# On the Chern-Moser-Weyl tensor of real hypersurfaces

## TL;DR

This paper derives an explicit, simplified formula for the Chern-Moser-Weyl tensor of nondegenerate real hypersurfaces in complex space, and applies it to analyze CR invariants on ellipsoids, resolving open questions and conjectures.

## Contribution

It provides a new explicit formula for the Chern-Moser-Weyl tensor and applies it to demonstrate nontriviality of a CR invariant on ellipsoids, addressing open problems in CR geometry.

## Key findings

- The formula simplifies for pluriharmonic perturbations of the sphere.
- The CR invariant one-form is nontrivial on ellipsoids of revolution in C^3.
- Counterexample to Hirachi's recent conjecture.

## Abstract

We derive an explicit formula for the well-known Chern-Moser-Weyl tensor for nondegenerate real hypersurfaces in complex space in terms of their defining functions. The formula is considerably simplified when applying to "pluriharmonic perturbations" of the sphere or to a Fefferman approximate solution to the complex Monge-Amp\`ere equation. As an application, we show that the CR invariant one-form $X_{\alpha}$ constructed recently by Case and Gover is nontrivial on each real ellipsoid of revolution in $\mathbb{C}^3$, unless it is equivalent to the sphere. This resolves affirmatively a question posed by these two authors in 2017 regarding the (non-) local CR invariance of the $\mathcal{I}'$-pseudohermitian invariant in dimension five and provides a counterexample to a recent conjecture by Hirachi.