Un théorème de Frobenius singulier via l'arithmétique élémentaire

Cerveau, Dominique; Loray, Frank

doi:10.1006/jnth.1997.2202

Cited by 13 publications

(22 citation statements)

References 8 publications

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“…Natural examples of such groups of germ maps are given by holonomy groups and monodromy groups of integrable systems (foliations) under certain conditions. We prove some finiteness results for these groups extending previous results in [3]. Applications are given to the framework of germs of holomorphic foliations.…”

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confidence: 59%

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Irreducible holonomy groups and first integrals for holomorphic foliations

2020

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We study groups of germs of complex diffeomorphisms having a property called irreducibility. The notion is motivated by the similar property of the fundamental group of the complement of an irreducible hypersurface in the complex projective space. Natural examples of such groups of germ maps are given by holonomy groups and monodromy groups of integrable systems (foliations) under certain conditions. We prove some finiteness results for these groups extending previous results in [3]. Applications are given to the framework of germs of holomorphic foliations. We prove the existence of first integrals under certain irreducibility or more general conditions on the tangent cone of the foliation after a punctual blow-up.2000 Mathematics Subject Classification. Primary 37F75, 57R30; Secondary 32M25, 32S65.

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confidence: 59%

“…Using these ideas and some features from arithmetics, in [3] the authors prove a finiteness theorem for groups of germs of complex diffeomorphisms in one variable. This reads as follows: Theorem 1.1 ([3], Theorem 1 page 221).…”

Section: Introductionmentioning

confidence: 99%

Irreducible holonomy groups and first integrals for holomorphic foliations

2020

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“…We now apply the following lemma whose proof is found in [3] (page 222): This produces a convergent algorithm in the Krull topology. This already shows that G is formally linearizable.…”

Section:  mentioning

confidence: 99%

Irreducible holonomy groups and Riccati foliations in higher complex dimension

León¹,

Martelo²,

Scárdua³

2019

jsing

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We study groups of germs of complex diffeomorphisms having a property called irreducibility. The notion is motivated by a similar property of the fundamental group of the complement of an irreducible hypersurface in the complex projective space. Natural examples of such groups of germ maps are given by holonomy groups and monodromy groups of integrable systems (foliations) under certain conditions on the singular or ramification set. The case of complex dimension one is studied in [7] where finiteness is proved for irreducible groups under certain arithmetic hypothesis on the linear part. In dimension n ≥ 2 the picture changes since linear groups are not always abelian in dimension two or bigger. Nevertheless, we still obtain a finiteness result under some conditions in the linear part of the group, for instance if the linear part is abelian. Examples are given illustrating the role of our hypotheses. Applications are given to the framework of holomorphic foliations and analytic deformations of rational fibrations by Riccati foliations.

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“…Voici deux résultats dus à D. Cerveau et F. Loray [11] et D. Cerveau et J.-F. Mattei [12] que nous concentrons dans un seul et qui donnent un aperçu raisonnable des feuilletages F = F ω lorsque ω ν est non dicritique logarithmique. Théorème 4.…”

Section: Notations Définitions Et Rappelsunclassified

“…Modulo des conditions génériques portant sur ω ν , ces propriétés sont gardées en mémoire par les ω telles que In(ω) = ω ν . Ainsi, dans le cas non dicritique, si [P ν+1 = 0] ⊂ P n−1 C est réduit et à croisement normaux, alors F ω est encore défini par une 1−forme fermée méromorphe dès que les résidus λ i de ω ν = In(ω) satisfont une condition générique (satisfaite sur un ouvert dense) ; de même si [P ν+1 = 0] est irréductible de degré p s , avec p premier et n ≥ 3 [11] : en fait dans ce cas ω possède une intégrale première holomorphe non constante.…”

Section: Introductionunclassified