On the Milnor fibration for $f(\mathbf z)\bar g(\mathbf z)$

TL;DR

This paper investigates the Milnor fibration for a specific class of mixed functions formed by the product of a holomorphic function and its conjugate, establishing conditions under which such functions admit Milnor fibrations.

Contribution

It demonstrates that mixed functions of the form $f(z)ar g(z)$ have tubular and spherical Milnor fibrations under certain conditions, extending the understanding of Milnor fibrations to mixed functions.

Findings

$H(z,\bar z)=f(z)\bar g(z)$ admits Milnor fibrations under multiplicity conditions.

Examples show that Milnor fibrations may or may not exist without the Newton multiplicity condition.

The paper provides explicit cases illustrating the presence or absence of Milnor fibrations.

Abstract

We consider a mixed function of type $H(\mathbf z,\bar {\mathbf z})=f(\mathbf z)\bar g(\mathbf z)$ where $f$ and $g$ are convenient holomorphic functions which have isolated critical points at the origin and we assume that the intersection $f=g=0$ is a complete intersection variety with an isolated singlarity at theorigin. We assume also that $H$ satisfies the multiplicity condition.We will show that $H$ has a tubular Milnor fibration and also a spherical Milnor fibration. We give examples which does not satisfy the Newton multiplicity condition where one does not have Milnor fibration and the others have Milnor fibrations.