L\^e's polyhedron for line singularities

TL;DR

This paper investigates the topology of line singularities in complex hypersurfaces, describing how their Milnor fibers degenerate to the singular fiber using polyhedra that serve as deformation retracts, providing a new perspective on complex singularities.

Contribution

It introduces a novel approach using polyhedra to understand the degeneration of Milnor fibers in line singularities and extends this perspective to isolated singularities.

Findings

Polyhedra serve as deformation retracts of Milnor and singular fibers.

A continuous map relates the Milnor fiber to the singular fiber via these polyhedra.

Provides a non-local topological analysis of complex hypersurface singularities.

Abstract

We study the topology of a line singularity, which is a complex hypersurface with non-isolated singularity given by a complex line. We describe the degeneration of its Milnor fibre to the singular hypersurface by means of a pair of polyhedra, one in the Milnor fibre and other in the singular fibre, which are deformation retracts of the corresponding fibres; and a continuous map taking the Milnor fibre to the singular fibre and the first polyhedron to the second one, which restrict to a homeomorphism outside the polyhedra. In the same sense, we also study the topology of a complex isolated singularity hypersurface under a non-local viewpoint.