$\mathcal{I}^{\prime}$-curvatures in higher dimensions and the Hirachi conjecture

Abstract

We construct higher-dimensional analogues of the $\mathcal{I}^{\prime}$-curvature of Case and Gover in all CR dimensions $n \geq 2$. Our $\mathcal{I}^{\prime}$-curvatures all transform by a first-order linear differential operator under a change of contact form and their total integrals are independent of the choice of pseudo-Einstein contact form on a closed CR manifold. We exhibit examples where these total integrals depend on the choice of general contact form, and thereby produce counterexamples to the Hirachi conjecture in all CR dimensions $n \geq 2$.