Special Lagrangian Submanifolds with Isolated Conical Singularities. II. Moduli spaces

This is the second in a series of five papers studying special Lagrangiansubmanifolds (SLV m-folds) X in (almost) Calabi–Yau m-foldsM with singularities x₁, ..., x_n locally modelled on specialLagrangian conesC₁, ..., C_n in ^m with isolated singularities at 0.Readers are advised to begin with Paper V.This paper studies the deformation theory of compact SL m-folds X in Mwith conical singularities. We define the moduli space_Xof deformations of X in M, and construct a natural topology on it. Then we show that _X is locally homeomorphic to the zeroes of a smooth map : _XX between finite-dimensional vector spaces.Here the infinitesimal deformation space_X depends only on the topology of X, and the obstruction space_X only on the cones C₁, ..., C_n at x₁, ..., x_n. If the cones C_i are stable then _X is zero, and _Xis a smooth manifold. We also extend our results to families of almost Calabi–Yau structures on M.