The degeneration of the boundary of the Milnor fibre to the link of complex and real non-isolated singularities

TL;DR

This paper investigates the boundary behavior of Milnor fibres in real and complex singularities, establishing conditions for fiber bundle structures and describing degenerations to links, with explicit constructions for certain real analytic cases.

Contribution

It introduces the concept of vanishing zones, provides criteria for their fiber bundle structure, and constructs Lê's polyhedron for specific real analytic singularities.

Findings

Vanishing zones are fiber bundles over the link of the singular set under certain conditions.

Degeneration of Milnor fibre boundaries to links is characterized for complex conjugate singularities.

Explicit Lê's polyhedron construction for specific real analytic singularities.

Abstract

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 describe the degeneration of the boundary of the Milnor fibre to the link of $\fgbar$. As a milestone, we also construct a L\^e's polyhedron for real analytic singularities of the type $\fgbar: (\bC^2,0) \to (\bC,0)$ such that either $f$ or $g$ depends only on one variable.