A moduli curve for compact conformally-Einstein Kähler manifolds

Abstract

We classify quadruples $(M, g, m, \tau)$ in which (M, g) is a compact Kähler 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 $\tau \ne 0$, is an Einstein metric. It turns out that M then is the total space of a holomorphic $\mathbb{C}{\rm P}^1$ bundle over a compact Kähler–Einstein manifold (N, h). The quadruples in question constitute four disjoint families: one, well known, with Kähler metrics g that are locally reducible; a second, discovered by Bérard Bergery (1982), and having $\tau \ne 0$ everywhere; a third one, related to the second by a form of analytic continuation, and analogous to some known Kähler surface metrics; and a fourth family, present only in odd complex dimensions $m \ge 9$. Our classification uses a moduli curve, which is a subset $\mathcal{C}$, depending on m, of an algebraic curve in $\mathbb{R}^2$. 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 (u, v) coupled with some other data, among them a Kähler–Einstein base (N, h) for the $\mathbb{C}{\rm P}^1$ bundle M. The points (u, v) arising in this way from $(M, g, m, \tau)$ with compactM form a countably infinite subset of \mathcal{C}$.