ov 2 00 1 Lectures on special Lagrangian geometry

Calabi–Yau m-folds (M,J, ω,Ω) are compact complex manifolds (M,J) of complex dimension m, equipped with a Ricci-flat Kähler metric g with Kähler form ω, and a holomorphic (m, 0)-form Ω of constant length |Ω| = 2. Using Algebraic Geometry and Yau’s solution of the Calabi Conjecture, one can construct them in huge numbers. String Theorists (a species of theoretical physicist) are very interested in Calabi–Yau 3-folds, and have made some extraordinary conjectures about them, in the subject known as Mirror Symmetry. Special Lagrangian submanifolds, or SL m-folds, are a distinguished class of real m-dimensional minimal submanifolds that may be defined in C, or in Calabi–Yau m-folds, or more generally in almost Calabi–Yau m-folds. They are calibrated with respect to the m-form ReΩ. They are fairly rigid and wellbehaved, so that compact SL m-folds N occur in smooth moduli spaces of dimension b(N), for instance. They are important in String Theory, and are expected to play a rôle in the eventual explanation of Mirror Symmetry. This paper is a much shortened version of [22]. It 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. The author’s paper [22] contains essentially all the material below, together with introductions to holonomy groups, Kähler geometry, the Calabi Conjecture, Calabi–Yau manifolds, and calibrated geometry. Sections 2 and 3 discuss general properties of special Lagrangian submanifolds of C, and ways to construct examples. Then §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 §6 we describe a mostly conjectural picture of the singularities of compact SL m-folds, particularly in generic almost Calabi–Yau m-folds. Finally, §7 briefly introduces String Theory, Mirror Symmetry 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.