ScalingStacks

An influential early work related to the SYZ conjecture is the gluing description of Gross-Wilson [32] concerning the hyperkähler metric on K3 surfaces $X$ with elliptic fibrations $\pi:X\to\mathbb{CP}^{1}$.

By hyperkähler rotation, the special Lagrangian torus fibres can be viewed as the holomorphic elliptic curves in a different complex structure. In the generic situation, the elliptic fibration has 24 $I_{1}$-type singular fibres, namely the local singularity in the fibration is modelled on $(z_{1},z_{2})\mapsto z_{1}z_{2}$ complex geometrically. They fix Kähler classes $[\omega_{X}]$ on the K3 surface, and $[\omega_{\mathbb{P}^{1}}]$ on $\mathbb{CP}^{1}$, and describe the Calabi-Yau metrics $\omega_{\tau}$ in the Kähler class $\tau[\omega_{X}]+\pi^{*}[\omega_{\mathbb{P}^{1}}]$ for $0<\tau\ll 1$. Some key conceptual features are:

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  In the generic region, the metrics $\omega_{\tau}$ are up to exponentially small errors modelled on semiflat metrics, which can be explicitly described via the periods integrals of the elliptic curves. The subset of the K3 surface on which the semiflat metric asymptote breaks down, has length scale $O(\tau^{1/2}|\log\tau|^{1/2})$.

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  As $\tau\to 0$, the metrics $\omega_{\tau}$ converge in the Gromov-Hausdorff sense to a singular metric on $\mathbb{CP}^{1}$. The diameter of the K3 surfaces is of constant order $O(1)$. The length scale of generic elliptic curve fibres is $O(\tau^{1/2})$, which shrinks to zero size as $\tau\to 0$.

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  In the neighbourhood of the $I_{1}$ type singular fibres, the metrics are modelled on the Ooguri-Vafa metrics, which are explicit $S^{1}$-invariant incomplete hyperkähler metrics constructed by means of the Gibbons-Hawking ansatz.

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  The asymptotic geometry of the Ooguri-Vafa metric matches with the semiflat metric. This is a basic requirement for the gluing construction.

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  Near the nodal singular point of the $I_{1}$-fibre, the Ooguri-Vafa metric contains a small region approximated by the Taub-NUT metric, whose length scale is $O(\tau^{1/2}|\log\tau|^{-1/2})$. These regions concentrate almost the entire $L^{2}$-Riemannian curvature of the K3 surface.

The Gross-Wilson picture represents the best hope on the SYZ conjecture,²² 2 As a caveat sometimes overlooked in the literature, the hyperkähler rotation of the Gross-Wilson setting is not quite a polarized degeneration family, hence does not quite fit into our notion of large complex structure limit. In our perspective, Gross-Wilson is an inspiration, rather than an example of the SYZ conjecture. which includes an almost explicit description of the metric, and a special Lagrangian torus fibration exists globally on $X$. It is partially generalized to higher dimensional hyperkähler manifolds with holomorphic Lagrangian abelian variety fibrations. After hyperkähler rotation, these can be regarded as special Lagrangian fibrations. Tosatti et al. [72] [31] established the semiflat metric asymptote in the generic region, and Gromov-Hausdorff collapse to the base manifold.