On the geometry of non-collapsed polarized cscK surfaces

Abstract

We show that the Gromov--Hausdorff convergence of non-collapsed polarized constant scalar curvature Kähler (cscK) surfaces can be realized as convergence in a Hilbert scheme. We also derive uniform estimates of Bergman kernels on the effective regular set. As an application, we establish the Zariski openness of cscK metrics for certain smooth polarized families, following the approach of Donaldson.