Fibered Multilinks and singularities $f\bar{g}$

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 $f \bar g$havinganisolatedcriticalvalueat$0 \in C$. 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 $\frac{f \bar g}{|f \bar g|}$definedon$L_X \setminus L_{f \bar g}$ 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.