Conformally K\"ahler base metrics for Einstein warped products

Abstract

A Riemannian metric $\wht{g}$ with Ricci curvature $\wht{\ri}$ is called nontrivial quasi-Einstein, in the sense of Case, Shu and Wei, if it satisfies $(-a/f)\wht{\nab} df+\wht{\ri}=\lambda \wht{g}$, 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 $(g,\t)$ be a pair consisting of a K\"ahler metric $g$ which is locally K\"ahler irreducible, and a nonconstant Killing potential $\t$. Suppose the metric $\wht{g}=g/\t^2$ is nontrivial \bee on $M\setminus\t^{-1}(0)$, and the associated function $f$ is locally a function of $\t$. Then $(g,\t)$ 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 $CP^1$ 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 $\t^{-1}$ with nonzero constants. Conversely, in all even dimensions $n\geq 4$, there exist \sk pairs $(g,\t)$ and corresponding nonzero constants $K$ and $L$ for which $g/\t^2$ is nontrivial quasi-Einstein with $f=K\t^{-1}+L$. 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.