The purpose of this paper is to relate a metric version of the Strominger-Yau-Zaslow (SYZ) conjecture to non-archimedean (NA) pluripotential theory. One interpretation of the SYZ conjecture [41] is the following:

This interpretation puts the CY metrics at the forefront, in contrast to alternative softer viewpoints which emphasize the algebraic, symplectic or topological aspects. Here the generic region means a subset of $X_{t}$ with almost the full percentage of the CY measure on $X_{t}$. This represents a compromise, as the difficulty of finding a special Lagrangian fibration on the entire $X_{t}$ is well appreciated since [26].

The potential relevance of NA geometry to SYZ conjecture was suggested in Kontsevich and Soibelman [29][28]. NA pluripotential theory is taken much further by Boucksom et al [5][4][3][2]. Impressionistically,

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  NA geometry offers a natural language to describe the degeneration of complex manifolds into real simplicial/tropical objects.

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  It systematically encodes the combinatorics of tropical geometry.

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  It encodes much of the birational geometry for the degeneration family.

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  Usual notions in Kähler geometry such as functions, line bundles, Kähler metrics, complex Monge-Ampère measures, have natural (albeit exotic looking) analogues in NA geometry.

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  There is a natural way for CY volume measures to converge into a measure on a NA space.

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  An analogue of the Calabi conjecture holds in the NA context: one can solve the NA Monge-Ampère equation.

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  Under additional hypotheses, the NA Monge-Ampère measure agrees with the real Monge-Ampère measure.

In short, NA spaces are the natural candidates for limits of CY metrics relevant for the SYZ conjecture. While NA pluripotential theory has close analogy with Kähler geometry, so far it has found no direct implication on the behaviour of degenerating CY metrics. The goal of this paper is to show that, assuming a comparison property between NA vs. real MA equations, then the crank of NA pluripotential theory can be turned to prove the metric SYZ conjecture in quite satisfactory generality, at least in an algebraic setup. This largely accomplishes the reduction of this metric SYZ conjecture to a problem in NA geometry.

We work over $\mathbb{C}$. To set the scene,

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  Let $S$ be a smooth affine algebraic curve, with a point $0\in S$. An algebraic degeneration family is given by a submersive projective morphism $\pi:X\to S\setminus\{0\}$ with smooth connected $n$-dimensional fibres $X_{t}$ for $t\in S\setminus\{0\}$. This is in contrast with the formal setting over the punctured formal disc $\text{Spec}(K)$ with $K=\mathbb{C}(\!(t)\!)$. An algebraic degeneration induces a formal degeneration by base change.

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  A polarisation is given by an ample line bundle $L$ over $X$.

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  We say $\pi$ is a degeneration family of Calabi-Yau manifolds if there is a trivialising section $\Omega$ of the canonical bundle $K_{X}$. Over a small disc $\mathbb{D}_{t}$ around $0\in S$, this induces holomorphic volume forms $\Omega_{t}$ on $X_{t}$ via $\Omega=dt\wedge\Omega_{t}$. The normalised Calabi-Yau measure on $X_{t}$ is the probability measure

  $$d\mu_{t}=\frac{\Omega_{t}\wedge\overline{\Omega}_{t}}{\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}}.$$

  (1)

  The Calabi-Yau metrics $\omega_{CY,t}$ on $X_{t}$ are the unique Kähler metrics in the class $\frac{1}{|\log|t||}c_{1}(L)$ such that

  $$\frac{\omega_{CY,t}^{n}}{\int_{X_{t}}\omega_{CY,t}^{n}}=d\mu_{t}.$$

  (2)

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  We say $\pi:X\to S\setminus\{0\}$ is a maximal degeneration of Calabi-Yau manifolds if the essential skeleton has the maximal dimension $n$, and the degeneration family admits a semistable snc model over $S$. Maximal degenerations are also known as large complex structure limits (cf. section 3.1 for more details).

We now briefly explain the context of the comparison property between NA vs. real MA equations, referring the details to section 3. Given a polarized algebraic maximal degeneration family of CY manifolds, one can canonically associate a NA object called the Berkovich space $X_{K}^{an}$, which can be viewed as the inverse limit of all the dual intersection complexes $\Delta_{\mathcal{X}}$ of snc models over the formal disc. There is a Lebesgue measure $d\mu_{0}$ supported on the essential skeleton $Sk(X)\subset X_{K}^{an}$, which is the natural limit of normalised CY measures on $X_{t}$ in a suitable sense. A central result of NA pluripotential theory due to Boucksom-Favre-Jonsson [4] is the solution of the NA Calabi conjecture. In this setting, it provides a unique (up to scale) continuous semipositive metric $\left\lVert\cdot\right\rVert_{CY}$ on $X_{K}^{an}$ with the polarisation $L$, which solves the NA MA equation

where $(L^{n})=\int_{X_{t}}c_{1}(L)^{n}$. The NA MA measure is defined by intersection theory, so this is not even a partial differential equation as it stands, although the theory shares many features of Kähler geometry. Given a model $\mathcal{L}$ of the line bundle $L$, then the semipositive metric can be represented by a potential function $\phi_{0}$ on $X_{K}^{an}$, similar to the usual relation in Kähler geometry between Hermitian metrics on line bundles and potential functions.

Now given any snc model $\mathcal{X}$ over the formal disc, there is a retraction map $r_{\mathcal{X}}:X_{K}^{an}\to\Delta_{\mathcal{X}}$ onto the dual intersection complex, and $Sk(X)$ sits inside $\Delta_{\mathcal{X}}$ as a simplicial subcomplex, with dimension $n$ by the maximal degeneration assumption. The $n$-dimensional open faces $\text{Int}(\Delta_{J})$ of $Sk(X)$ have a canonical integral affine structure. In local affine coordinates, the Lebesgue measure $d\mu_{0}$ is a constant multiple of $dx_{1}\ldots dx_{n}$. Assuming $\mathcal{L}$ lives over $\mathcal{X}$, the semipositivity of $\left\lVert\cdot\right\rVert_{CY}$ implies that the potential $\phi_{0}$ is a convex function on $\text{Int}(\Delta_{J})$, similar to the fact that Kähler metrics are locally represented by psh functions. Consequently, the restriction of $\phi_{0}$ to $\text{Int}(\Delta_{J})\subset Sk(X)$ has a well defined real MA measure $MA_{\mathbb{R}}(\phi_{0})$.

The comparison property hypothesis (cf. section 3.5) requires that there is some semistable snc model $\mathcal{X}$ over the formal disc as above, such that over all the $n$-dimensional open faces $\text{Int}(\Delta_{J})$ of $Sk(X)\subset\Delta_{\mathcal{X}}$, the function $\phi_{0}=\phi_{0}\circ r_{\mathcal{X}}$ factors through the retraction map. This means the information of the potential $\phi_{0}$ on the abstract looking space $X_{K}^{an}$ is largely contained in a convex function on these open faces. Morever, a recent result of Vilsmeier [45] implies that under this comparison hypothesis, then $\phi_{0}$ satisfies a real MA equation on these open faces:

In particular, the NA MA equation is closely related to a PDE after all. Such a relation between NA geometry and the real MA equation is in the spirit of the Kontsevich-Soibelman conjecture [29] about Gromov-Hausdorff limits of maximally degenerate CY metrics.

The main theorem of this paper is

The proof strategy is the following. It has already been understood in the author’s previous paper [32] that for the existence of the special Lagrangian fibration in the generic region, it is enough to prove that the Kähler potential for the CY metric on $X_{t}$ is $C^{0}$-close to a solution of the real Monge-Ampère equation at least in the generic region. The natural candidate of this real MA solution comes from the Boucksom-Favre-Jonsson solution of the NA MA equation. To bridge the gap between the NA space and the complex manifold $X_{t}$, we go through a Fubini-Study $C^{0}$-approximation of the potential, to get a Kähler metric whose potential is $C^{0}$ close to the BFJ solution in a suitable sense. Here it is crucial to preserve positivity. We then regularize it to make its volume form close to being CY in the generic region. We compare the regularized potential to the CY potential in $C^{0}$ by adapting an $L^{1}$-stability argument of Kolodziej; making this work on the highly degenerate complex manifolds $X_{t}$ requires a uniform Skoda estimate, proved in a companion paper [33].

In this strategy the appeal to NA geometry is for the following reasons.

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  NA geometry is a natural language to discuss intrinsic properties of the degeneration family independent of the choice of models.

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  The NA MA solution is a natural candidate solution to the real MA equation on the essential skeleton $Sk(X)$.

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  The NA framework effectively encodes the notion of positivity (psh properties). This gives one answer to the question: what is the appropriate analogous notion of Kähler potentials in the maximal degeneration limit?

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  The solution to the NA MA equation is known to be unique. This allows one to expect that the limit of the CY metrics on $X_{t}$ is unique at least in the generic region, without the need to pass to subsequences.

The organization of the paper is as follows. We collect some essential analytic ingredients in section 2; in particular we prove a uniform one-sided version of the $L^{1}$-stability estimate by adapting an argument of Kolodziej. Section 3 is an introduction of NA geometry for differential geometers, and the emphasis is on the relation with Kähler geometry. Section 4 works on the metric SYZ conjecture proper, and the key is to prove the $C^{0}_{loc}$-convergence of the local potentials of the CY metric towards the NA solution, assuming the comparison property. We end this section with some discussions about the closely related Kontsevich-Soibelman conjecture. Section 5 presents problems and speculations beyond the SYZ setting. It contains a heuristic general formula for NA MA measure, and suggests how the NA MA equation is related to non-maximal degenerations by proposing a generalised Calabi ansatz.