Special Lagrangian submanifolds with isolated conical singularities. III. Desingularization, the unobstructed case - Joyce, D (2004)
This is the third in a series of five papers studying special Lagrangian submanifolds (SLV m-folds) X in (almost) Calabi-Yau m-folds M with singularities x 1,..., x n locally modelled on special Lagrangian cones C 1,..., C n in ℂ m with isolated singularities at 0. Readers are advised to begin with Paper V. This paper and Paper IV construct desingularizations of X, realizing X as a limit of a family of compact, nonsingular SL m-folds Ñ t in M for small t > 0. Suppose L 1,..., L n are Asymptotically Conical SL m-folds in ℂ m, with L i asymptotic to the cone C i at infinity. We shrink L i by a small t > 0, and glue t L i into X at x i for i = 1,..., n to get a 1-parameter family of compact, nonsingular Lagrangian m-folds N t for small t > 0. Then we show using analysis that when t is sufficiently small we can deform N t to a compact, nonsingular special Lagrangian m-fold Ñ t, via a small Hamiltonian deformation. This \tilde Ñ t depends smoothly on t, and as t → 0 it converges to the singular SL m-fold X, in the sense of currents. This paper studies the simpler cases, where by topological conditions on X and L i we avoid various obstructions to existence of Ñ t. Paper IV will consider more complex cases when these obstructions are nontrivial, and also desingularization in families of almost Calabi-Yau m-folds.