Special Lagrangian Submanifolds with Isolated Conical Singularities. v. Survey and Applications

Abstract

This is the last in a series of five papers studying compact special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x₁, . . . , x_n locally modelled on special Lagrangian cones C₁, . . . , C_n in ℂ^m with isolated singularities at 0. Readers are advised to begin with this paper.

We survey the major results of the previous four papers, giving brief explanations of the proofs. We apply the results to describe the boundary of a moduli space of compact, nonsingular SL m-folds N in M. We prove the existence of special Lagrangian connected sums N₁#...#N_k of SL m-folds N₁, . . . , N_k in M. We also study SL 3-folds with T²-cone singularities, proving results related to ideas of the author on invariants of Calabi-Yau 3-folds, and the SYZ Conjecture.

Let X be a compact SL m-fold with isolated conical singularities x_i and cones C_i for i = 1, . . . , n. The first paper studied the regularity of X near its singular points, and the the second the moduli space of deformations of X. The third and fourth papers construct desingularizations of X, realizing X as a limit of a family of compact, nonsingular SL m-folds N^t in M for small t > 0. Let L_i be an asymptotically conical SL m-fold in ℂ^m asymptotic to C_i at infinity. We make N^t by gluing tL_i into X at x_i for i = 1, . . . , n.