Differentiable equisingularity of holomorphic foliations

Mol, Rogério; Rosas, Rudy

doi:10.5427/jsing.2019.19f

Cited by 5 publications

(4 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [23,Proposition 9.5] the authors prove that the tangency excess is a C ∞ invariant, and after [24] the algebraic multiplicity of a holomorphic foliation is a C 1 invariant. Hence the χ-number of a holomorphic foliation is a C 1 invariant.…”

Section: The χ-Number Of a Foliationmentioning

confidence: 99%

On Milnor and Tjurina numbers of foliations

Fernández-Pérez¹,

Barroso²,

Saravia-Molina³

2021

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: The χ-Number Of a Foliationmentioning

confidence: 99%

On Milnor and Tjurina numbers of foliations

Fernández-Pérez¹,

Barroso²,

Saravia-Molina³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The first one is the tangency excess index (Definition 2.2), an invariant that computes the contributions of the orders of tangencies of saddle-node singularities along the divisor of the reduction of singularities. For a germ of complex analytic vector field, this is C ∞ -invariant [12]. In the real case, the fact that a vector field is a topological real generalized curve implies that its tangency excess index is even.…”

Section: Introductionmentioning

confidence: 99%

Separatrices for real analytic vector fields in the plane

Cabrera¹,

Mol²

2019

Preprint

Self Cite

View full text Add to dashboard Cite

Let X be a germ of real analytic vector field at (R 2 , 0) with an algebracally isolated singularity. We say that X is a topological generalized curve if there are no topological saddle-nodes in its reduction of singularities. In this case, we prove that if either the order ν0(X) or the Milnor number µ0(X) is even, then X has a formal separatrix, that is, a formal invariant curve at 0 ∈ R 2 . This result is optimal, in the sense that these hypotheses do not assure the existence of a convergent separatrix.

show abstract

“…Recently, second type foliations have been the object of some works. We should mention [10] -which deals with the "realization problem", that is, the existence of foliations with prescribed reduction of singularities and projective holonomy representations, [11] -which studies local polar invariants and applications to the study of the Poincaré problem for foliations -and [19] -where equisingularitiy properties are considered. Our main goal in this article is to give a characterization of second type foliations by means of residue-type indices, providing a generalization of Brunella's result.…”

Section: Introductionmentioning

confidence: 99%

Residue-type indices and holomorphic foliations

Fernández‐Pérez¹,

Mol²

2019

Annali Scuola Normale Superiore - Classe Di Scienze

Self Cite

View full text Add to dashboard Cite

We investigate residue-type indices for germs of holomorphic foliations in the plane and characterize second type foliations -those not containing tangent saddlenodes in the reduction of singularities -by an expression involving the Baum-Bott, variation and polar excess indices. These local results are applied in the study of logarithmic foliations on compact complex surfaces.2010 Mathematics Subject Classification. 32S65, 37F75.

show abstract

Differentiable equisingularity of holomorphic foliations

Cited by 5 publications

References 10 publications

On Milnor and Tjurina numbers of foliations

On Milnor and Tjurina numbers of foliations

Separatrices for real analytic vector fields in the plane

Residue-type indices and holomorphic foliations

Contact Info

Product

Resources

About