A brief introduction on residue theory of holomorphic foliations

TL;DR

This survey explores the residue theory of holomorphic foliations, highlighting its applications in classifying foliations, solving the Poincaré problem, and studying minimal sets, with emphasis on Chern classes and Baum-Bott residues.

Contribution

It provides a comprehensive overview of residue theory in holomorphic foliations, including recent advances and applications, emphasizing the use of Chern classes and Baum-Bott residues.

Findings

Residue theory aids in classifying holomorphic foliations.

Residues help solve the Poincaré problem.

Residues are useful in analyzing minimal sets.

Abstract

This is a survey paper dealing with holomorphic foliations, with emphasis on residue theory and its applications. We start recalling the definition of holomorphic foliations as a subsheaf of the tangent sheaf of a manifold. The theory of Characteristic Classes of vector bundles is approached from this perspective. We define Chern classes of holomorphic foliations using the Chern-Weil theory and we remark that the Baum-Bott residue is a great tool that help us to classify some foliations. We present throughout the survey several recent results and advances in residue theory. We finish by presenting some applications of residues to solve for example the Poincar\'e problem and the existence of minimal sets for foliations.