A moduli curve for compact conformally-Einstein Kahler manifolds

We classify quadruples $(M,g,m,tau)$ in which $(M,g)$ is a compact Kahler manifold of complex dimension $m>2$ with a nonconstant function $tau$ on $M$ such that the conformally related metric $g/tau^2$, defined wherever $tau e 0$, is Einstein. It turns out that $M$ then is the total space of a holomorphic $CP^1$ bundle over a compact Kahler-Einstein manifold $(N,h)$. The quadruples in question constitute four disjoint families: one, well-known, with Kahler metrics $g$ that are locally reducible; a second, discovered by Berard Bergery (1982), and having $tau e 0$ everywhere; a third one, related to the second by a form of analytic continuation, and analogous to some known Kahler surface metrics; and a fourth family, present only in odd complex dimensions $mge 9$. Our classification uses a {it moduli curve}, which is a subset $mathcal{C}$, depending on $m$, of an algebraic curve in $R^2$. A point $(u,v)$ in $mathcal{C}$ is naturally associated with any $(M,g,m,tau)$ having all of the above properties except for compactness of $M$, replaced by a weaker requirement of ``vertical compactness. One may in turn reconstruct $M,g$ and $tau$ from this $(u,v)$ coupled with some other data, among them a Kahler-Einstein base $(N,h)$ for the $CP^1$ bundle $M$. The points $(u,v)$ arising in this way from $(M,g,m,tau)$ with compact $M$ form a countably infinite subset of $mathcal{C}$.