Special Kähler-Ricci potentials on compact Kähler manifolds

By a special Kähler-Ricci potential on a Kähler manifold we mean a nonconstant real-valued C ∞ function τ such that J (∇τ) is a Killing vector field and, at every point with d τ ≠ 0, all nonzero tangent vectors orthogonal to ∇τ and J (∇τ) are eigenvectors of both ∇ d τ and the Ricci tensor. For instance, this is always the case if τ is a nonconstant C ∞ function on a Kähler manifold ( M, g ) of complex dimension m > 2 and the metric g ˜ = g /τ 2 , defined wherever τ ≠ 0, is Einstein. (When such τ exists, ( M , g ) may be called almost-everywhere conformally Einstein .) We provide a complete classification of compact Kähler manifolds ( M , g ) with special Kähler-Ricci potentials, showing, in particular, that in any complex dimension m ≧ 2 they form two separate classes: in one, M is the total space of a holomorphic ℂP 1 bundle; in the other, M is biholomorphic to ℂP m . We then use this classification to prove a structure theorem for compact Kähler manifolds of any complex dimension m > 2 which are almost-everywhere conformally Einstein.