On the integrability of holomorphic vector fields

TL;DR

This paper investigates the conditions under which a germ of a holomorphic foliation in three complex dimensions admits a holomorphic first integral, linking topological and algebraic properties of the vector field.

Contribution

It establishes criteria for the existence of holomorphic first integrals for generic vector fields in three complex variables, advancing understanding of integrability in complex dynamical systems.

Findings

Identifies topological conditions for integrability.

Determines algebraic constraints for the existence of first integrals.

Provides a framework for analyzing holomorphic foliations in complex dimension three.

Abstract

We determine topological and algebraic conditions for a germ of holomorphic foliation $\mathcal F(X)$ induced by a generic vector field $X$ on $(\mathbb{C}^{3},0)$ to have a holomorphic first integral, i.e., a germ of holomorphic map $F \colon(\mathbb{C}^{3},0)\longrightarrow(\mathbb{C}^{2},0)$ such that the leaves of $\mathcal F(X)$ are contained in the level curves of $F$.