On Normal Forms of Singular Levi-Flat Real Analytic Hypersurfaces

TL;DR

This paper establishes a normal form for singular Levi-flat real analytic hypersurfaces near singularities, showing they can be simplified to a polynomial form with specific properties using holomorphic transformations.

Contribution

It introduces a method to normalize Levi-flat hypersurfaces with isolated singularities to a polynomial form matching the Milnor number, extending understanding of their local structure.

Findings

Existence of a holomorphic change of coordinates simplifying the hypersurface

Normal form polynomial degree relates to Milnor number

Preservation of initial polynomial jet in normalization

Abstract

Let $F(z)=\mathcal{R}e(P(z)) + h.o.t$ be such that $M=(F=0)$ defines a germ of real analytic Levi-flat at $0\in\mathbb{C}^{n}$, $n\geq{2}$, where $P(z)$ is a homogeneous polynomial of degree $k$ with an isolated singularity at $0\in\mathbb{C}^{n}$ and Milnor number $\mu$. We prove that there exists a holomorphic change of coordinate $\phi$ such that $\phi(M)=(\mathcal{R}e(h)=0)$ where $h(z)$ is a polynomial of degree $\mu+1$ and $j^{k}_{0}(h)=P$.