2014/10/23 by Michael T. Lock, Jeff Viaclovsky, Lock, Michael T. +1
Arts and Humanities · Mathematics · #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #French Literature and Critical Theory #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1410.6461
openalex publication_date 2014/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
There are many known examples of scalar-flat Kähler ALE surfaces, all of which have group at infinity either cyclic or contained in \rmSU(2). The main result in this paper shows that for any non-cyclic finite subgroup Γ⊂ \rmU(2) containing no complex reflections, there exist scalar-flat Kähler ALE metrics on the minimal resolution of ℂ2 / 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.