A brief survey on residue theory of holomorphic foliations

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 Class 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 along the survey several recent results and advances in residue theory. We finish the work present some applications of residues to solve for example the Poincaré problem and the existence of minimal sets for foliations.