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.