Multiplicity, regularity and blow-spherical equivalence of complex analytic sets

TL;DR

This paper introduces blow-spherical equivalence for complex analytic sets, providing new insights into their classification, regularity, and multiplicity, with applications to Zariski's multiplicity conjecture and complex curve classification.

Contribution

It presents the concept of blow-spherical equivalence and applies it to classify complex analytic sets, offering partial solutions to longstanding conjectures.

Findings

Blow-spherical regular sets are smooth.

A classification of complex analytic curves is achieved.

A reduction of Zariski's multiplicity conjecture for homogeneous sets.

Abstract

This paper is devoted to study multiplicity and regularity as well as to present some classifications of complex analytic sets. We present an equivalence for complex analytical sets, namely blow-spherical equivalence and we receive several applications with this new approach. For example, we reduce to homogeneous complex algebraic sets a version of Zariski's multiplicity conjecture in the case of blow-spherical homeomorphism, we give some partial answers to the Zariski's multiplicity conjecture, we show that a blow-spherical regular complex analytic set is smooth and we give a complete classification of complex analytic curves.