Analytic Moduli of Plane Branches and Holomorphic Flows

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.

This version supersedes the previous one: the title has changed as a consequence of major improvements and new results. This work must be considered a joint work of both authors from now on

References in corpus (2)

• The embedding flows of $C^{\infty }$ hyperbolic diffeomorphisms
• Dimension of the moduli space of a curve in the complex plane