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

Abstract

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

Singular open book structures from real mappings

“…Several recent papers considered this situation with respect to the existence of the Milnor fibration (1), see e.g. [2,4,18]. Let (b) This fibration has a regularity property in the neighbourhood of the link, namely, it is induced by a more refined fibration with codimension fibres.…”

“…Suppose that , where the singular locus was of type A. We discuss in Section 3 other particular cases and the relations to [18].…”

Milnor fibrations of meromorphic functions

“…It is shown in [13] that Φ is a fibration of the multilink L f ∪ −L g , where −L g means L g with the opposite orientation.…”

“…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.…”

The boundary of the Milnor fibre of complex and real analytic non-isolated singularities

“…is the projection of a locally trivial fibration, where ∆ is the polydisk in the ambient space C N ⊃ X defined by [15] that if f and g are holomorphic functiongerms defined on a holomorphic germ of surface X with no common irreducible components and if the real analytic map-germ fḡ : (X, 0) → (C) has an isolated critical value, then fḡ has the Thom a f condition and therefore it has a Milnor fibration in the tube. They also conjectured that this was true in higher dimensions and latter they gave a proof to this conjecture in [16], which turned out to be false, as pointed by A. Parusinski with a counter-example (see [17]).…”