Formulas for Residues of Type Camacho-Sad and Applications

TL;DR

This paper derives formulas for Camacho-Sad residues in holomorphic foliations, relates them to geometric properties, and provides criteria for invariance and formulas for hypersurface singularities.

Contribution

It introduces new formulas for residues, links them to geometric invariants, and establishes conditions for invariance and singularity analysis in complex foliations.

Findings

Formulas for Camacho-Sad residues in holomorphic foliations.

Relation between residues and degrees of invariant subvarieties.

Conditions for a curve to be invariant under a projective foliation.

Abstract

In this paper, we provide formulas for the sum of residues of type Camacho-Sad of a holomorphic foliation with respect to an invariant analytic subvariety. As application, in context of projective foliations, we obtain a formula that relates the sum these residues with the degree and other characteristics of the invariant subvariety. Furthermore, we establish sufficient conditions ensuring that an irreducible curve is invariant by a projective foliation on $\mathbb{P}^2$. In addition, we provide an adjunction formula for hypersurfaces with non-isolated singularities and obtain explicit formulas for the Milnor number of these hypersurfaces.