ScalingStacks

1.1.1. Kummer construction [03FV]

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1.1.1. Kummer construction

We start with a flat orbifold 𝕋4/ℤ2\mathbb{T}^{4}/\mathbb{Z}_{2} given by the quotient of a flat 44-torus by the involution x↦−xx\mapsto-x. It has 16 orbifold singularities. One can resolve these singularities by gluing 16 Eguchi-Hanson spaces onto XX, which are complete hyperkähler ALE metrics defined on the cotangent bundle of S2S^{2}. By varying the flat structure on 𝕋4/ℤ2\mathbb{T}^{4}/\mathbb{Z}_{2} and the gluing parameters, one obtains an open set in the moduli space of all hyperkähler metrics on the K3⁡3\K 3 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 𝕋4/ℤ2\mathbb{T}^{4}/\mathbb{Z}_{2}, 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 K3⁡3\K 3 surface, and the bubbles are ALE gravitational instantons, classified by Kronheimer in [Kro89].

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