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.