On homotopy types of complements of analytic sets and Milnor fibres

TL;DR

This paper demonstrates that for any complex analytic germ, there exists a hypersurface singularity with trivial monodromy whose Milnor fiber is homotopic to the complement of the germ, revealing new topological insights.

Contribution

It constructs hypersurface singularities with trivial monodromy and Milnor fibers homotopic to analytic set complements, including examples with simply connected and non-formal fibers.

Findings

Existence of hypersurface singularities with trivial monodromy

Milnor fibers homotopic to complements of analytic germs

Example of a non-formal, simply connected Milnor fiber

Abstract

We prove that for any germ of complex analytic set in $\CC^n$ there exists a hypersurface singularity whose Milnor fibration has trivial geometric monodromy and fibre homotopic to the complement of the germ of complex analytic set. As an application we show an example of a quasi-homogeneous hypersurface singularity, with trivial geometric monodromy and simply connected and non-formal Milnor fibre.