On real center singularities of complex vector fields on surfaces

TL;DR

This paper generalizes the Poincaré-Lyapunov center theorem to broader classes of real analytic and holomorphic vector fields on surfaces, establishing new conditions for the existence of first integrals near singularities.

Contribution

It extends classical results to cases with many periodic orbits and holomorphic foliations, providing new proofs and applications in complex surface dynamics.

Findings

Proved generalized Poincaré-Lyapunov center theorems for real and holomorphic vector fields.

Established conditions under which singularities admit analytic first integrals.

Highlighted potential for further research in complex foliation singularities.

Abstract

One of the various versions of the classical Lyapunov-Poincar\'e center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations (\cite{moussu}). In this paper we consider generalizations for two main frameworks: (i) planar real analytic vector fields with "many" periodic orbits near the singularity and (ii) germs of holomorphic foliations having a suitable singularity in dimension two. In this paper we prove versions of Poincar\'e-Lyapunov center theorem, including for the case of holomorphic vector fields. We also give some applications, hinting that there is much more to be explored in this framework.