Survey on the metric SYZ conjecture and non-Archimedean geometry

Abstract

We survey the metric aspects of the Strominger–Yau–Zaslow conjecture on the existence of special Lagrangian fibrations on Calabi–Yau manifolds near the large complex structure limit. We will discuss the diverse motivations for the conjectural picture, what the best hopes are, and a number of subtleties. The bulk of the survey highlights the role of pluripotential theory, and non-Archimedean geometry in particular, with a list of open questions.