Separating sets, metric tangent cone and applications for complex algebraic germs

TL;DR

This paper explores the role of separating sets in complex surface germs, their prevalence in higher dimensions, and their connection to metric tangent cones, revealing new insights into Lipschitz geometry and complex germ classification.

Contribution

It establishes the ubiquity of separating sets in higher-dimensional complex germs and links them to metric tangent cone geometry, demonstrating their impact on Lipschitz classification.

Findings

Separating sets are common in higher-dimensional complex germs.

The relationship between separating sets and metric tangent cones is clarified.

Inner Lipschitz type can vary even with constant topology in complex surface germs.

Abstract

An explanation is given for the initially surprising ubiquity of separating sets in normal complex surface germs. It is shown that they are quite common in higher dimensions too. The relationship between separating sets and the geometry of the metric tangent cone of Bernig and Lytchak is described. Moreover, separating sets are used to show that the inner Lipschitz type need not be constant in a family of normal complex surface germs of constant topology.