On characterization of smoothness of complex analytic sets

TL;DR

This paper explores the metric properties of singularities in complex analytic sets, establishing conditions under which such sets are smooth based on their local metric conicality and the topology of their links.

Contribution

It provides a higher-dimensional analogue of Mumford's theorem, linking smoothness to local metric conicality and the topology of the link in complex analytic sets.

Findings

A complex analytic set with an isolated singularity is smooth iff it is locally metrically conical and its link is a homotopy sphere.

Establishes a higher-dimensional analogue of Mumford's theorem on smoothness.

Connects topological and metric properties to characterize smoothness of complex analytic sets.

Abstract

The paper is devoted to metric properties of singularities. We investigate the relations among topology, metric properties and smoothness. In particular, we present some higher dimensional analogous of Mumford's theorem on smoothness of normal surfaces. For example, we prove that a complex analytic set, with an isolated singularity at $0$, is smooth at $0$ if and only if it is locally metrically conical at $0$ and its link at $0$ is a homotopy sphere.