The boundary of the Milnor fibre of complex and real analytic non-isolated singularities

TL;DR

This paper investigates the conditions under which certain real analytic map-germs derived from complex holomorphic functions have Milnor fibrations, describing their boundaries and providing formulas for their monodromy zeta functions.

Contribution

It establishes necessary and sufficient conditions for Milnor fibrations of these map-germs and characterizes their boundary as Waldhausen manifolds, introducing new geometric descriptions and formulas.

Findings

Conditions for Milnor fibration existence are identified.

Boundaries of Milnor fibres are Waldhausen manifolds.

A new A'Campo-type formula for monodromy zeta functions is derived.

Abstract

Let f and g be holomorphic function-germs vanishing at the origin of a complex analytic germ of dimension three. Suppose that they have no common irreducible component and that the real analytic map-germ given by the multiplication of f by the conjugate of g has an isolated critical value at 0. We give necessary and sufficient conditions for the real analytic map-germ to have a Milnor fibration and we prove that in this case the boundary of its Milnor fibre is a Waldhausen manifold. As an intermediate milestone we describe geometrically the Milnor fibre of map-germs given by the multiplication of a holomorphic and a anti-holomorphic function defined in a complex surface germ, and we prove an A'Campo-type formula for the zeta function of their monodromy.