Complex polynomial vector fields with many algebraic orbits

TL;DR

This paper generalizes Darboux's theorem on polynomial vector fields with many algebraic orbits, linking it to Reeb stability and extending results to higher dimensions and codimensions.

Contribution

It provides new interpretations and generalizations of Darboux's theorem for complex polynomial vector fields and holomorphic foliations across various dimensions.

Findings

Existence of rational first integrals with infinitely many algebraic orbits.

Extension of Darboux's theorem to higher codimension foliations.

Results based on the geometry of complex projective spaces and singularity analysis.

Abstract

We state some generalizations of a theorem due to G. Darboux, which originally states that a polynomial vector field in the complex plane exhibits a rational first integral and has all its orbits algebraic provided that it exhibits infinitely many algebraic orbits. In this paper, we give an interpretation of this result in terms of the classical Reeb stability theorems, for compact leaves of (non-singular) smooth foliations. Then we give versions of Darboux's theorem, assuring, for a (non-singular) holomorphic foliation of any codimension, the existence of an open set of compact leaves provided that the measure of the set of compact leaves is not zero. As for the case of polynomial vector fields in the complex affine space of dimenion $m\geq2$, we prove suitable versions of the above results, based also on the very special geometry of the complex projective space of dimension $m$, and on the nature of the singularities of such vector fields we consider.