# Residue-type indices and holomorphic foliations

## TL;DR

This paper studies residue indices for plane holomorphic foliations, characterizes second type foliations via indices, and applies findings to logarithmic foliations on complex surfaces.

## Contribution

It introduces a new characterization of second type foliations using residue-type indices and applies these results to complex surface foliations.

## Key findings

- Characterization of second type foliations through residue indices
- Expression involving Baum-Bott, variation, and polar excess indices
- Application to logarithmic foliations on compact complex surfaces

## Abstract

We investigate residue-type indices for germs of holomorphic foliations in the plane and characterize second type foliations - those not containing tangent saddle-nodes in the reduction of singularities - by an expression involving the Baum-Bott, variation and polar excess indices. These local results are applied in the study of logarithmic foliations on compact complex surfaces.