An integral affine structure on a manifold of dimension $n$ is given by a torsion-free flat connection with the monodromy reduced to $GL(n,{{\bf Z}})$. There are two basic situations in which integral affine structures occur naturally. One is the case of classical integrable systems described briefly in Section 3. Most interesting for us is a class of examples arising from analytic manifolds over non-archimedean fields which is discussed in Section 4. It is motivated by the approach to Mirror Symmetry suggested in [KoSo]. We recall it in Section 5. From our point of view manifolds with integral affine structure appear in Mirror Symmetry in two ways. One considers the Gromov-Hausdorff collapse of degenerating families of Calabi-Yau manifolds. The limiting space can be interpreted either as a contraction (see Section 4.1) of an analytic manifold over a non-archimedean field of Laurent series ${\bf C}((t))$, or as a base of a fibration of a Calabi-Yau manifold by Lagrangian tori (with respect to the symplectic Kähler 2-form). On a dense open subset of the limiting space one gets two integral affine structures associated with two interpretations, the non-archimedean one and the symplectic one. Mirror dual family of degenerating Calabi-Yau manifolds should have metrically the same Gromov-Hausdorff limit, with the roles of two integral affine structures interchanged.

Very interesting question arises: how to reconstruct these families of Calabi-Yau manifolds from the corresponding manifolds with integral affine structures? This question was one of the main motivations for present work.

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).

The relationship between K3 surfaces and singular affine structures on $S^{2}$ is of very general origin. Starting with a projective analytic Calabi-Yau manifold $X$ over a complete non-archimedean local field $K$ one can canonically construct a PL manifold $Sk(X)$ called the skeleton of $X$. If $X$ is a generic K3 surface then $Sk(X)$ is $S^{2}$. We discuss skeleta in Section 6.6. The group of birational automorphisms of $X$ acts on $Sk(X)$ by integral PL transformations. For $X=K3$ we obtain an action of an arithmetic subgroup of $SO(1,18)$ on $S^{2}$. Further examples should come from Calabi-Yau manifolds with large groups of birational automorphisms.

We have already discussed the content of the paper. Let us summarize it. The paper is naturally divided into three parts. Part 1 is devoted to generalities on integral affine structures and examples, including Mirror Symmetry. Motivated by string theory we use term A-model (resp. B-model) for examples arising in symplectic (resp. analytic) geometry.

In Part 2 we discuss the concept of singular integral affine structure, including an affine version of Gauss-Bonnet theorem. The latter implies that if all singularities of an integral affine structure on $S^{2}$ are standard (so-called focus-focus singularities) then there are exactly $24$ singular points. Part 2 also contains a statement of the Lifting Problem and discussion of flat coordinates on the moduli space of complex Calabi-Yau manifolds. We expect that under mild conditions on the singular integral affine structure there exists a solution of the Lifting Problem, which is unique as long as we fix periods (see Sections 7.3 and 7.4 for more details).

Most technical Part 3 contains a solution of the Lifting Problem for K3 surfaces. We construct the corresponding analytic K3 surface as a ringed space. The sheaf of analytic functions is defined differently near a singular point and far from the singular set. It turns out that the “naive” candidate for the sheaf on the complement of the singular set has to be modified before we can glue it with the model sheaf near each singular point. This modification procedure involves a new set of data (we call them lines). We also discuss the group of automorphisms of the canonical sheaf which preserve the symplectic form. We use this group in order to modify the “naive” sheaf along each line.

The paper has two Appendices. First one contains some background on analytic spaces, while the second one is devoted to Torelli theorem.

Acknowledgements. We are grateful to Ilya Zharkov and Mark Gross for useful discussions. Second author thanks Clay Mathematics Institute for supporting him as a Fellow and IHES for excellent research and living conditions.