Lectures on Calabi-Yau and special Lagrangian geometry

TL;DR

This paper provides an accessible introduction to Calabi-Yau and special Lagrangian geometry, surveys recent advances on singularities, and discusses their implications for the SYZ Conjecture, aimed at graduate students and researchers.

Contribution

It offers a comprehensive, self-contained overview of Calabi-Yau and special Lagrangian geometry, including recent results on singularities and their role in mirror symmetry.

Findings

Analysis of singularities modeled on cones

Deformation theory of special Lagrangian submanifolds

Results on singularities in Calabi-Yau 3-folds

Abstract

This paper gives a leisurely introduction to Calabi-Yau manifolds and special Lagrangian submanifolds from the differential geometric point of view, followed by a survey of recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. It is aimed at graduate students in Geometry, String Theorists, and others wishing to learn the subject, and is designed to be fairly self-contained. It is based on lecture courses given at Nordfjordeid, Norway and MSRI, Berkeley in June and July 2001. We introduce Calabi-Yau m-folds via holonomy groups, Kahler geometry and the Calabi Conjecture, and special Lagrangian m-folds via calibrated geometry. `Almost Calabi-Yau m-folds' (a generalization of Calabi-Yau m-folds useful in special Lagrangian geometry) are explained and the deformation theory and moduli spaces of compact special Lagrangian submanifolds in (almost) Calabi-Yau m-folds is described. In the final part we consider isolated singularities of special Lagrangian m-folds, focussing mainly on singularities locally modelled on cones, and the expected behaviour of singularities of compact special Lagrangian m-folds in generic (almost) Calabi-Yau m-folds. String Theory, Mirror Symmetry and the SYZ Conjecture are briefly discussed, and some results of the author on singularities of special Lagrangian fibrations of Calabi-Yau 3-folds are described.