Let $B^{sm}$ be a manifold with ${\bf Z}$-affine structure. The sheaf of ${\bf Z}$-affine functions $Aff_{{\bf Z}}:=Aff_{{{\bf Z}},B^{sm}}$ gives rise to an exact sequence of sheaves of abelian groups

Let $K$ be a complete non-archimedean field with a valuation map $val$. We give two equivalent definitions of a $K$-affine structure on $B^{sm}$ compatible with a given ${\bf Z}$-affine structure.

Since $B^{sm}$ carries a ${\bf Z}$-affine structure, we have an associated $GL(n,{{\bf Z}})\ltimes{{\bf R}}^{n}$-torsor on $B^{sm}$, whose fiber over a point $x$ consists of all ${\bf Z}$-affine coordinate systems at $x$.

Equivalence of two definitions from above is obvious in local ${\bf Z}$-affine coordinates. The reason is that the set of automorphisms of the exact sequence of groups

identical on $K^{\times}$ coincides with the group $GL(n,{{\bf Z}})\ltimes(K^{\times})^{n}$.

Finally, we can formulate the Fixed Point Property for $K$-affine structures (see Section 3.1 for ${\bf Z}$-affine case):
Fixed Point Property for $K$-affine structures. In the notation of the end of Section 3.1, for any $b\in B^{sing}$ and sufficiently small neighborhood $U$ of $b$ the lifted monodromy representation $\pi_{1}(U)\to GL(n,{{\bf Z}})\ltimes(K^{\times})^{n}$ has fixed vectors in $K^{\times n}$, and the ${{\bf R}}$-affine span of the corresponding (under the valuation map) vectors in ${{\bf R}}^{n}$ coincides with the set of fixed points of the monodromy representation $\pi_{1}(U)\to GL(n,{{\bf Z}})\ltimes{{\bf R}}^{n}$.

Starting from this section till the end of the paper (except of the Section 11.7) we will assume the following
Zero Charactersistic Assumption. $K$ is a complete non-archimedean local field such that its residue field has characteristic zero.

Let $X$ be a $K$-analytic manifold of dimension $n$ and we are given a continuous map $\pi:X\to B$, where $B$ is a topological space. Then $B^{sm}$ carries a ${\bf Z}$-affine structure (Theorem 1). Suppose that there is an open $K$-analytic submanifold $U\subset X$ such that $\pi^{-1}(B^{sm})\subset U$ and there is a nonwhere vanishing analytic form $\Omega\in\Gamma(U,\Omega_{X}^{n})$. We are going to define a ${\bf Z}$-affine function $Val(\Omega)$ similarly to the definition of the function $Val(\varphi)$ in Section 4.1. Namely, in local coordinates $(z_{1},...,z_{n})$ we consider the expression $\varphi:=\Omega/\bigwedge_{1\leq i\leq n}(dz_{i}/z_{i})$. This is an invertible function, and we define $Val(\Omega)$ as $Val(\varphi)$. The independence on the choice of coordinates follows from the following lemma

Proof: By Lemma 1 from Section 4.1 we know that $z_{i}^{\prime}$ as any invertible function can be written in form $c_{i}z^{I^{(i)}}(1+o(1))$ for some nonzero $c_{i}\in K$ and a multi-index $I^{(i)}\in{\bf Z}^{n}$. Vectors $I^{(1)},\dots,I^{(n)}$ form a basis of ${\bf Z}^{n}$, as follows from the condition that $z_{1}^{\prime},\dots,z_{n}^{\prime}$ form a coordinate system. Therefore, after applying the change of coordinates $z_{i}\mapsto c_{i}z^{I^{(i)}}$ preserving form $\bigwedge_{i}dz_{i}/z_{i}$ up to sign, we may assume that $z_{i}^{\prime}=(1+o(1))z_{i}$. The Jacobian matrix of the transformation $(z_{i})\to(z_{i}^{\prime})$ is the identity matrix plus terms of size $o(1)$. Therefore its determinant has norm equal to 1. $\blacksquare$

Now we make the following
Constant Norm Assumption. The function $Val(\varphi)$ is locally constant.

Proof. Let us write in local coordinates $\Omega=\varphi(z_{1},...z_{n})\bigwedge_{1\leq i\leq n}{dz_{i}\over z_{i}}$. Define residue $Res(\Omega)\in K$ as the constant term $\varphi_{0}$ in the Laurent expansion $\varphi(z_{1},...z_{n})=\sum_{I\in{\bf Z}^{n}}\varphi_{I}z^{I}$. It is easy to see that $Res(\Omega)$ does not depend (up to a sign) on the choice of local coordinates. For non-vanishing everywhere $\Omega$ satisfying Constant Norm Assumption we have $\exp(-Val(\varphi))=|\varphi|=|\varphi_{0}|$. Therefore we have $Res(\Omega)\neq 0$.

Let us return to the proof of the Theorem. Let $F$ be the sheaf of abelian groups $F\subset\pi_{\ast}({\cal O}_{X}^{\times})$ consisting of $f$ such that $Val(f)=0$. Then we have an exact sequence of sheaves

where ${\cal O}_{K}$ denotes the constant sheaf with the fiber being the ring of integers of $K$. Indeed we embed $K^{\times}/{\cal O}_{K}^{\times}$ into $\pi_{\ast}({\cal O}_{X}^{\times})/F$ as constant functions. The projection $\pi_{\ast}({\cal O}_{X}^{\times})/F\to(T_{X}^{\ast})^{{\bf Z}}$ assigns to the function $f$ the linear part of the corresponding ${\bf Z}$-affine function $Val(f)$.

Notice that if $U\subset B^{sm}$ is a connected domain then any $f\in\Gamma(U,F)$ can be written (non-canonically) as $f=a(1+r)$, where $a\in{\cal O}_{K}^{\times}$ and $r=o(1)$ in $\pi^{-1}(U)$.

We define an epimorphism of sheaves $p_{\Omega}:F\twoheadrightarrow{\cal O}_{K}^{\times}$ by formula

Here $\exp$ and $\log$ are understood as infinite convergent series (in order to make sense of them we use Zero Characteristic Assumption).

It is easy to see that $p_{\Omega}$ is well-defined. Then the exact sequence of sheaves

defines a $K$-affine structure on $B^{sm}$ compatible with $Aff_{{{\bf Z}},B^{sm}}^{can}$. This concludes proof of the Theorem. $\blacksquare$

Notice that the above proof gives an explicit construction of the $K$-affine structure. We will denote it by $Aff_{K,B^{sm}}^{\Omega}$. It is easy to see that this $K$-affine structure does not change if we make a rescaling $\Omega\mapsto c\Omega,c\in K^{\times}$.

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

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

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.

Here we are going to discuss a relation between $K$-affine structures and so-called flat coordinates on the moduli space of complex structures on Calabi-Yau manifolds. We assume the picture of collapse from Section 5.1.