1.1.1. Kummer construction [03FV]
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1.1.1. Kummer construction
We start with a flat orbifold given by the quotient of a flat -torus by the involution . It has 16 orbifold singularities. One can resolve these singularities by gluing 16 Eguchi-Hanson spaces onto , which are complete hyperkähler ALE metrics defined on the cotangent bundle of . By varying the flat structure on and the gluing parameters, one obtains an open set in the moduli space of all hyperkähler metrics on the surface where the areas of the exceptional curves are small. As these areas go to zero the corresponding hyperkähler metrics naturally converge back to the flat orbifold , and the Eguchi-Hanson spaces appear as bubbles under rescaling. For a rigorous proof we refer readers to [LS94], [Don12] and the references therein. This is a typical example of singularity formation in the non-collapsing situation. In general the Gromov-Hausdorff limit will be an orbifold hyperkähler surface, and the bubbles are ALE gravitational instantons, classified by Kronheimer in [Kro89].