# Holomorphic vector fields tangent to foliations in dimension three

## TL;DR

This paper investigates holomorphic vector fields in three complex dimensions tangent to foliations, establishing conditions for hyperbolicity and showing that tangent vector fields to multiple foliations preserve an analytic surface.

## Contribution

It provides new results on the structure of vector fields tangent to foliations, including conditions for hyperbolicity and the invariance of surfaces when tangent to multiple foliations.

## Key findings

- Foliations are of complex hyperbolic type under certain singularity reduction hypotheses.
- Vector fields tangent to three independent foliations are tangent to a pencil of foliations.
- Such vector fields leave invariant a germ of an analytic surface.

## Abstract

This article studies germs of holomorphic vector fields at the origin of C3 that are tangent to holomorphic foliations of codimension one. Two situations are considered. First, we assume hypotheses on the reduction of singularities of the vector field - for instance, that the final models belong to a family of vector fields whose linear parts have eigenvalues satisfying a condition of non-resonance - in order to conclude that the foliation is of complex hyperbolic type, that is, without saddle-nodes in its reduction of singularities. In the second part, we prove that a vector field that is tangent to three independent foliations is tangent to a whole pencil of foliations - hence, to infinitely many foliations - and, as a consequence, it leaves invariant a germ of analytic surface. This final part is based on a local version of a well-known characterization of pencils of foliations of codimension one in projective spaces.