtain results on hyperbolicity of projective manifolds in full generality (even if lots of spectacular results have been obtained in the last decades in special cases). Last, if time permits, we shall overview (in a toy case) McQuillan’s celebrated proof of the the fact the a projective surface of general type with positive second Segre number is "almost" hyperbolic: again this is a combination of jet differentials and holomorphic foliations