Analytic varieties invariant by foliations and Pfaff systems

TL;DR

This paper surveys the theory of singular holomorphic foliations and Pfaff systems on complex manifolds, focusing on invariant analytic varieties and classical problems of algebraic integration, degree, and genus bounds.

Contribution

It provides a comprehensive overview of recent advances in invariant varieties, integrating classical and modern results on integrability and algebraic bounds.

Findings

Bounds on degree and genus of invariant varieties

Characterization of integrability via meromorphic first integrals

Connections to classical problems of Darboux, Poincaré, and Painlevé

Abstract

In this work we shall present a survey on problems and results on singular holomorphic foliations and Pfaff systems on complex manifolds assuming that these objects possess invariant analytic varieties. We will focus on recent results which have been motivated by classical works of Darboux, Poincar\'e and Painlev\'e on the problem of algebraic integration of singular polynomial differential equations. We present results on Poincar\'e and Painlev\'e problem of bounding the degree and the genus of analytic varieties invariant by holomorphic foliations and Pfaff systems. We shall discuss the general ideas of the theory of integrability characterizing the existence of meromorphic first integrals for complex analytic Pfaff equations.