Special K\"ahler-Ricci potentials on compact K\"ahler manifolds

A special K"ahler-Ricci potential on a K"ahler manifold is any nonconstant $C^{\mathrm{\infty }}$ function $\tau$ such that $J(\mathrm{\nabla }\tau )$ is a Killing vector field and, at every point with $d\tau \mathrm{\ne }0$, all nonzero tangent vectors orthogonal to $\mathrm{\nabla }\tau$ and $J(\mathrm{\nabla }\tau )$ are eigenvectors of both $\mathrm{\nabla }d\tau$ and the Ricci tensor. For instance, this is always the case if $\tau$ is a nonconstant $C^{\mathrm{\infty }}$ function on a K"ahler manifold $(M,g)$ of complex dimension $m>2$ and the metric $\tilde{g}=g\mathrm{/}{\tau }^2$, defined wherever $\tau\ne 0$,isEinstein\mathrm{.}(Whensuch$\tau$exists,$(M,g)$ may be called {\it almost-everywhere conformally Einstein}.) We provide a complete classification of compact K"ahler manifolds with special K"ahler-Ricci potentials and use it to prove a structure theorem for compact K"ahler manifolds of any complex dimension $m>2$ which are almost-everywhere conformally Einstein.