Some remarks on meromorphic first integrals

TL;DR

This paper provides a straightforward proof and additional insights into the conditions under which germs of codimension one foliations possess meromorphic first integrals, especially when an invariant real hypersurface is involved.

Contribution

It offers a simplified proof and complements to a key result on meromorphic first integrals for certain foliations, enhancing understanding of their existence conditions.

Findings

Simplified proof of the existence of meromorphic first integrals.

Additional insights into foliations with invariant real hypersurfaces.

Clarification of conditions for meromorphic integrals in codimension one foliations.

Abstract

We give a simple proof, with some complements, of a result of Cerveau and Lins Neto, concerning the existence of meromorphic first integrals for germs of codimension one foliations with an invariant real hypersurface.