Sliding invariants and classification of singular holomorphic foliations in the plane

TL;DR

This paper introduces the set of slidings, a new invariant, to classify non-dicritical holomorphic foliations in the plane with non-rational Camacho-Sad indices, establishing finite determinacy and classification results.

Contribution

The paper presents a new invariant called the set of slidings, enabling a complete classification of certain holomorphic foliations and proving their finite determinacy.

Findings

Complete classification of non-dicritical foliations with non-rational Camacho-Sad indices.

Finiteness of the invariant determining the foliation class.

Finite determination of isoholonomic and dicritical foliations with the same Dulac maps.

Abstract

By introducing a new invariant called the set of slidings, we give a complete strict classification of the class of germs of non-dicritical holomorphic foliations in the plan whose Camacho-Sad indices are not rational. Moreover, we will show that, in this class, the new invariant is finitely determined. Consequently, the finite determination of the class of isoholonomic non-dicritical foliations and absolutely dicritical foliations that have the same Dulac maps are proved.