Milnor Fibrations and the Thom Property for maps $f \bar g$

TL;DR

This paper proves that certain complex map-germs with isolated critical values have the Thom property, extending known results to a broader class and establishing the existence of Milnor fibrations for these maps.

Contribution

It extends Hironaka's theorem to map-germs of the form f ḡ, showing they have the Thom property and Milnor fibrations, which was previously unknown.

Findings

Every map-germ f ḡ with isolated critical value has the Thom a_{f ḡ}-property.

Such map-germs admit Milnor-Lê fibrations on a Milnor tube.

The fibrations are locally trivial and explicitly described near the link K.

Abstract

We prove that every map-germ ${f \bar g}: (\C^n,\0) {\to}(\C,0)$ with an isolated critical value at 0 has the Thom $a_{f \bar g}$-property. This extends Hironaka's theorem for holomorphic mappings to the case of map-germs $f \bar g$ and it implies that every such map-germ has a Milnor-L\^e fibration defined on a Milnor tube. One thus has a locally trivial fibration $\phi: \mathbb S_\e \setminus K \to \mathbb S^1$ for every sufficiently small sphere around $\0$, where $K$ is the link of $f \bar g$ and in a neighbourhood of $K$ the projection map $\phi$ is given by $f \bar g / | f \bar g|$.