ScalingStacks

1.1.2. Codimension- 1 collapse [03FW]

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1.1.2. Codimension-11 collapse

In [Fos16] Foscolo constructed a family of hyperkähler metrics on a K3⁡3\K 3 surface that collapses to the flat orbifold 𝕋3/ℤ2\mathbb{T}^{3}/\mathbb{Z}_{2}. The collapse has bounded curvature away from finitely many points, and is given by shrinking the fibers of an S1S^{1}-fibration. In the simplest case, curvature blow-up occurs at the 8 singular points of 𝕋3/ℤ2\mathbb{T}^{3}/\mathbb{Z}_{2}, where the bubbles are given by complete hyperkähler spaces with cubic volume growth, which in this case are ALF-D2D_{2} spaces. See [CC15, Min11] for a partial classification of hyperkähler ALF spaces. Let us also point out that the results of [Fos16] have motivated the study of codimension-11 collapse of G2G_{2}-manifolds to 33-dimensional Calabi-Yau manifolds in [FHN17].

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