Special Lagrangian submanifolds with isolated conical singularities. II. Moduli spaces

This is the second 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 C1,..., C-n in C-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 M with conical singularities. We define the moduli space M-X of deformations of X in M, and construct a natural topology on it. Then we show that MX is locally homeomorphic to the zeroes of a smooth map Phi: l(X') --> O-X' between finite-dimensional vector spaces. Here the infinitesimal deformation space l(X') depends only on the topology of X, and the obstruction space O-X' only on the cones C-1,..., Cn at x(1),..., x(n). If the cones C-i are stable then O-X' is zero, and MX is a smooth manifold. We also extend our results to families of almost Calabi-Yau structures on M.