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 ℂ^m 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 ℂ^m, 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.