On the reconstruction problem in mirror symmetry

Let π:M→B be a Lagrangian torus fibration with singularities such that the fibers are of Maslov index zero, and unobstructed. The paper constructs a rigid analytic space M0∨ over the Novikov field which is a deformation of the semi-flat complex structure of the dual torus fibration over the smooth locus B0⊂B of π. Transition functions of M0∨ are obtained via A∞ homomorphisms which capture the wall-crossing phenomenon of moduli spaces of holomorphic disks.