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 and tau is a nonconstant function on M such that the conformally related metric g/tau(2), defined wherever tau not equal 0, is an Einstein metric. It turns out that M then is the total space of a holomorphic CP1 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 not equal 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 m >= 9. Our classification uses a moduli curve, which is a subset C, depending on m, of an algebraic curve in R-2. A point (u, v) in 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 (u, v) coupled with some other data, among them a Kahler-Einstein base (N, h) for the CP1 bundle M. The points (u, v) arising in this way from (M, g, m, tau) with compact M form a countably infinite subset of C.