Fibered Multilinks and singularities $f \bar g$

TL;DR

This paper extends Milnor's fibration theorem to certain complex singularities involving $f ar g$, providing new criteria and results for fibred multilinks in complex analytic singularities, especially in dimension two.

Contribution

It generalizes Milnor's theorem to singularities of the form $f ar g$, establishes conditions for fibred multilinks, and offers a combinatorial criterion and realization theorem for these multilinks.

Findings

Extended Milnor's fibration theorem to $f ar g$ singularities.

Established equivalence between isolated critical value and fibred multilink.

Provided a combinatorial criterion for fibred multilinks.

Abstract

In this article we extend Milnor's fibration theorem for complex singularities to the case of singularities $f \bar g:(X,P) \to (C,0))$ defined on a complex analytic singularity germ $(X,P)$, with $f, g$ holomorphic and $f \bar g$ having an isolated critical value at $0 \in C$. This can also be regarded as a result for meromorphic germs. Then we strenghten this fibration theorem when $X$ has complex dimension 2, obtaining a fibration theorem for multilinks that extends previous work by Pichon. We prove that the multilink $L_{f \bar g}$ in $L_X$ (the link of $X$), is fibred iff the map $f \bar g$ has an isolated critical value at $0 \in C$, and in this case the map $\frac{f \bar g}{|f \bar g|}$ defined on $L_X \setminus L_{f \bar g}$ is a multilink fibration.We also give a combinatorial criterium, easy to verify, to decide when is $L_{f \bar g}$ a fibred multilink. We finally prove a realization theorem for fibred multilinks.