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

$$Hom_{{\cal C}(Y)}(E_{1},E_{2})=\Gamma(Y,Hom_{{\cal O}_{Y}}(E_{1},E_{2})\widehat{\otimes}_{{\cal O}_{Y}}\widehat{\Omega}_{Y}^{\ast}),$$

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.