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On some indices of foliations and applications

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

TL;DR

This work builds a unified framework linking the Milnor number, GSV-index, multiplicity, and Tjurina number for foliations on $\mathbb{P}^2_{\mathbb{C}}$, both locally along invariant curves and globally. It provides new local formulas for the Tjurina number, proves a foliation-independent relation between multiplicity and Tjurina number along reduced curves, and connects these with the GSV-index through precise equalities and bounds. The authors derive global consequences, delivering new proofs of classical results by Cerveau–Lins Neto and Soares, establishing bounds for the global Tjurina number, and addressing the Alcántara–Mozo-Fernández conjecture, including a negative answer for irreducible and reduced curves. Central to the approach are the GSV-index, balanced divisors of separatrices, and the χ-number, which together yield sharp inequalities (notably a bound of $\mu_p(\mathcal{F})/(\tau_p(\mathcal{F},\mathcal{B}_0)+\chi_p(\mathcal{F})-\mu_p(\mathcal{F},\mathcal{B}_{\infty})+1) < 4/3$ under suitable hypotheses) and explicit global formulas for $\tau(\mathcal{F},C)$. The results advance the understanding of how local index data constrain global foliation geometry on projective planes.

Abstract

In this paper we establish a relationship between the Milnor number, the $χ$-number, and the Tjurina number of a foliation with respect to an effective balanced divisor of separatrices. Moreover, using the Gómez-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 Alcántara and Mozo-Fernández about foliations on the complex projective plane having a unique singularity.

On some indices of foliations and applications

TL;DR

This work builds a unified framework linking the Milnor number, GSV-index, multiplicity, and Tjurina number for foliations on , both locally along invariant curves and globally. It provides new local formulas for the Tjurina number, proves a foliation-independent relation between multiplicity and Tjurina number along reduced curves, and connects these with the GSV-index through precise equalities and bounds. The authors derive global consequences, delivering new proofs of classical results by Cerveau–Lins Neto and Soares, establishing bounds for the global Tjurina number, and addressing the Alcántara–Mozo-Fernández conjecture, including a negative answer for irreducible and reduced curves. Central to the approach are the GSV-index, balanced divisors of separatrices, and the χ-number, which together yield sharp inequalities (notably a bound of under suitable hypotheses) and explicit global formulas for . The results advance the understanding of how local index data constrain global foliation geometry on projective planes.

Abstract

In this paper we establish a relationship between the Milnor number, the -number, and the Tjurina number of a foliation with respect to an effective balanced divisor of separatrices. Moreover, using the Gómez-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 Alcántara and Mozo-Fernández about foliations on the complex projective plane having a unique singularity.

Paper Structure

This paper contains 16 sections, 16 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Basic tools
  3. GSV-index
  4. The multiplicity of a foliation along a separatrix.
  5. The Tjurina number of a foliation along an invariant reduced curve
  6. Local formulas for the Tjurina number of a foliation
  7. The GSV-index, the multiplicity and the Tjurina number of a foliation
  8. Values set of fractional ideals and the Tjurina number of a foliation
  9. Multiplicity of a foliation along a divisor of separatrices
  10. Proof of Theorem \ref{['DG']}
  11. Proof of Theorem A
  12. Applications to global foliations
  13. New proof of a global result of Cerveau-Lins Neto
  14. New proof of a global result of Soares
  15. Bounds of the global Tjurina number of a foliation
  16. ...and 1 more sections

Key Result

Theorem A

Let $\mathcal{F}$ be a germ of a singular holomorphic foliation at $(\mathbb{C}^2,p)$, and $\mathcal{B}$ be an effective primitive balanced divisor of separatrices for $\mathcal{F}$ at $p$. Then Moreover, if $\mathcal{F}$ is of second type then $\frac{\mu_{p}(\mathcal{F})}{\tau_{p}(\mathcal{F},\mathcal{B})}<\frac{4}{3}.$ In particular, if $\mathcal{F}$ is a non-dicritical foliation of second type

Theorems & Definitions (34)

  • Theorem A
  • Proposition 3.1
  • proof
  • Example 3.2
  • Proposition 4.1
  • proof
  • Corollary 4.2
  • proof
  • Corollary 4.3
  • proof
  • ...and 24 more