Lectures on special Lagrangian geometry

By D. Joyce. 2001

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expected to play a role in the eventual explanation of Mirror Symmetry. This article is intended as an introduction to special Lagrangian geometry, and a survey of the author's research on the singularities of SL m-folds, of directions in which the subject might develop in the next few years, and of possible applications of it to Mirror Symmetry and the SYZ Conjecture. Sections 2 and 3 discuss general properties of special Lagrangian submani- folds of C m , and ways to construct examples. Then Section 4 defines Calabi-Yau and almost Calabi-Yau manifolds, and their special Lagrangian submanifolds. Section 5 discusses the deformation and obstruction theory of compact SL m- folds, and properties of their moduli spaces. In Section 6 we describe a theory of isolated conical singularities in compact SL m-folds. Finally, Section 7 briefly introduces String Theory, Mirror Symme- try and the SYZ Conjecture, a conjectural explanation of Mirror Symmetry of Calabi-Yau 3-folds, and discusses mathematical progress towards clarifying and proving the conjecture. Acknowledgments. This article is based on lecture courses given to the Summer School on Symplectic Geometry in Nordfjordeid, Norway, in June 2001, and to the Clay Institute's Summer School on the Global Theory of Minimal Surfaces,

Subfields: Applied Mathematics, Geometry and Topology

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