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.