Radial foliations in dimension three

TL;DR

This paper explores the properties of radial foliations in three dimensions, characterizing their singularity reduction and linking them to open book structures, with discussions on more general cases.

Contribution

It extends the concept of radial foliations from dimension two to three and characterizes their singularity reduction and geometric structure.

Findings

Radial foliations in dimension three are characterized by their singularity reduction.

The radial condition corresponds to an 'open book' geometric structure.

Discussion on the broader class of almost radial foliations.

Abstract

Radial germs of holomorphic foliations in dimension two have a characteristic property: they are the only singular foliations whose reduction of singularities has no singular points. We also know that they are desingularized by a single dicritical blowing-up. Let us say that a foliated space ((C3, 0),E,F) is almost radial when it has a reduction of singularities without singular points; it will be "radial" under a certain additional condition on the morphism of reduction of singularities. We show that the radial condition corresponds to the "open book" situation. We end the paper with a discussion on the general almost radial case.