ScalingStacks

We survey the recent progress on the metric aspect of the Stominger-Yau-Zaslow conjecture, which concerns the existence of special Lagrangian fibrations on Calabi-Yau manifolds near the large complex structure limit. The paper is based substantially on [52][53][51] and various subsequent talks. A rough outline of the contents of chapter 2,3,4,5,6 is as follows:

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  We trace the diverse motivations of the SYZ conjecture from mirror symmetry, minimal surfaces, Riemannian geometry, complex and non-archimedean geometry. We discuss the most optimistic interpretation and its difficulties, before moving on to a more cautious weak version.

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  We review the meaning of the large complex structure and essential skeleton from a complex geometric perspective, before a brief survey on the Kontsevich-Soibelman conjecture.

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  We explain the various analytic ingredients. A brief overview of Yau’s solution of the Calabi conjecture is given, to explain why new ideas are needed to understand the large complex structure limit. Our primary focus is on complex pluripotential theory, which is the analytic core of [52][53], alongside Savin’s small perturbation theorem in elliptic PDE, and the regularity theory of real Monge-Ampère equation.

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  We include a minimalistic overview of non-archimedean pluripotential theory: the notion of Berkovich spaces, semipositive metrices, and the non-archimedean Calabi metric. We also explain the intuition of the conjectural ‘comparison property’, which is needed to give a differential geometric interpretation of the non-archimedean Calabi metric.

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  We outline the proof of the recent progress [52][53], emphasizing on the intuition, the subtleties, and the open problems.