Meromorphic functions, bifurcation sets and fibred links

For a polynomial $f$ in two complex variables, we prove that the multi-link at infinity of the 0-fiber $f^{-1}(0)$ is a fibred multi-link if and only if all the values different from 0 are regular at infinity.

In this article we investigate mixed polynomials and present conditions that can be applied on a specific class of polynomials in order to prove the existence of the Milnor Fibration, Milnor-L\^e Fibration and the equivalence between them. We prove for this class the of functions that the Milnor-L\^e fiber on a regular value is homeomorphic to the Milnor-L\^e fiber on a critical value. We develop a criterion to verify the transversality property and apply it to a special case...

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 previo...

In this paper we present new results about the topology of the Milnor fibrations of analytic function-germs with a special attention to the topology of the fibers. In particular, we provide a short review on the existence of the Milnor fibrations in the real and complex cases. This allows us to compare our results with the previous ones.

We consider a real analytic map $F=(f_1,...,f_k) : (\mathbb{R}^n,0) \rightarrow (\mathbb{R}^k,0)$, $2 \le k \le n-1$, that satisfies Milnor's conditions (a) and (b) introduced by D. Massey. This implies that every real analytic $f_I=(f_{i_1},...,f_{i_l}) : (\mathbb{R}^n,0) \rightarrow (\mathbb{R}^l,0)$, induced from $F$ by projections where $1 \le l \le n-2$ and $I=\{i_1,...,i_l\}$, also satisfies Milnor's conditions (a) and (b). We give several relations between the Euler ch...

In this note, we prove the connectivity of the Milnor fiber for a mixed polynomial $f(\mathbf z,\bar{\mathbf z})$, assuming the existence of a sequence of smooth points of $f^{-1}(0)$ converging to the origin. This result gives also a another proof for the connectivity of the Milnor fiber of a non-reduced complex analytic function which is proved by A. Dimca

We study the boundary of the Milnor fibre of real analytic singularities $f: (\bR^m,0) \to (\bR^k,0)$, $m\geq k$, with an isolated critical value and the Thom $a_f$-property. We define the vanishing zone for $f$ and we give necessary and sufficient conditions for it to be a fibre bundle over the link of the singular set of $f^{-1}(0)$. In the case of singularities of the type $\fgbar: (\bC^n,0) \to (\bC,0)$ with an isoalted critical value, $f, g$ holomorphic, we further descr...

Milnor's fibration theorem is about the geometry and topology of real and complex analytic maps near their critical points, a ubiquitous theme in mathematics. As such, after 50 years, this has become a whole area of research on its own, with a vast literature, plenty of different viewpoints, a large progeny and connections with many other branches of mathematics. In this work we revisit the classical theory in both the real and complex settings, and we glance at some areas of...

A meromorphic function on a compact complex analytic manifold defines a $\bc\infty$ locally trivial fibration over the complement of a finite set in the projective line $\bc\bp^1$. We describe zeta-functions of local monodromies of this fibration around atypical values. Some applications to polynomial functions on $\bc^n$ are described.

We present several aspects of the "topology of meromorphic functions", which we conceive as a general theory which includes the topology of holomorphic functions, the topology of pencils on quasi-projective spaces and the topology of polynomial functions.