Lipschitz regular complex algebraic sets are smooth

TL;DR

This paper proves that complex algebraic sets with Lipschitz regularity are necessarily smooth, extending Mumford's classical theorem without restrictions on dimension or isolated singularities.

Contribution

It establishes that Lipschitz regularity implies smoothness for complex algebraic sets in any dimension, generalizing previous results.

Findings

Lipschitz regularity guarantees smoothness in complex algebraic sets

No restriction on dimension or isolated singularities is necessary

Extends Mumford's theorem to a broader class of sets

Abstract

The classical Theorem of Mumford states that a topologically regular complex algebraic surface in $\mathbb{C}^3$ with an isolated singular point is smooth. We proof that any Lipschitz regular complex algebraic set is smooth. No restriction on the dimension is needed. No restriction of singularity to be isolated is needed.