ScalingStacks

We aim to show $\left\lVert\phi_{CY,J,t}-\bar{\phi}_{J,0}\circ\text{Log}_{\mathcal{X}}\right\rVert_{C^{0}}$ on the slightly shrinked $U_{J,t}$ converges to zero along the subsequence. We know $\left\lVert\bar{\phi}_{CY,t}-\bar{\phi}_{J,0}\right\rVert_{C^{0}}\to 0$ along the subsequence, and from Remark 17, we know $|\phi_{CY,J,t}-\bar{\phi}_{CY,J,t}\circ\text{Log}_{\mathcal{X}}|$ is small except on a subset with small measure. Removing this small measure problem, is however rather subtle, and requires a global argument.

The strategy is:

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  (‘Extension problem’) Find a global Kähler metric $\omega_{\psi,t}$ on $X_{t}$ whose local potentials on $U_{J,t}$ agree with $\bar{\phi}_{J,0}\circ\text{Log}_{\mathcal{X}}$ up to $C^{0}$ small error, and whose potential with respect to a fixed Fubini-Study metric is bounded independent of $t$.

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  (Potential stability estimate) We can then consider the potential $\phi_{CY,rel}$ of the Calabi-Yau metric $\omega_{CY,t}$ relative to $\omega_{\psi,t}$. A small upper bound for $\phi_{CY,rel}$ on $U_{J,t}$ follows from $\phi_{CY,J,t}-\bar{\phi}_{CY,J,t}\circ\text{Log}_{\mathcal{X}}\leq\frac{C}{|\log|t||^{1/2}}$. We also know a small lower bound on the $\phi_{CY,rel}$ holds except on a set with very small measure, and then an application of Theorem 4.7 concludes a small lower bound on $\inf\phi_{CY,rel}$. We emphasize that the global positivity of Kähler metrics is essential for this argument.

  The net conclusion is that $\phi_{CY,rel}$ is $C^{0}$-small on a slightly shrinked version of $U_{J,t}$. This amounts to the smallness of $\left\lVert\phi_{CY,J,t}-\bar{\phi}_{J,0}\circ\text{Log}_{\mathcal{X}}\right\rVert_{C^{0}}$, which is our goal.

The extension problem is about patching local potentials to global Kähler potentials, and the difficulty is to achieve psh property in the non-generic region. The core problem, which is not satisfactorily solved in general, is

The extension problem is solved in an ad hoc way for the Fermat family, and constitutes the most technical part of [52].⁵⁵ 5 Technically, the paper [52] does not use the language of dual complexes and essential skeletons, but proceed via explicit charts controlled by tropical geometry. Recall the Fermat family embeds into an ambient projective space $\mathbb{CP}^{n+1}$. Our strategy is to produce the extension $\omega_{\psi,t}$ as a toric Kähler metric on $\mathbb{CP}^{n+1}$, and then restrict to $X_{t}$, which guarantees the global positivity. Ensuring that $\omega_{\psi,t}$ agrees with the local convex functions up to $C^{0}$-small error is a delicate matter, that involves the explicit tropical hypersurface combinatorics, exploits the large amount of discrete symmetry of the Fermat family, and uses a double Legendre transform construction [52].