  This is the fourth in a series of five papers math.DG/0211294,
math.DG/0211295, math.DG/0302355, math.DG/0303272 studying compact special
Lagrangian submanifolds (SL 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 the final paper math.DG/0303272 which surveys the series, gives
examples, and applies the results to prove some conjectures.
  The first paper math.DG/0211294 studied the regularity of X near its singular
points, and the second math.DG/0211295 the moduli space of deformations of X.
The third paper math.DG/0302355 and this one construct desingularizations of X,
realizing X as a limit of a family of compact, nonsingular SL m-folds \tilde
N^t in M for small t>0.
  Let L_1,...,L_n be Asymptotically Conical SL m-folds in C^m, with L_i
asymptotic to C_i at infinity. We shrink L_i by t>0, and glue tL_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 for
small t we can deform N^t to a compact, nonsingular SL m-fold \tilde N^t via a
small Hamiltonian deformation. As t --> 0 this \tilde N^t converges to X, in
the sense of currents.
  The third paper math.DG/0302355 studied simpler cases, where by topological
conditions on X and L_i we avoid obstructions to existence of \tilde N^t. This
paper considers more complex cases when these obstructions are nontrivial, and
also desingularization in smooth families of almost Calabi-Yau m-folds M^s for
s in F, rather than a single almost Calabi-Yau m-fold M.
