The CR Killing operator and Bernstein-Gelfand-Gelfand construction in CR geometry

TL;DR

This paper develops tractor calculus for compatible almost CR structures on contact manifolds, explicitly describes the first BGG operator in some cases, and explores the relationship between the CR Killing operator and CR geometric analysis.

Contribution

It introduces a modified tractor connection that generates the CR Killing operator as its first BGG operator, revealing new insights into CR geometry and deformations.

Findings

Explicit expression for the first BGG operator in certain cases

Identification of the CR Killing operator as a BGG operator of a modified connection

Discussion of the CR Killing operator's relation to ACHE metrics and the CR obstruction tensor

Abstract

We elaborate the tractor calculus for compatible almost CR structures (also known as strictly pseudoconvex partially integrable almost CR structures) on contact manifolds, and as an application, express the first BGG invariant differential operator $D_0$ explicitly in some cases, i.e., for some tractor connections. An interesting outcome is the fact that the "modified" adjoint tractor connection $\tilde{\nabla}$ governing infinitesimal deformations of parabolic geometries generates what we call the CR Killing operator as its first BGG operator, and actually, it does not agree with the first BGG operator of the normal (unmodified) adjoint tractor connection. The relationship between the CR Killing operator and analysis of ACHE (asymptotically complex hyperbolic Einstein) metrics, or more specifically the CR obstruction tensor, is also discussed.