On the Chern-Moser-Weyl tensor of real hypersurfaces

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 ${\mathcal{I}}^{\mathrm{\prime }}$-pseudohermitian invariant in dimension five and provides a counterexample to a recent conjecture by Hirachi.