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, . . . , xn locally modelled on special Lagrangian cones C1, . . . , Cn in ℂm 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#...#Nk of SL m-folds N1, . . . , Nk 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 xi and cones Ci 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 Nt in M for small t > 0. Let Li be an asymptotically conical SL m-fold in ℂm asymptotic to Ci at infinity. We make Nt by gluing tLi into X at xi for i = 1, . . . , n.