A sm\"org\r{a}sbord of scalar-flat K\"ahler ALE surfaces

TL;DR

This paper constructs new scalar-flat K"ahler ALE metrics on minimal resolutions of quotients by non-cyclic finite subgroups of U(2), expanding known examples and demonstrating existence of extremal metrics with symmetries.

Contribution

It proves the existence of scalar-flat K"ahler ALE metrics for a broad class of non-cyclic groups, including those with no complex reflections, and explores their symmetries and deformations.

Findings

Existence of scalar-flat K"ahler ALE metrics for non-cyclic groups in U(2).

Metrics admit holomorphic isometric circle actions.

Existence of extremal K"ahler metrics on compactifications.

Abstract

There are many known examples of scalar-flat K\"ahler ALE surfaces, all of which have group at infinity either cyclic or contained in ${\rm{SU}}(2)$. The main result in this paper shows that for any non-cyclic finite subgroup $\Gamma \subset {\rm{U}}(2)$ containing no complex reflections, there exist scalar-flat K\"ahler ALE metrics on the minimal resolution of $\mathbb{C}^2 / \Gamma$, for which $\Gamma$ occurs as the group at infinity. Furthermore, we show that these metrics admit a holomorphic isometric circle action. It is also shown that there exist scalar-flat K\"ahler ALE metrics with respect to some small deformations of complex structure of the minimal resolution. Lastly, we show the existence of extremal K\"ahler metrics admitting holomorphic isometric circle actions in certain K\"ahler classes on the complex analytic compactifications of the minimal resolutions.