Foliations and algebraic geometry – Foliage

The members of this project are mainly united by the use of the theory of holomorphic foliations applied to several questions of algebraic geometry. Three directions can be seen to structure our project: 1) Hyperbolicity 2) Analytic and birational geometry 3) Arithmetic geometry. 1) We will study how the positivity of the canonical bundle of a variety, or more generally of a foliation, impacts the existence of entire curves, tangent to the foliation. We of course have in view the (difficult and still largely open) conjectures of Bombieri-Lang, Vojta, Green-Griffiths (when the canonical bundle is positive) who predict the degeneracy of entire curves or the scarcity of rational points, but also conjectures who predict their existence when the canonical bundle is trivial. The degeneracy of parabolic leaves of foliations on surfaces of general type and the description of the Green-Griffiths locus, defined by base loci of jet differentials, have already shown that the use of holomorphic foliations plays a crucial role. The proofs of algebraic degeneracy of entire curves require precise informations studied in theme 2) – foliations with positive canonical bundle. 2) Complex projective varieties are classified according to the properties of their canonical bundle. Similar ideas have started to be used in the context of foliations. Foliations compatible with the numerical properties of the canonical bundle are of particular interest e.g. in the problem of the abundance conjecture. In analogy with the case of complex manifolds, one is naturally lead to study three geometries defined by the “sign” of the canonical bundle of the foliation. Building bricks of algebraic foliations are (conjecturally and very roughly) foliations with ample canonical bundle, trivial canonical bundle or ample anticanonical bundle. The generalization of the Minimal Model Program to algebraic foliations is promising with already important results recently obtained on surfaces. 3) There are many fascinating conjectural connections between the algebraic integrability properties of foliations defined over number fields and their behaviour under reduction modulo p, or the distribution of algebraic points on their complex leaves. These conjectures reveal intriguing connections between foliation theory, and various classical problems in diophantine geometry and transcendence theory. Since twenty years and the birth of Arakelov geometry, transcendence methods have become more geometric and we now can try to study more involved geometric situations in particular with singular holomorphic foliations.