Some QCH Kahler surfaces with zero scalar curvature

TL;DR

This paper classifies certain well-known scalar-flat Kähler surfaces, showing they are QCH Kähler, and identifies their geometric types within a parametric family, including Taub-Nut and Burn's metrics.

Contribution

It demonstrates that specific scalar-flat Kähler surfaces are QCH Kähler and characterizes their geometric types based on a parameter, extending understanding of their structure.

Findings

Family of generalized Taub-Nut Kähler surfaces are orthotoric for k in (-1,1)

They are Calabi type for k in {-1,1}

Burn's metric is of Calabi type

Abstract

In this paper we prove that some well known K\"ahler surfaces with zero scalar curvature are QCH K\"ahler. We prove that family of generalized Taub-Nut K\"ahler surfaces parametrized by $k\in[-1,1]$ is of orthotoric type for $k\in(-1,1)$ and of Calabi type for $k\in\{-1,1\}$ and the Burn's metric is of Calabi type.