On flags of holomorphic foliations associated with singular second-order ordinary differential equations

TL;DR

This paper studies the geometric structures called 2-flags of holomorphic foliations linked to second-order ODEs in complex three-dimensional space, classifies such equations, and introduces a method to construct these structures.

Contribution

It provides a classification of second-order equations admitting 2-flags and a general construction method for such foliation germs with specific singular set properties.

Findings

Classification of second-order ODEs with 2-flags

Construction method for germs of 2-flags

Generic association of higher-order equations with 2-flags

Abstract

We consider germs of holomorphic vector fields at the origin of $\mathbb{C}^3$, with non-isolated singularities that are tangent to a holomorphic foliation of codimension one. This configuration is known as a $2$-flag of foliations. The focus is on cases where this geometric structure originates from second-order ordinary differential equations. We investigate the behavior of the singular sets associated with the foliations under consideration. Furthermore, we present a classification for second-order equations that admit a $2$-flag of foliations. Finally, we propose a general method for constructing germs of $2$-flags of foliations at the origin of $\mathbb{C}^n$, with suitable properties of the singular sets, and we conclude by demonstrating that under generic assumptions, every equation of order greater than or equal to two is associated formally with a germ of 2-flag of holomorphic foliations at $(\mathbb{C}^3,0)$.