Local Brunella's Alternative I. RICH Foliations

Abstract

This paper is devoted to studying the structure of codimension one singular holomorphic foliations on $({\mathbb C}^3,0)$ without invariant germs of analytic surface. We focus on the so-called CH-foliations, that is, foliations without saddle nodes in two dimensional sections. Considering a reduction of singularities, we detect the possible existence of "nodal components", which are a higher dimensional version of the nodal separators in dimension two. If the foliation is without nodal components, we prove that all the leaves in a neighborhood of the origin contain at least one germ of analytic curve at the origin. We also study the structure of nodal components for the case of "Relatively Isolated CH-foliations" and we show that they cut the dicritical components or they exit the origin through a non compact invariant curve. This allows us to give a precise statement of a local version of Brunella's alternative: if we do not have an invariant surface, all the leaves contain a germ of analytic curve or it is possible to detect the nodal components in the generic points of the singular curves before doing the reduction of singularities.