Let $K$ be as in Section 7.2, $B\supset B^{pre-sing}$ be a space with singular ${\bf Z}$-affine structure (see Section 6.3), and an extension of ${\bf Z}$-affine structure on $B\setminus B^{pre-sing}$ to a $K$-affine structure satisfying fixed point property (see 7.1). We assume that ${\bf Z}$-affine structure cannot be extended to a larger open set $U\supset B\setminus B^{pre-sing},U\neq B\setminus B^{pre-sing}$. Slightly abusing notation we will denote $B\setminus B^{pre-sing}$ simply by $B^{sm}$. We want to have a $K$-analytic space $X$, meromorphic non-zero top degree form $\Omega$ and a continuous proper (and maybe also Stein) map $\pi:X\to B$ such that:

1. 1.

   $B^{pre-sing}$ coincides with $B^{sing}$, and ${\bf Z}$-affine structure on $B^{sm}$ arising from the projection $\pi$ coincides with the given one;

2. 2.

   the restriction $\Omega_{|\pi^{-1}(B^{sm})}$ is a nowhere vanishing analytic form which satisfies the Constant Norm Assumption;

3. 3.

   the $K$-affine structure on $B^{sm}$ arising from the pair $(X,\Omega)$ coincides with the initial one.

We call the problem of finding such data Lifting Problem.

Question. What restrictions on the behavior of the $K$-affine structure near $B^{pre-sing}=B^{sing}$ should we impose in order to guarantee the existence of a solution of the Lifting Problem?

Let $B=B^{sm}$ be a flat torus (see Section 3.2.1). Then the Lifting Problem has a solution (canonical up to rescaling of $\Omega$) for any compatible $K$-affine structure. More precisely, the groupoid of Tate tori and isomorphisms between them is equivalent to the groupoid of $K$-affine structures on real flat tori.

In Sections 8-11 we are going to discuss a solution of the Lifting Problem for K3 surfaces. In that case $B^{sing}\neq\emptyset$.

If we restrict ourselves only to the smooth part $B^{sm}$ (i.e. we allow non-compact $X$) then there is a canonical solution of this “reduced” Lifting Problem. In other words one can construct a smooth $K$-analytic space $X^{\prime}$ with an analytic top degree form $\Omega^{\prime}$ and a map $\pi^{\prime}:X^{\prime}\to B^{sm}$ satisfying the above conditions 1–3. Let us explain this construction assuming that $B^{sm}$ is oriented.

First of all we notice that the orientation of $B^{sm}$ gives a reduction to $SL(n,{{\bf Z}})\ltimes(K^{\times})^{n}$ of the structure group of the torsor defining the $K$-affine structure. The reduced group naturally acts by automorphisms of the fibration $\pi_{can}:({\bf G}_{m}^{an})^{n}\to{{\bf R}}^{n}$ preserving the form $\bigwedge_{1\leq i\leq n}{dz_{i}\over z_{i}}$. The action on $({\bf G}_{m}^{an})^{n}$ is induced from the action on monomials. Namely, the inverse to an element $(A,\lambda_{1},...,\lambda_{n})\in SL(n,{{\bf Z}})\ltimes(K^{\times})^{n}$ acts on monomials as

$$z^{I}=z_{1}^{I_{1}}\dots z_{n}^{I_{n}}\mapsto\left({\textstyle\prod_{i=1}^{n}}\lambda_{i}^{I_{i}}\right)\,\,z^{A(I)}\,\,\,.$$

The action of the same element on ${{\bf R}}^{n}$ is given by the similar formula

$$x=(x_{1},\dots,x_{n})\mapsto A(x)-(val(\lambda_{1}),\dots,val(\lambda_{n}))\,\,\,.$$

Let $B^{sm}=\cup_{\alpha}U_{\alpha}$ be an open covering by coordinate charts $U_{\alpha}\simeq V_{\alpha}\subset{{\bf R}}^{n}$ such that for any $\alpha,\beta$ we are given elements $g_{\alpha,\beta}\in SL(n,{{\bf Z}})\ltimes(K^{\times})^{n}$ satisfying the $1$-cocycle condition for any triple $\alpha,\beta,\gamma$. Then the space $X^{\prime}$ is obtained from $\pi_{can}^{-1}(V_{\alpha})$ by gluing by means of the transformations $g_{\alpha,\beta}$. The form $\bigwedge_{1\leq i\leq n}{dz_{i}\over z_{i}}$ gives rise to a nowhere vanishing analytic top degree form $\Omega^{\prime}$ on $X^{\prime}$. Thus we have obtained a solution of the reduced Lifting Problem. The sheaf $\pi_{\ast}({\cal O}_{X^{\prime}}):={\cal O}_{B^{sm}}^{can}$ is called the canonical sheaf.

In the case $B^{pre-sing}\neq\emptyset$ this solution seems to be a “wrong” one, i.e. it cannot be extended to a solution $\pi:X\to B$, where $X$ and $B$ are compact. In the case of K3 surfaces we will show later how to modify it in order to obtain a “true” solution of the Lifting Problem.