On the Kummer construction for Kcsc metrics

TL;DR

This paper establishes conditions under which singularities in a compact constant scalar curvature Kähler orbifold can be resolved to produce a smooth Kähler manifold with constant scalar curvature, extending previous results and providing explicit examples.

Contribution

It generalizes prior work by identifying sufficient conditions for desingularization of Kähler orbifolds with nontrivial holomorphic vector fields.

Findings

Derived sufficient conditions for desingularization

Extended previous results to broader classes of orbifolds

Presented explicit examples illustrating the theory

Abstract

Given a compact constant scalar curvature Kaehler orbifold, with nontrivial holomorphic vector fields, whose singularities admit a local ALE Kaehler Ricci-flat resolution, we find sufficient conditions on the position of the singular points to ensure the existence of a global constant scalar curvature Kaehler desingularization. This generalizes the results previously obtained by the first author with F. Pacard. A series of explicit examples is discussed.