Kähler geometry of scalar flat metrics on Line bundles over polarized Kähler-Einstein manifolds

Abstract

In view of a better understanding of the geometry of scalar flat Kähler metrics, this paper studies two families of scalar flat Kähler metrics constructed in [10] by A. D. Hwang and M. A. Singer on $\mathbb C^{n+1}$ and on $\mathcal O(-k)$. For the metrics in both the families, we prove the existence of an asymptotic expansion for their $ε$-functions and we show that they can be approximated by a sequence of projectively induced Kähler metrics. Further, we show that the metrics on $\mathbb C^{n+1}$ are not projectively induced, and that the Burns-Simanca metric is characterized among the scalar flat metrics on $\mathcal O(-k)$ to be the only projectively induced one as well as the only one whose second coefficient in the asymptotic expansion of the $ε$-function vanishes.