Survey on the metric SYZ conjecture and non-archimedean geometry

TL;DR

This survey explores the metric aspects of the SYZ conjecture on special Lagrangian fibrations in Calabi-Yau manifolds, emphasizing non-archimedean geometry and pluripotential theory, and discusses open questions in the field.

Contribution

It provides a comprehensive overview of the current state, motivations, and challenges related to the SYZ conjecture, highlighting the role of non-archimedean geometry.

Findings

Highlights the importance of pluripotential theory in SYZ conjecture

Connects non-archimedean geometry with Calabi-Yau metrics

Identifies key open problems in the field

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.