Invariant Hypersurfaces and Nodal Components of Foliations

TL;DR

This paper generalizes a known property of singular foliations in complex two-dimensional spaces, showing that invariant hypersurfaces and nodal components exist in higher dimensions after singularity reduction.

Contribution

It extends the existence of invariant hypersurfaces passing through nodal components from 2D to arbitrary dimensions in complex spaces.

Findings

Invariant hypersurfaces exist in higher dimensions after singularity reduction.

Nodal components are associated with invariant hypersurfaces in complex foliations.

The generalization applies to any ambient dimension, not just two-dimensional cases.

Abstract

It is known that there is at least an invariant analytic curve passing through each of the components in the complement of nodal singularities, after the reduction of singularities of a germ of singular foliation in ${\mathbb C}^2,0$}. Here, we state and prove a generalization of this property to any ambient dimension.