# Analytic Moduli of Plane Branches and Holomorphic Flows

## TL;DR

This paper investigates the behavior of singular plane branches under holomorphic flows, offering a new geometric approach to Zariski's moduli problem and exploring conjugacy classes within these structures.

## Contribution

It introduces an elementary, geometric, and dynamical method to address Zariski's moduli problem for singular branches and examines the conjugacy properties within analytic classes.

## Key findings

- Provided a new solution to Zariski's moduli problem
- Identified an analytic class that is not complete
- Showed not all elements in a class are conjugate via a flow

## Abstract

We study the behaviour (in the infinitesimal neighbourhood of the singularity) of a singular plane branch under the action of holomorphic flows. The techniques we develop provide a new elementary, geometric and dynamical solution to Zariski's moduli problem for singular branches in $({\mathbb C}^{2},0)$. Furthermore, we study whether elements of the same class of analytic conjugacy are conjugated by a holomorphic flow; in particular we show that there exists an analytic class that is not complete: meaning that there are two elements of the class that are not analytically conjugated by a local diffeomorphism embedded in a one-parameter flow.