Deformation theory of scalar-flat Kähler ALE surfaces

Abstract

We prove a Kuranishi-type theorem for deformations of complex structures on ALE Kähler surfaces. This is used to prove that for any scalar-flat Kähler ALE surface, all small deformations of complex structure also admit scalar-flat Kähler ALE metrics. A local moduli space of scalar-flat Kähler ALE metrics is then constructed, which is shown to be universal up to small diffeomorphisms (that is, diffeomorphisms which are close to the identity in a suitable sense). A formula for the dimension of the local moduli space is proved in the case of a scalar-flat Kähler ALE surface which deforms to a minimal resolution of ${\mathbb{C}}^2/\Gamma$, where $\Gamma$ is a finite subgroup of $\mathrm{U}(2)$ without complex reflections.

35 pages. Revised version to appear in the American Journal of Mathematics

References in corpus (1)

• Smoothing singular extremal Kähler surfaces and minimal Lagrangians