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.
6.2.1 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 , 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 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 to , we must guarantee the global positivity. The difficulty lies in the non-generic regions where the complex structure on 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 with , which contains the essential skeleton.
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The notion of semipositive metric is built into NA geometry.