https://hal-univ-poitiers.archives-ouvertes.fr/hal-03475769Campana, FredericFredericCampanaIECL - Institut Élie Cartan de Lorraine - UL - Université de Lorraine - CNRS - Centre National de la Recherche ScientifiqueAlgebraicity of foliations on complex projective manifolds, applicationsAlgébricité de feuilletages sur les variétés projectives complexes, applicationsHAL CCSD2021[MATH.MATH-AG] Mathematics [math]/Algebraic Geometry [math.AG][MATH.MATH-CV] Mathematics [math]/Complex Variables [math.CV]Campana, Frederic2021-12-11 12:48:282022-06-26 03:24:452021-12-23 11:30:56enPreprints, Working Papers, ...https://hal-univ-poitiers.archives-ouvertes.fr/hal-03475769/documentapplication/pdf1This is an expository text, originally intended for the ANR ‘Hodgefun’ workshop, twice reported, organised at Florence,villa Finaly, by B. Klingler. We show that holomorphic foliations on complex projective manifolds have algebraic leaves under a certain positivity property: the‘non pseudoeffectivity’ of their duals. This permits to construct certain rational fibrations with fibres either rationally connected, or with trivial canonical bundle, of central importance in birational geometry. A considerable extension of the range of applicability is due to the fact that this positivity is preserved by the tensor powers of the tangent bundle. The results presented here are extracted from [10], which is inspired by the former results ([28], [6], [11]). In order to make things as simple as possible, we present here only the projective versions of these results, although most of them can be easily extended to the logarithmic or ‘orbifold’ context.