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On Milnor and Tjurina numbers of foliations

Arturo Fernández-Pérez, Evelia R. García Barroso, Nancy Saravia-Molina

TL;DR

The paper develops a unified framework connecting Milnor and Tjurina numbers for singular plane foliations with balanced divisors of separatrices. It introduces the $\\chi$-number as a $C^{1}$-invariant and derives a polar-excess formula that ties $\\mu_p(\\mathcal{F})$ to separatrix data and $\\chi_p(\\mathcal{F})$, extending Cano’s plane-curve relations to the dicritical setting. It then generalizes the Gómez-Mont-Seade-Verjovski index to formal invariant curves and relates it to polar excess, multiplicities, and Tjurina numbers, yielding a comprehensive suite of equalities and inequalities linking Milnor, Tjurina, and various residue-type indices (BB, CS, Var). The results culminate in global bounds for the Milnor sums on projective planes, providing tools to bound local singularities from global invariants. Overall, the work deepens the understanding of how analytic and topological invariants of foliations interact with divisors of separatrices across both dicritical and non-dicritical regimes, with concrete criteria for generalized curve foliations and explicit Milnor-Tjurina relationships.

Abstract

We study the relationship between the Milnor and Tjurina numbers of a singular foliation $\mathcal{F}$, in the complex plane, with respect to a balanced divisor of separatrices $\mathcal{B}$ for $\mathcal{F}$. For that, we associate with $\mathcal{F}$ a new number called the $χ$-number and we prove that it is a $C^{1}$ invariant for holomorphic foliations. We compute the polar excess number of $\mathcal{F}$ with respect to a balanced divisor of separatrices $\mathcal{B}$ for $\mathcal{F}$, via the Milnor number of the foliation, the multiplicity of some hamiltonian foliations along the separatrices in the support of $\mathcal{B}$ and the $χ$-number of $\mathcal{F}$. On the other hand, we generalize, in the plane case and the formal context, the well-known result of Gómez-Mont given in the holomorphic context, which establishes the equality between the GSV-index of the foliation and the difference between the Tjurina number of the foliation and the Tjurina number of a set of separatrices of $\mathcal{F}$. Finally, we state numerical relationships between some classic indices, as Baum-Bott, Camacho-Sad, and variational indices of a singular foliation and its Milnor and Tjurina numbers; and we obtain a bound for the sum of Milnor numbers of the local separatrices of a holomorphic foliation on the complex projective plane.

On Milnor and Tjurina numbers of foliations

TL;DR

The paper develops a unified framework connecting Milnor and Tjurina numbers for singular plane foliations with balanced divisors of separatrices. It introduces the -number as a -invariant and derives a polar-excess formula that ties to separatrix data and , extending Cano’s plane-curve relations to the dicritical setting. It then generalizes the Gómez-Mont-Seade-Verjovski index to formal invariant curves and relates it to polar excess, multiplicities, and Tjurina numbers, yielding a comprehensive suite of equalities and inequalities linking Milnor, Tjurina, and various residue-type indices (BB, CS, Var). The results culminate in global bounds for the Milnor sums on projective planes, providing tools to bound local singularities from global invariants. Overall, the work deepens the understanding of how analytic and topological invariants of foliations interact with divisors of separatrices across both dicritical and non-dicritical regimes, with concrete criteria for generalized curve foliations and explicit Milnor-Tjurina relationships.

Abstract

We study the relationship between the Milnor and Tjurina numbers of a singular foliation , in the complex plane, with respect to a balanced divisor of separatrices for . For that, we associate with a new number called the -number and we prove that it is a invariant for holomorphic foliations. We compute the polar excess number of with respect to a balanced divisor of separatrices for , via the Milnor number of the foliation, the multiplicity of some hamiltonian foliations along the separatrices in the support of and the -number of . On the other hand, we generalize, in the plane case and the formal context, the well-known result of Gómez-Mont given in the holomorphic context, which establishes the equality between the GSV-index of the foliation and the difference between the Tjurina number of the foliation and the Tjurina number of a set of separatrices of . Finally, we state numerical relationships between some classic indices, as Baum-Bott, Camacho-Sad, and variational indices of a singular foliation and its Milnor and Tjurina numbers; and we obtain a bound for the sum of Milnor numbers of the local separatrices of a holomorphic foliation on the complex projective plane.
Paper Structure (8 sections, 31 theorems, 122 equations, 3 figures)

This paper contains 8 sections, 31 theorems, 122 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Basic tools
  3. The $\chi$-number of a foliation
  4. Polar intersection, polar excess and Milnor numbers
  5. The Gómez-Mont-Seade-Verjovsky index
  6. Tjurina number
  7. Milnor and Tjurina numbers and some residue-type indices
  8. Bound for the Milnor sum of an algebraic curve

Key Result

Theorem A

Let $\mathcal{F}$ be a singular foliation at $(\mathbb C^2,p)$ and let $\mathcal{B}=\sum_{B}a_B B$ be a balanced divisor of separatrices for $\mathcal{F}$. Then where for each separatrix $B$, ${F}_B$ is a balanced divisor of separatrices for $\mathcal{F}$ adapted to $B$. Moreover, if $\mathcal{F}$ is a foliation of the second type, then

Figures (3)

  • Figure 1: Dicritical foliation $\mathcal{F}_k$
  • Figure 2: Dulac foliation $\mathcal{F}: \omega=(ny+x^{n})dx-xdy$
  • Figure 3: Foliation ${\mathcal{F}}: \omega=4xydx+(y-2x^{2})dy$

Theorems & Definitions (66)

  • Theorem A
  • Corollary B
  • Theorem C
  • Corollary D
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 3.1
  • proof
  • ...and 56 more