ScalingStacks

The above strategy contains many problems:

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  As discussed in section 5.6, it is unknown how to directly formulate the real MA equation on $Sk(X)$, nor do we know the precise class of convex functions needed for such formulations.

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  The essential skeleton is a simplicial complex, and $X_{t}$ is a complex manifold. These are conceptually very different objects, and we need a topology to unify both sides.

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  Pluripotential theoretic arguments require the global positivity (i.e. psh property) of Kähler potentials (cf. Remark 6). Thus when we graft the real MA solution from $Sk(X)$ to $X_{t}$, we must guarantee the global positivity. The difficulty lies in the non-generic regions where the complex structure on $X_{t}$ is highly singular.

These problems point naturally towards NA geometry:

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  The NA MA-real MA comparison property is a natural way to produce solutions.

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  The hybrid topology is a natural topology to compare $X_{t}$ with $X_{K}^{an}$, which contains the essential skeleton.

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  The notion of semipositive metric is built into NA geometry.

Let $(\mathcal{X},\mathcal{L})$ be a semistable snc model with $\mathcal{L}|_{X}=L$. The NA pluripotential theory provides a continuous semipositive metric $\left\lVert\cdot\right\rVert_{CY}=\left\lVert\cdot\right\rVert_{\mathcal{L}}e^{-\phi_{0}}$ on $L$ over $X_{K}^{an}$ solving the NA MA equation (16), which we assume henceforth satisfies the NA MA-real MA comparison property, so $\phi_{0}$ solves the real MA equation over the $n$-dimensional open faces $\text{Int}(\Delta_{J})$ of the essential skeleton $Sk(\mathcal{X})$ (cf. section 5.5).

It remains to show

Prop. 6.4 says that the local potential of $\omega_{\psi,t}$ and $\phi_{0}\circ\text{Log}_{\mathcal{X}}$ differ negligibly in the $t\to 0$ limit in the $C^{0}$-sense. Ideally, one would like to use some version of $L^{1}$-volume stability to conclude the $C^{0}$-smallness of the relative potential between $\omega_{\psi,t}$ and $\omega_{CY,t}$. Unfortunately, due to the difficulty of regularization in the non-generic region, there is very little control on the volume density of $\omega_{\psi,t}$ in the non-generic region, and we cannot conclude a Skoda type estimate like (11) for $\omega_{\psi,t}$. This technical problem causes an asymmetry between $\omega_{\psi,t}$ and $\omega_{CY,t}$, and only ‘one half’ of the $L^{1}$-volume stability estimate (cf. section 4.3) applies, which is why we designed Theorem 4.7.

After the dust settles, Theorem 4.7 implies that $\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}$ concentrates near its minimum value (normalized to be zero) on a subset with almost $100\%$ of the Calabi-Yau measure (cf. [53, Prop 4.4, Cor. 4.6]). More precisely, for any given small number $\kappa,\lambda\ll 1$, then for sufficiently small $t$, the measure

On a slightly shrinked version of $U_{J,t}$, this can be improved to the $C^{0}$-control

by a slightly tricky application of the mean value inequality (cf. [53, Thm. 4.7]). Since $\kappa$ is arbitrary, this achieves Prop. 6.5, which verifies the hypothesis of Prop. 6.5, whence the weak metric version of the SYZ conjecture.

Recall the logarithm map $\text{Log}_{t}:(\mathbb{C}^{*})^{n}\to\mathbb{R}^{n}$.

Consider an open convex subset $U\subset\mathbb{R}^{n}$, and let $\phi$ be a psh function on $\text{Log}_{t}^{-1}(U)\subset(\mathbb{C}^{*})^{n}$.

An important intuition is that on sufficiently collapsed toric regions inside $(\mathbb{C}^{*})^{n}$, bounded Kähler potentials have a strong tendency to be approximated by convex functions.

In our applications, the psh functions $\phi$ arise from the local potentials $\phi_{CY,J,t}$ of the Calabi-Yau metrics $\omega_{CY,t}$ on toric charts inside $X_{t}$. Since $\omega_{CY,t}$ has uniformly bounded potential with respect to Fubini-Study reference metrics (cf. Thm. 4.6), it is easy to arrange the local potentials $\phi_{CY,J,t}$ on toric charts to be uniformly bounded, whence the convex functions $\bar{\phi}_{CY,J,t}$ are also uniformly bounded. Using the Lipschitz bound, by Arzela-Ascoli, we can extract a collection of subsequential limits as $t\to 0$. By construction $\bar{\phi}_{CY,J,t}\to\bar{\phi}_{J,0}$ in the $C^{0}_{loc}$ sense on the interior of the $n$-dimensional faces of $Sk(X)$.

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].

To complete the circle, we need

The strategy is to pass the complex MA equation on $U_{J,t}$ to the limit. This is feasible, morally because the complex MA operator is weakly continuous under the $C^{0}$-convergence of potentials. In our setting, an extra subtlety is that the sequence of potentials are defined on different manifolds, and the modification of the usual arguments are carried out in [52, section 5.1].

The a priori limit strategy does not explicitly appeal to NA geometry. Its principal remaining difficulty is the extension problem. Based on the experience with the Fermat example, we anticipate that extension to toric metrics on ambient toric varieties is a useful technique, and the problem may have a substantially combinatorial aspect. As we emphasized in section 5.6, an explicit class of convex potentials would also be essential for a direct formulation of the real MA equation, which is likely needed for more refined questions such as the affine structure and the singular set of the real MA metric on $Sk(X)$ (cf. the Kontsevich-Soibelman conjecture in section 3.3).

In contrast, the NA pluripotential theory is built around the central concept of NA semipositive metrics on $X_{K}^{an}$, which extend up to $C^{0}$-small error to Kähler potentials on $X_{t}$ via the Fubini-Study approximation. In that respect, NA pluripotential theory may be viewed as a disguised solution of the extension problem. To make contact with differential geometric applications, however, requires some additional hypothesis such as the NA MA-real MA comparison property. Comparing the difficulties in the two strategies, we speculate that proving the NA MA-real MA comparison property requires a more concrete characterization of NA semipositive metrics, perhaps of explicitly combinatorial nature.