Local Brunella's Alternative I. RICH Foliations

Cano, Felipe; Ravara-Vago, Marianna; Soares, Marcio G.

doi:10.1093/imrn/rnu011

Cited by 8 publications

(9 citation statements)

References 14 publications

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“…It can be stated as follows: Given a codimension one holomorphic foliation F at (C 3 , 0), either F admits an invariant analytic surface through p, or any leaf of F in a punctured neighborhood of the origin contains an analytic curve through 0, except for the point 0 itself. In the joint papers [20,21] with Marianna Ravara and Marcio Soares, Felipe proved that for some special foliations the local version of Brunella's conjecture has a positive answer.…”

Section: Study Of Dicriticalness Property For Holomorphic Foliationsmentioning

confidence: 99%

Singular Felipe

Hernández-Ros

Sánchez

2019

RACSAM

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We review the remarkable and singular career of Felipe Cano in the study of Singularities. Keywords Singularities • Reduction of singularities • Foliations • Vector fields Mathematics Subject Classification 32Sxx • 14Exx • 14Bxx This issue contains original papers dedicated to Felipe Cano after a Conference in occasion of his 60th birthday, which took place in Santander in June 2017. The title of the Conference was "Singular Felipe", and we are going to explain the reasons for it. If we consult the Oxford Dictionary, there are several meanings for the word singular: 1. Unique. 2. Exceptionally good, remarkable. 3. Strange or eccentric in some aspect. In our opinion, and we are sure that also in the opinion of all his friends, the three meanings are well adapted to Felipe Cano. Clearly Felipe is singular, unique, some of us could say hopefully, because if there where two, we would all become crazy. Seriously, after a lot of years knowing Felipe and working with him, we all know that there are few people comparable to him, he is exceptionally good as person and as mathematician, and this justifies the second meaning of "singular". At last if we ask to his students in mathematics in Valladolid, either undergraduate or graduate, all of them would use the word eccentric, or even something worst, to describe the pedagogical methods of Felipe. But as far as we know these eccentric methods work perfectly. Felipe is also singular by the number of people with whom he has collaborated: he has joint publications with 26 mathematicians,

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Section: Study Of Dicriticalness Property For Holomorphic Foliationsmentioning

confidence: 99%

Singular Felipe

Hernández-Ros

Sánchez

2019

RACSAM

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“…Invariant Hypersurfaces and Partial Separatrices. Following [2,3,6], we recall here the description of the set of invariant hypersurfaces of a desingularized GH-foliated space (M, F ), in terms of the so called "partial separatrices".…”

Section: 3mentioning

confidence: 99%

“…Let N be the union of the nodal type irreducible components Γ of Sing(F ). A connected component B of N is a nodal separating block when it only intersects nodal and real saddle type irreducible components of the singular locus ( in [5,6] we call these sets "uninterrupted nodal components"). The nodal separator set S is the union of all nodal separating blocks.…”

Section: Introductionmentioning

confidence: 99%

Invariant Hypersurfaces and Nodal Components of Foliations

Cano¹,

Mattéi²,

Ravara-Vago³

2019

Preprint

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It is known that there is at least an invariant analytic curve passing through each of the components in the complement of nodal singularities, after the reduction of singularities of a germ of singular foliation in C 2 , 0. Here, we state and prove a generalization of this property to any ambient dimension. Contents 1. Introduction 2. Preliminaries 3. Standard Ambient Spaces 4. Desingularized GH-Foliated Spaces 4.1. Simple Points 4.2. Reduction of Singularities 4.3. Invariant Hypersurfaces and Partial Separatrices 5. Nodal Separating Blocks 5.1. Nodally Reduced Foliated Spaces 5.2. Connection Outside of the Separator Set 6. The complement of Nodal Separating blocks 6.1. Combinatorial Strata Structures 6.2. Abstract Datum of Nodal Strata 6.3. Parity Results 6.4. Number of Connected Components 7. Combinatorially Simply connected Ambient Spaces 7.1. The Topological Simplicial Complex 7.2. Combinatorial Strata Sructure After Blow-up 7.3. Stability of Simple Connectedness 8. Invariant Hypersurfaces and Separator Set 8.1. Structure of the proof 8.2. Two-Equireduction 8.3. Reduction to dimension two 8.4. End of the proof 9. Appendix: Simply Connectedness 10. Appendix: Strong Desingularization in Dimension Three References

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“…We have two possibilities (a) λ = 0. This is a complex hyperbolic singularity of corner type, following the terminology of [8]. (b) λ = 0.…”

Section: Recall On Local Invariants and Reduction Of Singularitiesmentioning

confidence: 99%

“…The terminology comes from previous papers [5,22,8]. Note that F is of second type if and only if the strong separatrices correspond to complex hyperbolic trace points and all the corners are also complex hyperbolic.…”

Section: Recall On Local Invariants and Reduction Of Singularitiesmentioning

confidence: 99%

Local polar invariants for plane singular foliations

Cano¹,

Corral²,

Mol³

2015

Preprint

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In this survey paper, we take the viewpoint of polar invariants to the local and global study of non-dicritical holomorphic foliations in dimension two and their invariant curves. It appears a characterization of second type foliations and generalized curve foliations as well as a description of the GSV -index in terms of polar curves. We also interpret the proofs concerning the Poincaré problem with polar invariants.

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Local Brunella's Alternative I. RICH Foliations

Cited by 8 publications

References 14 publications

Singular Felipe

Singular Felipe

Invariant Hypersurfaces and Nodal Components of Foliations

Local polar invariants for plane singular foliations

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