On some indices of foliations and applications

TL;DR

This paper explores relationships between various indices of foliations, providing new proofs, formulas, and bounds for invariants like the Milnor, Tjurina, and χ-numbers, with applications to foliations on complex projective planes.

Contribution

It establishes new relationships and formulas for foliation invariants, offers alternative proofs for known results, and addresses a conjecture on foliations with a single singularity.

Findings

Derived a local formula for the Tjurina number of a foliation.

Provided bounds for the global Tjurina number on the complex projective plane.

Confirmed the conjecture about foliations with a unique singularity.

Abstract

In this paper we establish a relationship between the Milnor number, the $\chi$-number, and the Tjurina number of a foliation with respect to an effective balanced divisor of separatrices. Moreover, using the G\'omez-Mont--Seade--Verjovsky index, we prove that the difference between the multiplicity and the Tjurina number of a foliation with respect to a reduced curve is independent of the foliation. We also derive a local formula for the Tjurina number of a foliation with respect to a reduced curve. From a global point of view, these results lead to the following consequences: we provide a new proof of a global result regarding the multiplicity of a foliation due to Cerveau-Lins Neto and a new proof of a Soares's inequality for the sum of the Milnor number of an invariant curve of a foliation. Additionally, we obtain bounds for the global Tjurina number of a foliation on the complex projective plane. Finally, we provide an answer to the conjecture posed by Alc\'antara and Mozo-Fern\'andez about foliations on the complex projective plane having a unique singularity.