Foliations with Algebraic Limit Sets

Camacho, César; Neto, Alcides Lins; Sad, Paulo

doi:10.2307/2946610

Cited by 65 publications

(99 citation statements)

References 6 publications

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“…The proof of the theorem is in two steps : first we prove the existence of an algebraic leaf L € F (i.e., a leaf whose closure is an algebraic curve), using the theory of harmonic measures [Gar], then we analyze the structure of L in order to apply the constructions of [CLS2].…”

Section: Foliations On the Complex Projective Plane With Many Parabolmentioning

confidence: 99%

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Foliations on the complex projective plane with many parabolic leaves

Brunella¹

1994

Annales De L’institut Fourier

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Section: Foliations On the Complex Projective Plane With Many Parabolmentioning

confidence: 99%

“…Because G is abelian and contains hyperbolic germs (even when L Ĉ *), we may apply the methods of [CLS2] …”

Section: Holonomy Of L and Linearization Of Fmentioning

confidence: 99%

Foliations on the complex projective plane with many parabolic leaves

Brunella¹

1994

Annales De L’institut Fourier

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“…In the codimension n = 1, case these are subgroups of germs of one variable holomorphic maps and there is a wellestablished dictionary relating topological and dynamical properties of (the leaves of) the foliation to algebraic properties of the group. This is clear in works as (Camacho et al 1992), (Il'yashenko 1978), (Nakai 1994) and (Scherbakov 2006). All these facts are compiled in some works relating the existence of suitable "transverse structures" for the foliation with…”

Section: Introductionmentioning

confidence: 95%

On groups of formal diffeomorphisms of several complex variables

Martelo

Scárdua

2012

An. Acad. Bras. Ciênc.

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In this note we announce some results in the study of groups of formal or germs of analytic diffeomorphisms in several complex variables. Such groups are related to the study of the transverse structure and dynamics of Holomorphic foliations, via the holonomy group notion of a foliation's leaf. For dimension one, there is a well-established dictionary relating analytic/formal classification of the group, with its algebraic properties (finiteness, commutativity, solvability, among others). Such system of equivalences also characterizes the existence of suitable integrating factors, i.e., invariant vector fields and one-forms associated to the group. Our aim is to state the basic lines of such dictionary for the case of several complex variables groups. Our results are applicable in the construction of suitable integrating factors for holomorphic foliations with singularities. We believe they are a starting point in the study of the connection between Liouvillian integration and transverse structures of holomorphic foliations with singularities in the case of arbitrary codimension. The results in this note are derived from the PhD thesis "Grupos de germes de difeomorfismos complexos em várias variáveis e formas diferenciais" of the first named author (Martelo 2010).

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“…-H is an abelian group. Since it contains a hyperbolic attractor by hypothesis, H can be made linear in some appropriate coordinate and therefore it is possible to construct a Darboux integral for F (see [1]). -H is a nonabelian group.…”

Section: Solvable Holonomy Groups and Foliationsmentioning

confidence: 99%

“…s be the complements in γ s of two opposite half lines re iθ0 , re i(θ0+π) , 0 ≤ r ≤ r 0 , and U (1) , U (2) be obtained by saturating Γ (2) , and add to our collection of affine coordinates the elements {t U (1) , Γ…”

Section: Theoremmentioning

confidence: 99%