It will be helpful (although not necessary) for the reader of this section to be familiar with basic facts of non-archimedean analysis (see [BGR]). For any smooth manifold $Y$ with integral affine structure we will construct a sheaf ${\cal O}_{Y}$ of ${{\bf C}}_{\varepsilon}$-algebras on $Y$. Stalks ${\cal O}_{Y,y}$ of this sheaf are noetherian algebras, and one can define the notion of coherent sheaves of ${\cal O}_{Y}$-modules. If $Y={{\bf R}}^{n}/{{\bf Z}}^{n}$ is the torus with the standard integral affine structure then the category of coherent ${\cal O}_{Y}$-modules will be equivalent (by a non-archimedean version of GAGA) to the category of coherent sheaves on an abelian variety over the field ${{\bf C}}_{\varepsilon}$.

We start with the local picture. We denote by $v:{{\bf C}}_{\varepsilon}\to{{\bf R}}\cup\{+\infty\}$ a (non-discrete) valuation defined by $v(\sum_{\lambda_{1}<\lambda_{2}<...}c_{i}e^{-\lambda_{i}/\varepsilon})=-\lambda_{1}$ if $c_{1}\neq 0$ and $v(0)=+\infty$.

It follows from the definition that if $f\in{\cal O}_{{{\bf R}}^{n}}(U)$ and $(z_{1},...,z_{n})\in({{\bf C}}_{\varepsilon}^{\ast})^{n}$ then the series $\sum_{k_{1},...,k_{n}}a_{k_{1}...k_{n}}z_{1}^{k_{1}}...z_{n}^{k_{n}}$ converges in the adic topology as long as $(v(z_{1}),...,v(z_{n}))\in U$.

We introduce an action of the group $GL(n,{{\bf Z}})\ltimes{{\bf R}}^{n}$ on $({{\bf R}}^{n},{\cal O}_{{{\bf R}}^{n}})$ such as follows:

a) $GL(n,{{\bf Z}})$ acts simultaneously by the linear change of coordinates and linear transformation of indices $(k_{1},...,k_{n})$ in the series;

b) translations $(t_{1},...,t_{n})\in{{\bf R}}^{n}$ act on the coordinates $(y_{1},...,y_{n})$ by the shift $(y_{1},...,y_{n})\mapsto(y_{1}+t_{1},...,y_{n}+t_{n})$, and on the series by the rescaling of coefficients

Using this action we define the sheaf ${\cal O}_{Y}$ for an arbitrary smooth manifold $Y$ with integral affine structure.

We claim that there is a canonically associated to $Y$ a rigid analytic space $Y^{an}$ defined over ${{\bf C}}_{\varepsilon}$. Here is the construction. Let us consider a covering of $Y$ by open subsets $U_{i}$ such that all non-empty intersections $U_{i_{1}i_{2}...i_{k}}:=U_{i_{1}}\cap...\cap U_{i_{k}}$ in some local affine coordinates are convex polyhedra whose faces have rational slopes. Every $U_{i_{1}i_{2}...i_{k}}$ can be identified with the intersection of finitely many half-spaces, such that their pre-images under the map $v^{n}:({{\bf C}}_{\varepsilon}^{\ast})^{n}\to{{\bf R}}^{n}$ are sets of the type $\{(z_{1},\dots,z_{n})|\,v(z_{1}^{k_{1}}...z_{n}^{k_{n}})\geq C\}$ for some rational $C>0$. It is known after Tate that such a system of inequalities defines an affinoid domain (i.e. a local model for a rigid analytic space over ${{\bf C}}_{\varepsilon})$.

It is easy to see that $Y^{an}$ is canonically defined, and that the category $Coh(Y^{an})$ of coherent analytic sheaves on $Y^{an}$ (in the sense on analytic geometry) is equivalent to the category of coherent ${\cal O}_{Y}$-modules (i.e. locally finitely generated ${\cal O}_{Y}$-modules).

To every algebraic variety ${\cal Y}$ over ${{\bf C}}_{\varepsilon}$ one can associate canonically a rigid analytic space ${\cal Y}^{an}$. If ${\cal Y}$ is projective then the category $Coh({\cal Y}^{an})$ is equivalent to the category $Coh({\cal Y})$ of algebraic coherent sheaves on ${\cal Y}$ (GAGA theorem).

Assume that $Y={{\bf R}}^{n}/\Lambda$ is an $n$-dimensional torus equipped with the standard integral affine structure induced by ${{\bf Z}}^{n}\subset{{\bf R}}^{n}$, and $\Lambda$ is a lattice commensurable with ${{\bf Z}}^{n}$. The following result can be derived from [Mum].

Let us return to the picture of metric collapse in the case of abelian varieties. Since the collapse was defined by rescaling of the lattice (see Section 2) one can prove that ${\cal Y}$ is isomorphic to the original abelian variety over ${{\bf C}}_{\varepsilon}$. Therefore in the case of abelian varieties we have two equivalent descriptions of the collapse: the one in terms of Riemannian geometry and the one in terms of analytic non-archimedean geometry.

There is a sheaf of abelian groups ${Af}_{Y}$ on $Y$ given by locally affine functions with integral slopes (such functions locally are given by $l=c+\sum_{1\leq i\leq n}m_{i}y_{i}$ where $m_{i}\in{{\bf Z}},c\in{{\bf R}}$). There is a morphism of sheaves $exp:{Af}_{Y}\to{\cal O}_{Y}^{\ast}$ given by $l\mapsto exp(l):=e^{-c/\varepsilon}\prod_{1\leq i\leq n}z_{i}^{m_{i}}$.

Let $(Y,g)$ be an AK-manifold (see Section 3.2). We are going to define a characteristic class $[g]$ of the metric, which will be an analog of the cohomology class of a Kähler form in complex geometry. Let $Af_{Y}\otimes{{\bf R}}$ be a sheaf of all real-valued locally affine functions. For a cover by convex sets $(U_{i})_{i\in I}$ one can choose smooth functions $K_{i}$ such that $g_{|U_{i}}=\partial^{2}K_{i}$. Then $K_{i}-K_{j}\in Af_{Y}\otimes{{\bf R}}(U_{i}\cap U_{j})$, defines a 1-cocycle whose cohomology class we denote by $[g]\in H^{1}(Y,Af_{Y}\otimes{{\bf R}})$. If the dual affine structure (see Section 3) is integral, we get a class $[g]$ in the subgroup $H^{1}(Y,Af_{Y})/torsion\subset H^{1}(Y,Af_{Y}\otimes{{\bf R}})$. We will call such classes integral. In this case $exp([g])\in H^{1}(Y^{an},{\cal O}_{Y}^{\ast})$ is the first Chern class of a line bundle on $Y^{an}$. By analogy with the Kähler geometry we expect that this line bundle is ample. In the case when $(Y,g)$ is a flat torus, the ampleness can be proven directly (see [BL]).

From now on we assume that $[g]$ is integral. Then by GAGA the category of analytic coherent sheaves on ${Y}^{an}$ is equivalent to the category of algebraic coherent sheaves on the corresponding algebraic projective variety ${\cal Y}$.

The sheaf ${\cal O}_{Y}$ admits a resolution $\widehat{\Omega}_{Y}^{\ast}$ by a soft sheaf of dg-algebras. Locally, for a small open $U\subset Y$, sections of $\widehat{\Omega}_{Y}^{\ast}$ are given by sums $\alpha=\sum_{i_{1},...,i_{n}}c_{i_{1}...i_{n}}z_{1}^{i_{1}}...z_{n}^{i_{n}}$ where $c_{i_{1}...i_{n}}=\sum_{j}c_{j,i_{1}...i_{n}}e^{-\lambda_{j,i_{1}...i_{n}}/\varepsilon},c_{j,i_{1}...i_{n}}\in\Omega^{\ast}(U)$ with the same convergence conditions as for the sheaf ${\cal O}_{Y}$. Differential is given by the de Rham differential acting on the coefficients $c_{j,i_{1}...i_{n}}$.

We define a dg-category ${\cal C}(Y)$ such as follows. Objects are finite complexes of locally free ${\cal O}_{Y}$-modules of finite rank. For any two such complexes $E_{1}$ and $E_{2}$ we define the space of morphisms as

where we use the completed tensor product in the r.h.s. Differential and grading on the spaces of morphisms are induced by those on $E_{1},E_{2},\widehat{\Omega}_{Y}^{\ast}$. We will treat ${\cal C}(Y)$ as an $A_{\infty}$-pre-category in which all sequences of objects are transversal and there is no higher compositions except $m_{1}$ and $m_{2}$.

For a given projective algebraic variety $V$ over a field, one can define canonically an equivalence class of $A_{\infty}$-categories $D^{b}(V)$. It is obtained by the following enhancement of the bounded derived category of coherent sheaves on $V$. Objects of this $A_{\infty}$-category are the same as of the derived category of coherent sheaves. In order to define the space of morphisms between two objects, one replaces them by arbitrary chosen acyclic resolutions by locally free sheaves (e.g. the Godement resolutions) and then takes the global sections of the space of morphisms between resolutions in the category of complexes of sheaves. In this way one obtains a dg-category. In the case of projective varieties over complex numbers, there is an alternative construction in terms of complexes of holomorphic vector bundles and Dolbeault forms. Different choices of resolutions lead to $A_{\infty}$-equivalent categories. We will loosely denote the ($A_{\infty}$-equivalence) class of these categories by $D^{b}(V)$.

Using the fact that spaces of morphisms of ${\cal C}(Y)$ are resolutions of the corresponding spaces of sheaves of ${\cal O}_{Y}$-modules, as well as GAGA theorem, one can prove the following result.