Formulae for indices of holomorphic foliations via reduction of singularities

TL;DR

This paper derives explicit formulas for key invariants of holomorphic foliations on complex surfaces during singularity reduction, extending previous results to all foliations and unifying various classical indices.

Contribution

It provides general explicit formulas for invariants like discrepancy vector and Milnor numbers for arbitrary foliations, expanding on prior work limited to generalized curve cases.

Findings

Explicit formulas for discrepancy vector, Milnor numbers, and indices along separatrices.

Unified numerical framework recovering classical results by Brunella and Cavalier-Lehmann.

Intrinsic characterizations of generalized curve and second type foliations using these invariants.

Abstract

We study numerical invariants associated with the reduction of singularities of holomorphic foliation germs on $(\mathbb{C}^2, 0)$. Building on our previous work on generalized curve foliations, we extend explicit formulas for several fundamental invariants to arbitrary foliations. In particular, we provide general expressions for the discrepancy vector, the Milnor and intrinsic Milnor numbers, and classical indices along a separatrix as Camacho-Sad, Variation, G\'omez-Mont-Seade-Verjovsky and also the Baum-Bott index. These extensions require a careful analysis of the contributions of saddle-nodes arising in the desingularization process. As applications, we recover results of Brunella and Cavalier-Lehmann, as well as a related statement appearing in [8], within a unified and purely numerical framework. Furthermore, we obtain intrinsic characterizations of generalized curve foliations in terms of indices and of second type foliations in terms of the discrepancy vector.