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Collapsing of K3 Surfaces and Special K\"ahler Structures
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abstract
We study the structure of $\mathfrak{M}_2$, the set of half-dimensional collapsing spaces of hyperk\"ahler metrics on K3 surfaces. We show that $\mathfrak{M}_2$ consists precisely of those underlying metric spaces of integral singular special K\"ahler structures (SKSs) on $ \mathbb{P}^1 $. Furthermore, we establish a bijection between integral singular SKSs on $\mathbb{P}^1 $ and Jacobian elliptic K3 surfaces with a marked holomorphic volume form. Additionally, we compute the number of Jacobian elliptic K3 surfaces that correspond to a given metric on $ \mathbb{P}^1 $.
Forward citations
Cited by 2 Pith papers
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Compactification of metric moduli space of $K3$ surfaces
The Gromov–Hausdorff compactification of unit-diameter hyperkähler K3 metrics is homeomorphic to the adjoint Satake compactification of the K3 period domain.
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Ricci-flat metrics on Calabi-Yau manifolds
A survey of the degeneration theory of Ricci-flat Kahler metrics on Calabi-Yau manifolds: smooth limits for semiample and nef-and-big classes, path-dependent counterexamples at the boundary, and a list of open conjectures.
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