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. OverviewWe 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:• 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.• 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. Metric SYZ conjectureThe Strominger-Yau-Zaslow conjecture is originally motivated by a combination of physical and differential geometric considerations, and stands at the crossroad of mirror symmetry, minimal surface theory, Riemannian geometry, complex Kähler geometry, and non-archimedean geometry. We will trace some historically significant developments, to see how the gradual realization of the analytic difficulties, has led to eclectic interpretations of the conjecture.