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

Abstract

This is the second in a series of five papers studying special Lagrangiansubmanifolds (SLV m-folds) X in (almost) Calabi–Yau m-folds M with singularities x 1 , ..., x n locally modelled on specialLagrangian cones C 1, ..., C n in $$\mathbb{C}$$ 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 Mwith 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 $$M$$ X is locally homeomorphic to the zeroes of a smooth map Φ: $$\ell $$ X′→ $${\mathcal{O}}$$ X′ between finite-dimensional vector spaces. Here the infinitesimal deformation space $$\ell $$ X′ depends only on the topology of X, and the obstruction space $${\mathcal{O}}$$ X′ only on the cones C 1, ..., C n at x 1, ..., x n . If the cones C i are stable then $${\mathcal{O}}$$ X′ is zero, and $$M$$ X is a smooth manifold. We also extend our results to families of almost Calabi–Yau structures on M.