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A complete normal form for everywhere Levi degenerate hypersurfaces in $\mathbb C^{3}$

Martin Kolar, Ilya Kossovskiy

TL;DR

The paper solves the invariant and equivalence problem for everywhere $2$-nondegenerate real-analytic hypersurfaces in $\\mathbb{C}^3$ by constructing a complete convergent normal form based on a Chern–Moser–style homological method, using the tube over the light cone as a rational model. A tailored weight system renders these hypersurfaces as perturbations of the model, and a chain-field framework ensures convergence of the formal normalization. The authors derive a sphericity criterion in terms of normal-form coefficients and describe the moduli space of such hypersurfaces as a quotient $\\mathcal D/\\!G$ of the distinguished normal-form data by the model’s stability group. This advances CR-geometry in the Levi-degenerate, infinite Catlin multitype regime and provides concrete tools for identifying equivalence to the model and for classifying nearby CR-structures. The work blends model-theoretic normalization, chain geometry, and PDE-based extension results to achieve analytic normalization and explicit moduli descriptions.

Abstract

$2$-nondegenerate real hypersurfaces in complex manifolds play an important role in CR-geometry and the theory of Hermitian Symmetric Domains. In this paper, we construct a complete convergent normal form for everywhere $2$-nondegenerate real-analytic hypersurfaces in complex $3$-space. We do so by developing the homological approach of Chern-Moser in the $2$-nondegenerate setting. This seems to be the first such construction for hypersurfaces of infinite Catlin multitype. Our approach is based on using a rational (nonpolynomial) model for everywhere $2$-nondegenerate hypersurfaces, which is the local realization due to Fels-Kaup of the well known tube over the light cone. As an application, we obtain, in the spirit of Chern-Moser theory, a criterion for the local sphericity (i.e. local equivalence to the model) for a $2$-nondegenerate hypersurface in terms of its normal form. As another application, we obtain an explicit description of the moduli space of everywhere $2$-nondegenerate hypersurfaces.

A complete normal form for everywhere Levi degenerate hypersurfaces in $\mathbb C^{3}$

TL;DR

The paper solves the invariant and equivalence problem for everywhere -nondegenerate real-analytic hypersurfaces in by constructing a complete convergent normal form based on a Chern–Moser–style homological method, using the tube over the light cone as a rational model. A tailored weight system renders these hypersurfaces as perturbations of the model, and a chain-field framework ensures convergence of the formal normalization. The authors derive a sphericity criterion in terms of normal-form coefficients and describe the moduli space of such hypersurfaces as a quotient of the distinguished normal-form data by the model’s stability group. This advances CR-geometry in the Levi-degenerate, infinite Catlin multitype regime and provides concrete tools for identifying equivalence to the model and for classifying nearby CR-structures. The work blends model-theoretic normalization, chain geometry, and PDE-based extension results to achieve analytic normalization and explicit moduli descriptions.

Abstract

-nondegenerate real hypersurfaces in complex manifolds play an important role in CR-geometry and the theory of Hermitian Symmetric Domains. In this paper, we construct a complete convergent normal form for everywhere -nondegenerate real-analytic hypersurfaces in complex -space. We do so by developing the homological approach of Chern-Moser in the -nondegenerate setting. This seems to be the first such construction for hypersurfaces of infinite Catlin multitype. Our approach is based on using a rational (nonpolynomial) model for everywhere -nondegenerate hypersurfaces, which is the local realization due to Fels-Kaup of the well known tube over the light cone. As an application, we obtain, in the spirit of Chern-Moser theory, a criterion for the local sphericity (i.e. local equivalence to the model) for a -nondegenerate hypersurface in terms of its normal form. As another application, we obtain an explicit description of the moduli space of everywhere -nondegenerate hypersurfaces.

Paper Structure

This paper contains 15 sections, 10 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Historic outline
  3. Overview of the results
  4. The normal form
  5. Applications of the normal form
  6. A formal normal form
  7. The light cone and its automorphisms
  8. The light cone as a model
  9. Normalization of initial terms of a CR-map
  10. Homological operator and formal normal form
  11. Convergence of the formal normal form and applications
  12. Proof of Theorem 4
  13. Chains in an everywhere $2$-nondegenerate hypersurface
  14. End of proof of Theorem 4
  15. Proofs of further results.

Key Result

theorem 1

Let $M$ be a real-analytic everywhere $2$-nondegenerate hypersurface in $\mathbb{C}^{3}$, and $p\in M$. Then, there exists a biholomorphic transformation $H:\,(\mathbb{C}^{3},p)\longrightarrow(\mathbb{C}^{3},0)$ mapping $M$ into a hypersurface in normal form. A normalizing transformation $H$ is dete

Theorems & Definitions (27)

  • definition 1.1
  • theorem 1
  • remark 1.2
  • theorem 2
  • remark 1.3
  • definition 1.4
  • theorem 3
  • definition 2.1
  • proposition 2.3
  • proof
  • ...and 17 more