Special Lagrangian submanifolds with isolated conical singularities. V. Survey and applications

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This is the last in a series of five papers math.DG/0211294, math.DG/0211295, math.DG/0302355, math.DG/0302356 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 math.DG/0211294 studied the regularity of X near its singular points, and the the second paper math.DG/0211295 the moduli space of deformations of X. The third and fourth papers math.DG/0302355, math.DG/0302356 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.