Inner Metric Geometry of Complex Algebraic Surfaces with Isolated Singularities

TL;DR

This paper constructs examples of complex algebraic surfaces with isolated singularities that are not metrically conic, using Metric Homology, revealing new insights into the inner metric geometry of such surfaces.

Contribution

It introduces a method to demonstrate the nonexistence of metric conic structures in complex algebraic surfaces with isolated singularities, expanding understanding of their geometric properties.

Findings

Examples of non-metrically conic singularities are provided.

The technique relates to development of Metric Homology.

Includes surfaces of Brieskorn as part of the class.

Abstract

We produce examples of complex algebraic surfaces with isolated singularities such that these singularities are not metrically conic, i.e. the germs of the surfaces near singular points are not bi-Lipschitz equivalent, with respect to the inner metric, to cones. The technique used to prove the nonexistence of the metric conic structure is related to a development of Metric Homology. The class of the examples is rather large and it includes some surfaces of Brieskorn.