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

TL;DR

This paper introduces higher-dimensional analogues of $ abla$-curvature in CR geometry, demonstrating their invariance properties and providing counterexamples to the Hirachi conjecture across all CR dimensions $n\,\geq\,2$.

Contribution

It constructs new $ abla$-curvatures in higher dimensions, analyzes their transformation properties, and disproves the Hirachi conjecture in all CR dimensions $n\geq 2$.

Findings

Total integrals are independent of pseudo-Einstein contact form

Counterexamples show dependence on general contact form

Disproof of the Hirachi conjecture in all CR dimensions $n\geq 2$

Abstract

We construct higher-dimensional analogues of the $\mathcal{I}^\prime$-curvature of Case and Gover in all CR dimensions $n\geq2$. 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 $n\geq2$.