Fundamental invariants of 2--nondegenerate CR geometries with simple models

TL;DR

This paper investigates the fundamental invariants of 2-nondegenerate CR geometries with simple models, identifying their sources and conditions for their vanishing, and explores examples through deformations of models.

Contribution

It identifies two sources of invariants in 2-nondegenerate CR geometries and characterizes when the local equivalence problem reduces to a Cartan connection.

Findings

Two sources of invariants: harmonic curvature and complex structure differences.

Vanishing invariants imply the local equivalence is given by a Cartan connection.

Nontrivial examples are constructed as deformations of models.

Abstract

This article studies the fundamental invariants of 2--nondegenerate CR geometries with simple models. We show that there are two sources of these invariants. The first source is the harmonic curvature of the parabolic geometry that appears (locally) on the leaf space of the Levi kernel. The second source is the difference between the complex structure on the complex tangent space of the CR geometry and the complex structure on the correspondence space to the underlying parabolic geometry. We show that the later fundamental invariants appear only when the model is generic and if they vanish, then the solution of the local equivalence problem of 2--nondegenerate CR geometries with simple models is provided by the Cartan connection of the underlying parabolic geometry. We show that nontrivial examples of CR geometries with the later fundamental invariants can be obtained as deformations of the models.