Lectures on special Lagrangian geometry

TL;DR

This paper introduces special Lagrangian submanifolds in complex and Calabi-Yau manifolds, surveys their singularities, and discusses their role in the SYZ Conjecture and Mirror Symmetry, aimed at graduate students and researchers.

Contribution

It provides a comprehensive overview of the theory, construction, and singularities of special Lagrangian submanifolds, including deformation and obstruction theories, with applications to the SYZ Conjecture.

Findings

Construction methods for special Lagrangian submanifolds described

Singularities modeled on special Lagrangian cones analyzed

Applications to SYZ Conjecture and Mirror Symmetry discussed

Abstract

We introduce special Lagrangian submanifolds in C^m and in (almost) Calabi-Yau manifolds, and survey recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. The paper is aimed at graduate students in Geometry, String Theorists, and others wishing to learn the subject. Special Lagrangian m-folds in C^m are defined, and ways of constructing them described. 'Almost Calabi-Yau manifolds' (a generalization of Calabi-Yau manifolds useful in special Lagrangian geometry) are introduced, and the deformation theory, obstruction theory, and moduli spaces of compact special Lagrangian m-folds in (almost) Calabi-Yau m-folds are explained. Then we consider singular special Lagrangian submanifolds which are locally modelled on special Lagrangian cones with an isolated singularity at 0. Compact singular special Lagrangian submanifolds of this type have a well-behaved deformation theory, and can often be realized as limits of families of compact, nonsingular special Lagrangian submanifolds. Applications of this to the SYZ Conjecture and Mirror Symmetry of Calabi-Yau 3-folds are discussed.