Conformally K\"ahler base metrics for Einstein warped products

A Riemannian metric $\text{wht}g$ with Ricci curvature $\text{wht}\text{ri}$ is called nontrivial quasi-Einstein, in the sense of Case, Shu and Wei, if it satisfies $(-a\mathrm{/}f)\text{wht}\text{nab}df+\text{wht}\text{ri}=\lambda \text{wht}g$, for a smooth nonconstant function $f$ and constants $\lambda$ and $a>0$. If $a$ is a positive integer, by a result of Kim and Kim, such a metric forms a base for certain warped Einstein metrics. On a manifold $M$ of real dimension at least six, let $(g,\text{t})$ be a pair consisting of a K"ahler metric $g$ which is locally K"ahler irreducible, and a nonconstant Killing potential $\text{t}$. Suppose the metric $\text{wht}g=g\mathrm{/}{\text{t}}^2$ is nontrivial \bee on $M\setminus {\text{t}}^{-1}(0)$, and the associated function $f$ is locally a function of $\text{t}$. Then $(g,\text{t})$ is an \sk
pair, a notion defined by Derdzinski and Maschler. This implies that $M$ is biholomorphic to an open set in the total space of a $C{P}^1$ bundle whose base manifold admits a K"ahler-Einstein metric. If $M$ is additionally compact, it is a total space of such a bundle or complex projective space. Also, the function $f$ is affine in ${\text{t}}^{-1}$ with nonzero constants. Conversely, in all even dimensions $n\ge 4$, there exist \sk pairs $(g,\text{t})$ and corresponding nonzero constants $K$ and $L$ for which $g\mathrm{/}{\text{t}}^2$ is nontrivial quasi-Einstein with $f=K{\text{t}}^{-1}+L$. Additionally, a result of Case, Shu and Wei on the K"ahler reducibility of nontrivial K"ahler \bers is reproduced in dimension at least six in a more explicit form.