The authors rigorously build local hyper-Kähler model geometries, generalizing Ooguri-Vafa and multi-Ooguri-Vafa examples, from the Gaiotto-Moore-Neitzke Riemann-Hilbert formalism.
Collapsing Ricci-flat metrics on elliptic K3 surfaces
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abstract
For any elliptic K3 surface $\mathfrak{F}: \mathcal{K} \rightarrow \mathbb{P}^1$, we construct a family of collapsing Ricci-flat K\"ahler metrics such that curvatures are uniformly bounded away from singular fibers, and which Gromov-Hausdorff limit to $\mathbb{P}^1$ equipped with the McLean metric. There are well-known examples of this type of collapsing, but the key point of our construction is that we can additionally give a precise description of the metric degeneration near each type of singular fiber, without any restriction on the types of singular fibers.
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math.DG 1years
2025 1verdicts
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Hyper-K\"ahler manifolds from Riemann-Hilbert problems I: Ooguri-Vafa-like model geometries
The authors rigorously build local hyper-Kähler model geometries, generalizing Ooguri-Vafa and multi-Ooguri-Vafa examples, from the Gaiotto-Moore-Neitzke Riemann-Hilbert formalism.