Holomorphic foliations transverse to fibrations on hyperbolic manifolds

Abstract

We regard some aspects of the following general question: Given a singular holomorphic foliation F on a complex manifold M. what can he said about F in the case that M M special geometry (e.g M is hyperbolic (of. [11,12] or in case the F leaves are embedded in a geometrically special way (e.g. F almost everywhere transverse to some fibration M → B with compact hyperbolic fibers:). We prove that the foliation exhibits meromorphic first integral under natural necessary conditions. We also study the classification of one foliations which are complete with respect to some holomorphic fibrations according to the genus of the fiber. The is applied to a remaining case in [1] and to poivnomial foliations in (See Section 4.1) Applications to systems of implicit ordinary differential equations without movable singularities are also given. The techniques and results that we obtain illustrate the interest of using featuring from Several Complex Variables and Hyperbolic Geometry in Holomorphie Dynamics and Complex Differential Equations. This paper is concieved to be accessible also to non-speciallists on singular holomorphic foliations and it contains paragraphs including some basic material on the subject.