On the Chern-Moser-Weyl tensor of real hypersurfaces

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

An inputenc package was missing in previous version, to appear in J. Math. Soc. Japan