ScalingStacks

Very little is proven about the non-archimedean CY metric $\left\lVert\cdot\right\rVert_{CY}$ beyond existence and continuity. We now discuss the meaning of the following conjectural NA MA-real MA comparison property.

Notice $\text{Int}(\Delta_{J})\subset\Delta_{\mathcal{X}}$ inherits a natural integral affine structures. Since the restriction of $\phi_{0}$ is convex on these faces by Prop. 5.2, its real MA measure makes sense, and by Prop. 5.4 it satisfies the real MA equation on $\text{Int}(\Delta_{J})$

A few comments are in order:

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  The comparison property is a conjecture in algebraic/non-archimedean geometry, and does not a priori involve PDE concepts. Its PDE implications come a posteriori.

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  The Kontsevich-Soibelman picture (cf. section 3.3) expects that there is a solution of the real MA equation on the essential skeleton, away from some singular locus. The NA-MA equation via the comparison property is the only known systematic method to produce solutions.

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  In the context of toric invariant metrics on toric varieties, there are comparison results between NA MA measure and real MA equation, cf. [28, Prop. 4.4.4].

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  The reader may feel that NA pluripotential theory is a very long-winded way to solve the real MA equation. However, surprisingly enough, it is not even known how to formulate the real MA equation globally on $Sk(X)$ in general, not just on the $n$-dimensional faces but also on the lower dimensional faces.

  One problem is that $Sk(X)$ does not come with an obvious preferred affine structure, but only a piecewise affine structure, so there is no obvious coordinate independent definition of the real MA measure. It seems that the affine structure conjectured by Kontsevich and Soibelman would need to be solved simultaneously with the real MA equation, rather like free boundary PDE problems.

  Another problem is that solving the real MA equation requires first specifying the class of convex functions to be admitted as potentials, just like solving the complex Monge-Ampère equation requires first specifying the meaning of Kähler potentials. We do not currently know any direct way of defining the class of convex potentials on piecewise affine manifolds such as $Sk(X)$. The semipositive metrics on the Berkovich space $X_{K}^{an}$, abstract as it may be, is our only available substitute.

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  If one believes the NA Calabi-Yau metric $\left\lVert\cdot\right\rVert_{CY}$ is the potential theoretic limit of the Calabi-Yau metrics on $X_{t}$ in the hybrid topology, and that the information of $\left\lVert\cdot\right\rVert_{CY}$ can be recovered from data on $Sk(X)$, then one may be inclined to think that the potential of $\left\lVert\cdot\right\rVert_{CY}$ factors globally through some retraction map $X_{K}^{an}\to Sk(X)$, defined perhaps through some divisorial log terminal minimal model.

  Such a statement would need to confront the difficulty that the divisorial log terminal model is not necessarily unique, and in principle the retraction map depends on the choice of the model. It seems highly nontrivial how the NA MA equation would select a preferred retraction map.

  The formulation of the comparison property is more cautious than this. We allow the dual intersection complex $\Delta_{\mathcal{X}}$ to be strictly bigger than $Sk(X)$, and there is no assumption on the complement of the $n$-dimensional faces of $Sk(X)$. Regarding the problem above, if one is undecided between a finite number of candidate retraction maps, then one can pass to a common snc resolution (and perhaps pass to finite base change, to find a semistable snc resolution). Of course, the more we blow up the model $\mathcal{X}$, the weaker is the comparison property.

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  Remark 13. The very recent work of Pille-Schneider and Mazzon [59] proposes gluing the retraction maps associated to several divisorial log terminal models to obtain a map $X_{K}^{an}\to Sk(X)$. Their map still factors through the dual intersection complex of some larger snc model, hence is compatible with the comparison property.

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  Without the comparison property, it seems hard to give any differential geometric interpretation to $\left\lVert\cdot\right\rVert_{CY}$ at all, since $X_{K}^{an}$ contains arbitrarily large dual intersection complexes, and thus is highly complicated. The heuristic intuition of this hypothetical scenario, is that the potential theoretic limit of the Calabi-Yau metrics would require infinitely many blow ups to describe. This is not yet ruled out by a theorem; we leave the reader to judge its plausibility.

The open question for algebraic geometers is