m > 3, or (0.2) with m 2, if and only if Q is a speci®c type of a rational function of t.A global classi®cation of all M, g, m, t with (0.1) and m > 3, or (0.2) and m 2, for which M is compact, can similarly be derived from a global classi®cation of compact Ka Èhler manifolds with special Ka Èhler±Ricci potentials. These classi®cation theorems both require extensive additional arguments, based on two different methods, and will therefore appear in separate papers [12,13].The simplest examples of quadruples M; g; m; t with (0.1) for which M is compact, m > 2, and t 0 somewhere in M, are provided by some Riemannian products that have apparently been known for decades; see § 25 below. Another family of compact examples with (0.1), representing all dimensions m > 2, and this time having t T 0 everywhere in M (so that g is globally conformally Einstein), was constructed by Lionel Be Ârard Bergery [2]; cf. also § 26 below. More recent extensions of the results of [2] can be found in [28].Compact Ka Èhler surfaces M; g with (0.1) have been studied by many authors. For instance, Page [21] found the only known example of an Einstein metric on a compact complex surface (namely, on M CP 2 # CP 2 ) which is globally conformally-Ka Èhler, but not Ka Èhler; its conformally-related Ka Èhler metric was independently discovered by Calabi [6,7,9]. Page's manifold thus satis®es (0.1) (and, in fact, (0.2)) with m 2, and has t T 0 everywhere in M. Examples of (0.2) with m 2 such that M is compact and t vanishes somewhere are constructed, for minimal ruled surfaces M, in [16] and [27]. Other results concerning (0.1) for compact Ka Èhler surfaces M; g include LeBrun's structure theorem [20] for Hermitian Einstein metrics on compact complex surfaces and the variational characterization of such metrics in [25]. For non-compact examples, see [1,11].Condition (0.1) with m 2 is much less restrictive than for m > 3, as it implies (0.2) in the latter case, but not in the former. This re¯ects the fact that, at points where the scalar curvature s of a Ka Èhler-surface metric g is non-zero, the metric e g g=s 2 already satis®es a consequence of the Einstein condition (namely, vanishing of the divergence of the self-dual Weyl tensor), and is, up to a constant factor, the only metric conformal to g with this property; see [10, top of p. 417].We wish to thank the referee for suggesting changes that make the present paper easier to read and bringing Gudmundsson's paper [15] to our attention.

Abstract. A special Kähler-Ricci potential on a Kähler manifold is any nonconstant C ∞ function τ ι such that J(∇τ ι) is a Killing vector field and, at every point with dτ ι = 0, all nonzero tangent vectors orthogonal to ∇τ ι and J(∇τ ι) are eigenvectors of both ∇dτ ι and the Ricci tensor. For instance, this is always the case if τ ι is a nonconstant C ∞ function on a Kähler manifold (M, g) of complex dimension m > 2 and the metricg = g/τ ι 2 , defined wherever τ ι = 0, is Einstein. (When such τ ι exists, (M, g) may be called almost-everywhere conformally Einstein.) We provide a complete classification of compact Kähler manifolds with special Kähler-Ricci potentials and use it to prove a structure theorem for compact Kähler manifolds of any complex dimension m > 2 which are almost-everywhere conformally Einstein. §0. Introduction This paper, although self-contained, can also be viewed as the second in a series of three papers that starts with [8] and ends with [9].We call τ ι a special Kähler-Ricci potential on a Kähler manifold (M, g) if τ ι is a nonconstant Killing potential on (M, g) and, at every point (0.1) with dτ ι = 0, all nonzero tangent vectors orthogonal to v = ∇τ ι and to u = Jv are eigenvectors of both ∇dτ ι and the Ricci tensor r.(Cf.[8], §7; for more on Killing potentials, see §4 below.) The word 'potential' reflects the fact that (0.1) is closely related, although not equivalent, to the requirement that ∇dτ ι + χ r = σg for some C ∞ functions χ, σ (see [8], beginning of §7). This requirement is reminiscent of Kähler-Ricci solitons (some of which, in fact, do satisfy (0.1), cf.[10] and Remark 10.1 below); while, in complex dimensions m > 2, it implies that τ ι arises from a Hamiltonian 2-form on the underlying Kähler manifold ([2], §1.4). See also [5]. What further sparked our interest in (0.1) was its being, in cases such as (0.4) below, a consequence of the following assumption:(M, g) is a Kähler manifold of complex dimension m and τ ι is (0.2) a nonconstant C ∞ function on M such that the conformally related metricg = g/τ ι 2 , defined wherever τ ι = 0, is Einstein.When m > 2, (0.2) implies the seemingly stronger condition (0.3) M, g, m, τ ι satisfy (0.2) and dτ ι ∧ d∆τ ι = 0 everywhere in M 1991 Mathematics Subject Classification. Primary 53C55, 53C21; Secondary 53C25.

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Compact pseudo-Riemannian manifolds that have parallel Weyl tensor without being conformally flat or locally symmetric are known to exist in infinitely many dimensions greater than 4. We prove some general topological properties of such manifolds, namely, vanishing of the Euler characteristic and real Pontryagin classes, and infiniteness of the fundamental group. We also show that, in the Lorentzian case, each of them is at least 5-dimensional and admits a two-fold cover which is a bundle over the circle.