On the reconstruction problem in mirror symmetry

Let \pi: M \ra 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 M_0^\chk over the Novikov field which is a deformation of the semi-flat complex structure of the dual torus fibration over the smooth locus B_0 of \pi. Transition functions of M_0^\chk are obtained via A-\infty homomorphisms which capture the wall-crossing phenomenon of moduli spaces of holomorphic disks.