Our approach to the reconstruction of analytic Calabi-Yau manifolds from real manifolds with integral affine structure can be illustrated in the following toy-model example. Let $S^{1}={{\bf R}}/{{\bf Z}}$ be a circle equipped with the induced from ${{\bf R}}$ affine structure. We equip $S^{1}$ with the canonical sheaf ${\cal O}^{can}_{S^{1}}$ of Noetherian ${{\bf C}}((q))$-algebras. By definition, for an open interval $U\subset S^{1}$ algebra ${\cal O}^{can}_{S^{1}}(U)$ consists of formal series $f=\sum_{m,n\in{{\bf Z}}}a_{m,n}q^{m}z^{n},\,\,\,a_{m,n}\in{\bf C}$ such that $\inf_{a_{m,n}\neq 0}(m+nx)>-\infty$. Here $x\in{\bf R}$ is any point in a connected component of the pre-image of $U$ in ${\bf R}$, the choice of a different component $x\rightarrow x+k,\,\,k\in{\bf Z}$ corresponds to the substitution $z\mapsto q^{k}z$. The corresponding analytic space is the Tate elliptic curve $(E,{\cal O}_{E})$, and there is a continuous map $\pi:E\to S^{1}$ such that $\pi_{\ast}({\cal O}_{E})={\cal O}^{can}_{S^{1}}$.

In the case of K3 surfaces one starts with $S^{2}$. The corresponding integral affine structure is well-defined on the set $S^{2}\setminus\{x_{1},...,x_{24}\}\subset S^{2}$, where $x_{1},...,x_{24}$ are distinct points. Similarly to the above toy-model example one can construct the canonical sheaf ${\cal O}_{S^{2}\setminus\{x_{1},...,x_{24}\}}^{can}$ of algebras, an open $2$-dimensional smooth analytic surface $X^{\prime}$ with the trivial canonical bundle (Calabi-Yau manifold), and a continuous projection $\pi^{\prime}:X^{\prime}\to S^{2}\setminus\{x_{1},...,x_{24}\}$ such that $\pi^{\prime}_{\ast}({\cal O}_{X^{\prime}})={\cal O}_{S^{2}\setminus\{x_{1},...,x_{24}\}}^{can}$. The problem is to find a sheaf ${\cal O}_{S^{2}}$ whose restriction to $S^{2}\setminus\{x_{1},...,x_{24}\}$ is locally isomorphic to ${\cal O}_{S^{2}\setminus\{x_{1},...,x_{24}\}}^{can}$, an analytic compact K3 surface $X$, and a continuous projection $\pi:X\to S^{2}$ such that $\pi_{\ast}({\cal O}_{X})={\cal O}_{S^{2}}$. We call this problem (in general case) the Lifting Problem and discuss it in Section 7. Unfortunately we do not know the conditions one should impose on singularities of the affine structure, so that the Lifting Problem would have a solution. We consider a special case of K3 surfaces in Sections 8-11. Here the solution is non-trivial and depends on data which are not visible in the statement of the problem. They are motivated by Mirror Symmetry and consist, roughly speaking, of an infinite collection of trees embedded into $S^{2}\setminus\{x_{1},...,x_{24}\}$ with the tail vertices belonging to the set $\{x_{1},...,x_{24}\}$. The sheaf ${\cal O}_{S^{2}\setminus\{x_{1},...,x_{24}\}}^{can}$ has to be modified by means of automorphisms assigned to every edge of a tree and then glued together with certain model sheaf near each singular point $x_{i}$.

Informally speaking, we break $S^{2}\setminus\{x_{1},...,x_{24}\}$ endowed with the sheaf ${\cal O}_{S^{2}\setminus\{x_{1},...,x_{24}\}}^{can}$ into infinitely many infinitely small pieces and then glue them back together in a slightly deformed way. The idea of such a construction was proposed several years ago independently by K. Fukaya and the first author. The realization of this idea was hindered by a poor understanding of singularities of the Gromov-Hausdorff collapse and by the lack of knowledge of certain open Gromov-Witten invariants (“instanton corrections”). The last problem is circumvented here (and in fact solved) with the use of some pro-nilpotent Lie group (see Section 10).