Partial separatrices and local Brunella's alternative

TL;DR

This paper proves a conjecture about the structure of codimension one foliations in complex three-space, showing that under certain conditions, leaves contain analytic curves near the origin, with partial separatrices playing a key role.

Contribution

It establishes the conjecture for foliations with specific singularity reduction properties, introducing the concept of partial separatrices and analyzing Camacho-Sad indices.

Findings

Proves the local Brunella's alternative for a class of foliations.

Identifies nodal components as obstructions to analytic curves.

Uses partial separatrices to construct analytic curves near singularities.

Abstract

Here we state a conjecture concerning a local version of Brunella's alternative: any codimension one foliation in $({\mathbb C}^3,0)$ without germ of invariant surface has a neighborhood of the origin formed by leaves containing a germ of analytic curve at the origin. We prove the conjecture for the class of codimension one foliations whose reduction of singularities is obtained by blowing-up points and curves of equireduction and such that the final singularities are free of saddle-nodes. The concept of "partial separatrix" for a given reduction of singularities has a central role in our argumentations, as well as the quantitative control of the generic Camacho-Sad index in dimension three. The "nodal components" are the only possible obstructions to get such germs of analytic curves. We use the partial separatrices to push the leaves near a nodal component towards compact diacritical divisors, finding in this way the desired analytic curves.