Proposition 6.3 (003G) · ScalingStacks

003G

Proposition 6.3. (cf. [53, section 4.5]) Let $\phi_{0}$ be an Alexandrov solution of the real MA equation (17) on the interior of $\Delta_{J}$ . Suppose the Calabi-Yau metrics on $U_{J,t}$ admit local potential functions $\phi_{CY,J,t}$ such that $\omega_{CY,t}=dd^{c}\phi_{CY,J,t}$ , and $\left\lVert\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}\right\rVert_{C^{0}}\to 0$ as $t\to 0$ . Then after slightly shrinking $U_{J,t}$ , we have the $C^{\infty}$ -asymptote $\left\lVert\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}\right\rVert_{C^{k}_{loc}}\to 0.$

003H

Proof. (Sketch)

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The first ingredient is that by the regularity theory of real MA equation (cf. section 4.5), after deleting a subset of $\text{Int}(\Delta_{J})$ of Hausdorff $(n-1)$ -measure zero, then $\phi_{0}$ is smooth. After a slight shrinking of the remaining open set, then $\phi_{0}$ has $C^{k}$ bounds.
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The second ingredient is Savin’s small perturbation theorem (cf. section 4.8). After passing to the local universal cover, both $\phi_{CY,J,t}$ and $\phi_{0}\circ\text{Log}_{\mathcal{X}}$ solve a complex Monge-Ampère equation. The difference in their RHS vanishes in the $t\to 0$ limit in arbitrarily high $C^{k}$ norm, as a consequence of the volume form asymptote in section 3.1. Savin’s result then improves the $C^{0}$ closeness of $\phi_{CY,J,t}$ and $\phi_{0}\circ\text{Log}_{\mathcal{X}}$ to $C^{\infty}_{loc}$ closeness, after small shrinking of $U_{J,t}$ .

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