Special Lagrangian Submanifolds with Isolated Conical Singularities. IV. Desingularization, Obstructions and Families

Abstract

This is the fourth 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 ℂm with isolated singularities at 0. Readers are advised to begin with Paper V.Paper III and this one construct desingularizations of X, realizing X as a limitof a family of compact, nonsingular SL m-folds C t in M for small t > 0. Suppose L 1,..., L n are Asymptotically Conical SL m-folds in ℂm, withL i asymptotic to the cone C i at infinity. We shrink L i by a small t > 0, and gluetL i into X at x i for i= 1,..., n to get a 1-parameter family of compact, nonsingularLagrangian m-folds N t for small t> 0.Then we show using analysis that when t is sufficiently small we can deform N t toa compact, nonsingular special Lagrangian m-fold C t, via a small Hamiltonian deformation. This C t depends smoothly on t, and as t→ 0 it converges to the singular SL m-fold X, in the sense of currents.Paper III studied simpler cases, where by topological conditions on X and L i we avoid obstructions to the existence of C t. This paper considers more complex cases when theseobstructions are nontrivial, and also desingularization in families of almost Calabi–Yaum-folds M s for s∈F, rather than in a single almost Calabi–Yau m-fold M.