On the decomposition of a 2D-complex germ with non-isolated singularities

TL;DR

This paper presents a method to decompose a 2D complex germ with non-isolated singularities into semi-algebraic sets, revealing its geometric structure through a detailed classification of its components.

Contribution

It introduces a novel decomposition of 2D complex germs with non-isolated singularities into four semi-algebraic classes, providing a new understanding of their geometric and metric properties.

Findings

Decomposition into four classes: Riemannian cones, thickened tori, mapping tori, tubular neighborhoods.

Existence of semi-algebraic sets metrically conical over link manifolds.

Reconstruction of the germ up to bi-Lipschitz equivalence using the model.

Abstract

The decomposition of a two dimensional complex germ with non-isolated singularity into semi-algebraic sets is given. This decomposition consists of four classes: Riemannian cones defined over a Seifert fibered manifold, a topological cone over thickened tori endowed with Cheeger-Nagase metric, a topological cone over mapping torus endowed with Hsiang-Pati metric and a topological cone over the tubular neighbourhoods of the link's singularities. In this decomposition there exist semi-algebraic sets that are metrically conical over the manifolds constituting the link. The germ is reconstituted up to bi-Lipschitz equivalence to a model describing its geometric behavior.