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Defining equations of $7$-dimensional model CR hypersurfaces

Jan Gregorovič, David Sykes

TL;DR

This work delivers a complete normal form for 2-nondegenerate Levi rank-2 CR hypersurfaces in $\mathbb{C}^4$ by developing and exploiting bigraded and modified CR symbols, normalizing via the group $\mathrm{CU}(\mathbf{H})\cap G_{0,0}$, and solving for the higher-order terms in $\mathbf{S}(\zeta)$. The authors classify homogeneous models, derive explicit defining equations, and determine their infinitesimal symmetry algebras, including a novel 7-dimensional homogeneous instance whose symmetry mirrors the split real form of $\mathrm{Lie}(G_2)$. They show that the moduli space of 2-nondegenerate models in $\mathbb{C}^4$ has functional freedom parameterized by holomorphic data, and that not all such models arise as perturbations of homogeneous ones. By connecting to the framework in GKS2024, the paper provides concrete normal forms, invariants, and symmetry classifications that advance the local CR equivalence theory for Levi form rank-2 structures.

Abstract

We study CR hypersurfaces in $\mathbb{C}^4$ that are Levi degenerate with constant rank Levi form, and moreover finitely nondegenerate. Each of these can be described as a deformation of a model CR hypersurface by adding terms of higher natural weighted order to the model's defining equation. We obtain a complete normal form for models of real analytic uniformly $2$-nondegenerate CR hypersurfaces in $\mathbb{C}^4$, and present a detailed study of their local invariants. The normal form illustrates that $2$-nondegenerate models in $\mathbb{C}^4$ comprise a moduli space parameterized by two univariate holomorphic functions, which is in sharp contrast to the well known Levi-nondegenerate setting and the more recently discovered behavior of $2$-nondegenerate structures in $\mathbb{C}^3$. In further contrast to these previously studied settings, we demonstrate that not all $2$-nondegenerate structures in $\mathbb{C}^4$ arise as perturbations of homogeneous models. We derive defining equations for the homogeneous $2$-nondegenerate models, a set of $9$ structures, and find explicit formulas for their infinitesimal symmetries.

Defining equations of $7$-dimensional model CR hypersurfaces

TL;DR

This work delivers a complete normal form for 2-nondegenerate Levi rank-2 CR hypersurfaces in by developing and exploiting bigraded and modified CR symbols, normalizing via the group , and solving for the higher-order terms in . The authors classify homogeneous models, derive explicit defining equations, and determine their infinitesimal symmetry algebras, including a novel 7-dimensional homogeneous instance whose symmetry mirrors the split real form of . They show that the moduli space of 2-nondegenerate models in has functional freedom parameterized by holomorphic data, and that not all such models arise as perturbations of homogeneous ones. By connecting to the framework in GKS2024, the paper provides concrete normal forms, invariants, and symmetry classifications that advance the local CR equivalence theory for Levi form rank-2 structures.

Abstract

We study CR hypersurfaces in that are Levi degenerate with constant rank Levi form, and moreover finitely nondegenerate. Each of these can be described as a deformation of a model CR hypersurface by adding terms of higher natural weighted order to the model's defining equation. We obtain a complete normal form for models of real analytic uniformly -nondegenerate CR hypersurfaces in , and present a detailed study of their local invariants. The normal form illustrates that -nondegenerate models in comprise a moduli space parameterized by two univariate holomorphic functions, which is in sharp contrast to the well known Levi-nondegenerate setting and the more recently discovered behavior of -nondegenerate structures in . In further contrast to these previously studied settings, we demonstrate that not all -nondegenerate structures in arise as perturbations of homogeneous models. We derive defining equations for the homogeneous -nondegenerate models, a set of structures, and find explicit formulas for their infinitesimal symmetries.
Paper Structure (20 sections, 11 theorems, 93 equations, 5 tables)

This paper contains 20 sections, 11 theorems, 93 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Invariants of CR hypersurfaces: bigraded symbols and modified symbols
  3. A complete normal form for models in C4̂
  4. Expansions of Row 1, 2, and 6 in Table 3
  5. Expansions of Row 3 in Table 3
  6. Expansions of Row 4 in Table 3
  7. Expansions of Row 5 in Table 3
  8. Expansions of Row 7 in Table 3
  9. Distinguished realizations of modified symbols
  10. Formulas of Table 3, row 1
  11. Formulas of Table 3, row 2
  12. Formulas of Table 3, row 3
  13. Formulas of Table 3, row 4
  14. Formulas of Table 3, row 5
  15. Formulas of Table 3, row 6
  16. ...and 5 more sections

Key Result

Theorem 1.1

For every germ of a $2$-nondegenerate model hypersurface in $\mathbb{C}^4$, there are local holomorphic coordinates centered at $0$ such that it is defined by gen def fun, H from S2 formula, and S from S2 formula with parameters $\mathbf{H}$ and $\mathbf{S}(\zeta)$ given by one and only one of the f

Theorems & Definitions (19)

  • Theorem 1.1
  • Definition 2.1: introduced in porter2021absolute
  • Corollary 2.2
  • Definition 2.3: GKS2024
  • Definition 2.4
  • Corollary 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • proof
  • ...and 9 more