Meromorphic functions, bifurcation sets and fibred links

Bodin, Arnaud; Pichon, Anne

doi:10.4310/mrl.2007.v14.n3.a6

Cited by 18 publications

(30 citation statements)

References 12 publications

(32 reference statements)

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“…In Sect. 5, we compare the geometry of the Milnor-Lê fibration fḡ : L X \ Int(T ε,η ) → S 1 η with the Milnor fibration φ f /g : L X \L f g → S 1 of the meromorphic function f /g defined in [4] by…”

Section: Introductionmentioning

confidence: 99%

Fibred multilinks and singularities $${f \overline g}$$

Pichon

Seade

2008

Math. Ann.

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In this article we extend Milnor's fibration theorem to the case of functions of the form fḡ with f , g holomorphic, defined on a complex analytic (possibly singular) germ (X, 0). We further refine this fibration theorem by looking not only at the link of { fḡ = 0}, but also at its multi-link structure, which is more subtle. We mostly focus on the case when X has complex dimension two. Our main result (Theorem 4.4) gives in this case the equivalence of the following three statements:(i) The real analytic germ fḡ : (X, p) → (R 2 , 0) has 0 as an isolated critical value; (ii) the multilink L f ∪ −L g is fibered; and (iii) if π :X → X is a resolution of the holomorphic germ f g : (X, p) → (C, 0), then for each rupture vertex ( j) of the decorated dual graph of π one has that the corresponding multiplicities of f, g satisfy:Moreover one has that if these conditions hold, then the Milnor-Lê fibration fḡ : L X \(L f ∪ L g ) → S 1 η of fḡ is a fibration of the multilink L f ∪ −L g . We also give a combinatorial criterium to decide whether or not the multilink L f ∪ −L g is fibered.If the meromorphic germ f /g is semitame, then we show that the Milnor-Lê fibration

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Section: Introductionmentioning

confidence: 99%

Fibred multilinks and singularities $${f \overline g}$$

Pichon

Seade

2008

Math. Ann.

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“…The following lemmas are easily obtained by adapting the proofs of Lemmas 4.3 and 4.4 in [8] as already performed in [9] and [1] in close situations.…”

Section: Preliminary Lemmasmentioning

confidence: 99%

“…In fact, it is proved in [1] that if the meromorphic germ F = f /g is semitame (see the definition in Section 2), then…”

Section: Introductionmentioning

confidence: 99%

“…The statement that (2) is a fibre bundle was proved later in [6] by Lê Dũng Tráng in the more general setting of holomorphic maps defined on arbitrary complex analytic spaces, and we call it the Milnor-Lê fibration of f . Once we know that (2) is a fibre bundle, the arguments of [8,Chapter 5] show this is equivalent to the Milnor fibration (1).…”

Section: Introductionmentioning

confidence: 99%

“…The first of these questions was addressed in [12,1,13] from two different viewpoints, while the answer to the second question is the bulk of this article.…”

Section: Introductionmentioning

confidence: 99%

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