A smorgasbord of scalar-flat Kahler ALE surfaces

There are many known examples of scalar-flat Kahler ALE surfaces, all of which have group at infinity either cyclic or contained in SU(2). The main result in this paper shows that for any non-cyclic finite subgroup Gamma subset of U(2) containing no complex reflections, there exist scalar-flat Kahler ALE metrics on the minimal resolution of 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 Kahler ALE metrics with respect to some small deformations of complex structure of the minimal resolution. Lastly, we show the existence of extremal Kahler metrics admitting holomorphic isometric circle actions in certain Kahler classes on the complex analytic compactifications of the minimal resolutions.