6.2 Strategy I: non-archimedean geometry [004S]

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6.2 Strategy I: non-archimedean geometry

The remaining task is to achieve the local $C^{0}$ -convergence of local potentials to a solution of the real MA equation on the open $n$ -dimensional faces of $Sk(X)$ (cf. Prop. 6.3). The first strategy [53] is:

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Solve the real MA equation on $Sk(X)$ , independent of the CY metrics on $X_{t}$ .
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Then attempt to compare the solution with the potential of the CY metrics on $X_{t}$ . First, one needs to produce a Kähler metric on $X_{t}$ whose local potential is $C^{0}$ -close to the real MA solution on $Sk(X)$ in some topology. Then one needs some version of the $L^{1}$ -volume stability estimate (cf. section 4.3) to show the $C^{0}$ -smallness of the relative potential between this Kähler metric and the CY metric, at least in the generic region.

6.2.1 Motivation for NA geometry

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.

Remark 15.

A byproduct of the non-archimedean approach, is that the limit of Calabi-Yau local potentials is in fact independent of subsequence, since the non-archimedean analogue of the Calabi-Yau metric is known to be unique.

6.2.2 Grafting the real MA solution

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).

Proposition 6.4.

[53, Lemma 4.1, 4.2] Given any $\epsilon\ll 1$ , and let $t$ be small enough depending on $\epsilon$ . There is a Kähler metric $\omega_{\psi,t}$ , such that

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On $\text{Log}_{\mathcal{X}}^{-1}(\text{Int}(\Delta_{J}))$ , the local Kähler potentials $\phi_{J,t}$ of $\omega_{\psi,t}$ can be chosen to satisfy $|\phi_{J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}|<\epsilon$ .
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The total variation $\int_{X_{t}}|\frac{|\log|t||^{n}\omega_{\psi,t}^{n}}{(L^{n})}-d\mu_{t}|<\epsilon.$
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The Kähler potential of $\omega_{\psi,t}$ relative to a fixed Fubini-Study reference metric, is uniformly bounded independent of $t,\epsilon$ .

Proof.

(Sketch)

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We first $C^{0}$ approximate the NA metric $\left\lVert\cdot\right\rVert_{CY}$ by some NA Fubini-Study metric, which arises naturally as a hybrid topology limit of usual Fubini-Study metrics on $X_{t}$ (cf. section 5.3). The Fubini-Study metrics are positive, and by construction their local potentials differ from $\phi_{0}\circ\text{Log}_{\mathcal{X}}$ by an arbitrarily small amount in the $C^{0}$ sense.
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We do not have direct control on the volume form of the Fubini-Study metrics; the degrees of the associated projective embeddings are gigantic. In contrast, the volume form of the local potential $\phi_{0}\circ\text{Log}_{\mathcal{X}}$ has negligible difference from $\frac{(L^{n})}{|\log|t||^{n}}d\mu_{t}$ , by the volume asymptote in section 3.1 and the real MA equation (17).
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The idea is to perform a further regularization. We modify the Fubini-Study metric in the generic region of $X_{t}$ , so that it essentially agrees with $\phi_{0}\circ\text{Log}_{\mathcal{X}}$ in the generic region up to $C^{2}$ -small error. In this step we appealed also to the regularity theory of real MA equation. The end result is $\omega_{\psi,t}$ , which is Kähler by construction.
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In the non-generic region, we do not perform regularization. Since the generic region already takes up $99.9\%$ of the $\omega_{\psi,t}^{n}$ measure for $|t|\ll 1$ , the non-generic region has negligible total measure. We use this to argue for the total variation bound.

∎

6.2.3 $C^{0}$ -convergence of the potential

It remains to show

Proposition 6.5.

Up to slightly shrinking the domains, 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 $t\to 0$ .

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

d\mu_{t}(\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}\geq\kappa/4)<\lambda.

(21)

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

0\leq\phi_{CY,J,t}-\phi_{0}\circ\text{Log}_{\mathcal{X}}<\kappa,

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.

Remark 16.

The $C^{0}$ convergence statement only applies to the generic region. We do not know the answer to

Question 7.

Do the potentials of the CY metrics on $X_{t}$ converge to the NA CY metric $\left\lVert\cdot\right\rVert_{CY}$ on $X_{K}^{an}$ globally in the hybrid topology?