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

TL;DR

This paper studies the deformation theory of special Lagrangian submanifolds with isolated conical singularities, defining their moduli spaces and analyzing conditions for smoothness and stability.

Contribution

It introduces a framework for understanding the local structure of moduli spaces of singular special Lagrangian submanifolds, linking topology, cones, and stability conditions.

Findings

Moduli space locally homeomorphic to zeroes of a smooth map

Stable cones lead to smooth moduli spaces

Extension of results to families of Calabi-Yau structures

Abstract

This is the second in a series of five papers math.DG/0211294, math.DG/0302355, math.DG/0302356, math.DG/0303272 studying 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 proves some conjectures. In this paper we study 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 M_X is locally homeomorphic to the zeroes of a smooth map \Phi : I --> O between finite-dimensional vector spaces. Here the infinitesimal deformation space I depends only on the topology of X, and the obstruction space O only on the cones C_1,...,C_n at x_1,...,x_n. If the cones C_i are "stable" then O is zero and M_X is a smooth manifold. We also extend our results to families of almost Calabi-Yau structures on M. The first paper math.DG/0211294 laid the foundations for the series, and studied the regularity of X near its singular points. The third and fourth papers math.DG/0302355, math.DG/0302356 construct desingularizations of X, realizing X as the limit of a family N^t of compact, nonsingular SL m-folds in M.