This paper lays the foundations of a program to study mirror symmetry by studying the log structures of Illusie-Fontaine and Kato on degenerations of Calabi-Yau manifolds. The basic idea is that one can associate to certain sorts of degenerations of Calabi-Yau manifolds a log Calabi-Yau space, which is a log structure on the degenerate fibre. The log CY space captures essentially all the information of the degeneration, and hence all mirror statements for the "large complex structure limit" given by the degeneration can already be derived from the log CY space. In this paper we begin by discussing affine manifolds with singularities. Given such an affine manifold along with a polyhedral decomposition, we show how to construct a scheme consisting of a union of toric varieties. In certain non-degenerate cases, we can also construct log structures on these schemes. Conversely, given certain sorts of degenerations, one can build an affine manifold with singularities structure on the dual intersection complex of the degeneration. Mirror symmetry is then obtained as a discrete Legendre transform on these affine manifolds, thus providing an algebro-geometrization of the Strominger-Yau-Zaslow conjecture. The deepest result of this paper shows an isomorphism between log complex moduli of a log CY space and log Kähler moduli of its mirror.

Let D₀ = {x ∈ Rⁿ,H₀(x) ≤ 1} be a strictly convex domain in Rⁿ with n ≤ 3 and D_u = {x ∈ Rⁿ,H_u(x) ≤ 1}, u ∈ [η, η] be a continuous one-parameter deformation of D₀ with lattice-point counting function N_u(T) := {m ∈ Zⁿ : H_u(m) ≤ T²}. The main result of this paper is an estimate for large values of T of the variation of the counting function, N_u(T), over generic volume-preserving deformations D_u.