Classification of polynomial models without 2-jet determination in $\mathbb{C}^3$

TL;DR

The paper addresses the problem of classifying polynomial models of Levi-degenerate real hypersurfaces in $\mathbb{C}^3$ that fail 2-jet determination, by leveraging the Catlin multitype and a generalized Chern–Moser framework. It analyzes the graded symmetry algebra $\mathfrak{g}$, with emphasis on the exotic part $\mathfrak{g}_c$ and its interaction with rotations and tubular symmetries, to derive explicit normal forms for $\operatorname{Im} w$ in various interaction scenarios. The authors provide a complete description of models admitting exotic symmetries, including monomial and nonmonomial cases, by introducing $X$-pairs of chains and deriving concrete weighted-homogeneous representations, such as $\operatorname{Im} w = \sum_{N,K,m} \operatorname{Re}(\tau (z_1^p z_2^q)^N) |z_1|^{2k} |z_2|^{2l} (\operatorname{Re} z_1^\alpha z_2^\beta)^m$, with precise relations among the exponents. They also classify coexistence with real/imaginary rotations and tubular symmetries, yielding a finite set of local models and pinpointing solitary exotic symmetries that enforce $\dim\mathfrak{g}=3$. The results advance understanding of 2-jet determinacy obstructions and provide a structured catalogue of graded-symmetry types for Levi-degenerate CR manifolds in dimension three.

Abstract

An intriguing phenomenon regarding Levi-degenerate hypersurfaces is the existence of nontrivial infinitesimal symmetries with vanishing 2-jets at a point. In this work we consider polynomial models of Levi-degenerate real hypersurfaces in $\mathbb{C}^3$ of finite Catlin multitype. Exploiting the structure of the corresponding Lie algebra, we characterize completely models without 2-jet determination, including an explicit description of their symmetry algebras.

Theorems & Definitions (41)

• Definition 2.1: Catlin, C
• Lemma 2.2
• Lemma 2.3
• Definition 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Remark 2.8
• Definition 2.9
• Theorem 2.10