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.