A smörgåsbord of scalar-flat Kähler ALE surfaces

Abstract

There are many known examples of scalar-flat Kähler 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 Γ ⊂ U(2) containing no complex reflections, there exist scalar-flat Kähler ALE metrics on the minimal resolution of C 2 /Γ, for which Γ 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 Kähler ALE metrics with respect to some small deformations of complex structure of the minimal resolution. Lastly, we show the existence of extremal Kähler metrics admitting holomorphic isometric circle actions in certain Kähler classes on the complex analytic compactifications of the minimal resolutions.

On the ADM mass of Kähler scalar flat ALE metrics

“…The existence and geometric properties of Ricci and scalar flat metrics on ALE Kähler spaces with a given structure group at infinity have been subject of intense research in past few decades (see e.g. [8], [15], [14], [13], [7], [10], [16]). In this paper we study to which extent the machinery developed in [2] for constructing constant scalar curvature Kähler metrics on blow ups can be used and extended in this setting.…”