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A Morse-Bott normal form for real analytic Levi-flat hypersurfaces

Arturo Fernández-Pérez, Gustavo Marra

TL;DR

The paper addresses local classification of real-analytic Levi-flat hypersurfaces with Morse-Bott-type singularities by combining holomorphic foliation techniques with a Morse-Bott normalization. It proves the existence of a biholomorphism $\\Phi$ with $D\\Phi(0)=\\operatorname{id}_{n-c}\\star0\\operatorname{id}_{c}$ such that $\\Phi^{-1}(M)=\\{\\mathrm{Re}(x_1^2+\\cdots+x_{n-c}^2)=0\\}$, thereby generalizing Burns-Gong to the Morse-Bott setting and yielding a new normal form for a class of real-analytic quadratic Levi-flat hypersurfaces. The approach uses a holomorphic first integral for a foliation tangent to $M$ (via Alcides–Cerveau–Lins Neto) and then a holomorphic Morse-Bott lemma to straighten the defining function; the special case $n-c=2$ is handled with a blow-up and holonomy argument to guarantee a first integral. Overall, the work provides a canonical local model for Levi-flat singularities and strengthens connections between real-analytic and holomorphic data through complexification.

Abstract

We prove the existence of a normal form for a real-analytic Levi-flat hypersurface defined by the vanishing of the real part of a holomorphic function with a Morse-Bott singularity. As a consequence, we recover the Burns-Gong normal form for Levi-flat hypersurfaces with generic Morse singularities and provide a new normal form for a certain class of real analytic quadratic Levi-flat hypersurfaces.

A Morse-Bott normal form for real analytic Levi-flat hypersurfaces

TL;DR

The paper addresses local classification of real-analytic Levi-flat hypersurfaces with Morse-Bott-type singularities by combining holomorphic foliation techniques with a Morse-Bott normalization. It proves the existence of a biholomorphism with such that , thereby generalizing Burns-Gong to the Morse-Bott setting and yielding a new normal form for a class of real-analytic quadratic Levi-flat hypersurfaces. The approach uses a holomorphic first integral for a foliation tangent to (via Alcides–Cerveau–Lins Neto) and then a holomorphic Morse-Bott lemma to straighten the defining function; the special case is handled with a blow-up and holonomy argument to guarantee a first integral. Overall, the work provides a canonical local model for Levi-flat singularities and strengthens connections between real-analytic and holomorphic data through complexification.

Abstract

We prove the existence of a normal form for a real-analytic Levi-flat hypersurface defined by the vanishing of the real part of a holomorphic function with a Morse-Bott singularity. As a consequence, we recover the Burns-Gong normal form for Levi-flat hypersurfaces with generic Morse singularities and provide a new normal form for a certain class of real analytic quadratic Levi-flat hypersurfaces.

Paper Structure

This paper contains 7 sections, 5 theorems, 49 equations.

Table of Contents

  1. Introduction and Statement of the main result
  2. Singular Levi-flat hypersurfaces
  3. Complexification of a Levi-flat hypersurface
  4. Holomorphic foliations and Levi-flat hypersurfaces
  5. Holonomy and holomorphic first integrals
  6. Morse-Bott Lemma
  7. Proof of Theorem \ref{['main_theorem']}

Key Result

Theorem 1

Let $M=\{ F=0\}$ be a germ of a real analytic Levi-flat hypersurface at $(\mathbb{C}^n,0)$, $n\geq 2$ such that: Then there exists a germ of biholomorphism $\Phi\in \operatorname{Diff}(\mathbb{C}^n,0)$ such that where $\operatorname{id}_{n-c}\in\operatorname{GL}(n-c,\mathbb{C})$, $\operatorname{id}_{c}\in\operatorname{GL}(c,\mathbb{C})$, and

Theorems & Definitions (8)

  • Theorem 1
  • Lemma 2.1
  • Remark 2.1
  • Definition 2.1
  • Theorem 2.2: Cerveau-Lins Neto alcides
  • Lemma 2.3
  • Lemma 3.1
  • Example 1