Subsection 6.1 (004R) · ScalingStacks

6.1 Reduction to potential estimates

The common part of the strategy is to reduce the existence question of special Lagrangians to $C^{0}$ -estimate on the potential.

Given a fixed snc model $\mathcal{X}\to S$ , there is a logarithm map $\text{Log}_{\mathcal{X}}:X_{t}\to\Delta_{\mathcal{X}}$ defined up to $O(\frac{1}{|\log|t||})$ coordinate ambiguity. Given an $n$ -dimensional face $\Delta_{J}$ of $Sk(X)\subset\Delta_{\mathcal{X}}$ , we consider the preimage $U_{J,t}\subset X_{t}$ under the logarithm map, of a slightly shrinked version of the interior of $\Delta_{J}$ . We will take the liberty of shrinking $U_{J,t}$ several times, as long as the deleted sets have negligible Calabi-Yau measure in the $t\to 0$ limit. Since $U_{J,t}$ can be regarded as a torus invariant subset of $(\mathbb{C}^{*})^{n}$ , we can make sense of $C^{k}_{loc}$ norms uniformly in $t$ , by passing to the universal cover with the coordinates $\frac{\log z_{i}}{\log|t|}$ .

The first main step is to improve $C^{0}$ -estimate to $C^{\infty}$ -estimate.

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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 $\left\lVert\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}\right\rVert_{C^{k}_{loc}}\to 0.$

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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 $U_{J,t}$ .

∎

Prop. 6.3 implies the semiflat metric asymptote on the slightly shrinked $U_{J,t}$ , with $C^{\infty}$ small error in the $t\to 0$ limit:

\omega_{CY,t}\sim dd^{c}(\phi_{0}\circ\text{Log}_{\mathcal{X}})=\frac{\sqrt{-1}}{4\pi|\log|t||^{2}}\sum_{i,j}\frac{\partial^{2}\phi_{0}}{\partial x_{i}\partial x_{j}}d\log z_{i}\wedge d\overline{\log z_{j}}.

(19)

In terms of the Riemannian metric tensors,

g_{CY,t}\sim\frac{1}{2\pi|\log|t||^{2}}\text{Re}\{\sum_{1\leq i,j\leq n}\frac{\partial^{2}\phi_{0}}{\partial x_{i}\partial x_{j}}d\log z_{i}\otimes d\log\bar{z}_{j}\}.

(20)

In particular the Riemannian curvature stays uniformly bounded in $U_{J,t}$ . By the perturbation theory of special Lagrangians reviewed in section 4.6, the $T^{n}$ fibres of the logarithm map can be made into a special Lagrangian fibration by a $C^{\infty}$ small perturbation, on a slightly shrinked subset.

Suppose the assumption of Prop. 6.3 holds on all the $n$ -dimensional faces of $Sk(X)$ , then the union of all $U_{J,t}$ cover almost all the CY measure on $X_{t}$ , and the measure lost in the domain shrinking process is negligible. The weak metric version of the SYZ conjecture then follows.

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Remark 14. A subtlety is that the local regularity theory of real Monge-Ampère equation allows for Hausdorff codimension $1+\epsilon$ singularities. This means the codimension two singularity prediction in the Kontsevich-Soibelman conjecture cannot follow simply from the above argument. One must find a more global argument on $Sk(X)$ , not just on the interior of its $n$ -dimensional faces.