Feuilletages holomorphes de codimension 1: une étude locale dans le cas dicritique

Cerveau, Dominique; Neto, Alcides Lins; Ravara-Vago, Marianna

doi:10.48550/arxiv.1404.2069

Cited by 1 publication

(1 citation statement)

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“…We know that any non dicritical germ of codimension one foliation F in (C 3 , 0) always has an invariant germ of analytic surface, as proved in [7] (the result is also true in higher ambient dimension [10]). Following a local version of Brunella's alternative [15] and a conjecture of D. Cerveau [14] we ask whether any germ of codimension one foliation F over (C 3 , 0) without invariant germ of surface satisfies the following property:…”

Section: Introductionmentioning

confidence: 99%

Partial separatrices and local Brunella's alternative

Cano,

Ravara-Vago

2014

Preprint

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Here we state a conjecture concerning a local version of Brunella's alternative: any codimension one foliation in (C 3 , 0) without germ of invariant surface has a neighborhood of the origin formed by leaves containing a germ of analytic curve at the origin. We prove the conjecture for the class of codimension one foliations whose reduction of singularities is obtained by blowingup points and curves of equireduction and such that the final singularities are free of saddle-nodes. The concept of "partial separatrix" for a given reduction of singularities has a central role in our argumentations, as well as the quantitative control of the generic Camacho-Sad index in dimension three. The "nodal components" are the only possible obstructions to get such germs of analytic curves. We use the partial separatrices to push the leaves near a nodal component towards compact diacritical divisors, finding in this way the desired analytic curves.

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