Residue-type indices and holomorphic foliations

Fernández‐Pérez, Arturo; Mol, Rogério

doi:10.2422/2036-2145.201703_006

Cited by 3 publications

(3 citation statements)

References 25 publications

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“…Remark that ξ p (F ) ≥ 0 and, by definition, ξ p (F ) = 0 if and only if SN(F ) = ∅, that is, if and only if F is of second type. In several papers (see for example [10], [4]) the tangency excess of F is denoted by τ p (F ). In this paper, we denote it by ξ p (F ) since we keep the letter τ for the Tjurina number of a curve or a foliation.…”

Section: Basic Toolsmentioning

confidence: 99%

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On Milnor and Tjurina numbers of foliations

Fernández-Pérez¹,

Barroso²,

Saravia-Molina³

2021

Preprint

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We study the relationship between the Milnor and Tjurina numbers of a singular foliation F , in the complex plane, with respect to a balanced divisor of separatrices B for F . For that, we associated with 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 F with respect to a balanced divisor of separatrices B for F , via the Milnor number of the foliation, the multiplicity of some hamiltonian foliations along the separatrices in the support of B and the χnumber of 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 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.

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Section: Basic Toolsmentioning

confidence: 99%

“…We say that α is an exponent form for ω. The variational index or variation of F relative to C at p is In order to establish a relationship between the residue-type indices defined in this section, we use the following theorem given in [10,Theorem 5.2]. where the summation runs over all infinitely near points of F at p.…”

Section: Tjurina Numbermentioning

confidence: 99%

On Milnor and Tjurina numbers of foliations

Fernández-Pérez¹,

Barroso²,

Saravia-Molina³

2021

Preprint

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“…Proof. We apply the two-dimensional version of this result proved in [8]. Let i : P 2 → P 3 be a linear embedding generically transversal to F, set G = i * F and identify P 2 and the plane H = i(P 2 ).…”

Section: Logarithmic Foliationsmentioning

confidence: 99%

Second type foliations of codimension one

Cuzzuol

Mol

2018

Bol. Soc. Mat. Mex.

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In this article, for holomorphic foliations of codimension one at (C 3 , 0), we define the family of second type foliations. This is formed by foliations having, in the reduction process by blow-up maps, only well oriented singularities, meaning that the reduction divisor does not contain weak separatrices of saddle-node singularities. We prove that the reduction of singularities of a non-dicritical foliation of second type coincides with the desingularization of its set of separatrices.

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Differentiable Invariants of Holomorphic Foliations

Rosas

2022

Bull Braz Math Soc, New Series

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Residue-type indices and holomorphic foliations

Cited by 3 publications

References 25 publications

On Milnor and Tjurina numbers of foliations

On Milnor and Tjurina numbers of foliations

Second type foliations of codimension one

Differentiable Invariants of Holomorphic Foliations

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