On the Chern–Moser–Weyl tensor of real hypersurfaces

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