The C^1 invariance of the algebraic multiplicity of a holomorphic vector field

Rosas, Rudy

doi:10.5802/aif.2578

Cited by 4 publications

(3 citation statements)

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“…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

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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: The χ-Number Of a Foliationmentioning

confidence: 99%

On Milnor and Tjurina numbers of foliations

Fernández-Pérez¹,

Barroso²,

Saravia-Molina³

2021

Preprint

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show abstract

“…The final ingredient for the proof of Theorem II is the following result of [9]: Theorem 9.4. Let F and F ′ be germs at (C n , 0) of C 1 equivalent one dimensional foliations.…”

Section: ∞ Equivalences Of Foliations and Equisingularity Of The Set ...mentioning

confidence: 99%

“…In general, the validity of Theorem A outside the class of generalized curve foliations is a difficult open problem. Actually, such a result would imply the topological invariance of the algebraic multiplicity of a holomorphic foliation, which is also an open problem (see [8,9,10]). The desingularization of a germ of foliation F is closely related to the desingularization of its set of separatrices Sep(F) -including the purely formal ones -, although Theorem B is not always true.…”

Section: Introductionmentioning

confidence: 99%