A moduli curve for compact conformally-Einstein K\"ahler manifolds

We classify quadruples $(M,g,m,\tau )$ in which $(M,g)$ is a compact K"ahler manifold of complex dimension $m>2$ with a nonconstant function $\tau$ on $M$ such that the conformally related metric $g\mathrm{/}{\tau }^2$, defined wherever $\tau\ne 0$,isEinstein\mathrm{.}Itturnsoutthat$M$thenisthetotalspaceofaholomorphic$CP^1$bundleoveracompactK\ddot{a}hler-Einsteinmanifold$(N,h)$. The quadruples in question constitute four disjoint families: one, well-known, with K"ahler metrics $g$ that are locally reducible; a second, discovered by B'erard Bergery (1982), and having $\tau \mathrm{\ne }0$ everywhere; a third one, related to the second by a form of analytic continuation, and analogous to some known K"ahler surface metrics; and a fourth family, present only in odd complex dimensions $m\ge 9$. 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 K"ahler-Einstein base $(N,h)$ for the $C{P}^1$ bundle $M$. The points $(u,v)$ arising in this way from $(M,g,m,\tau )$ with compact $M$ form a countably infinite subset of $\mathcal{C}$.