The purpose of this section is to describe the framework in which the results concerning $A_{\infty}$-categories will be formulated. We would like to make few comments even before recalling a definition of the Fukaya category. There are two main problems with the definition. First, morphisms can be defined only for transversal Lagrangian submanifolds (in particular, the identity morphism is never defined). Second, since there are pseudo-holomorphic discs with the boundary on a given Lagrangian submanifold, one has to add a composition $m_{0}$ to the set of compositions $m_{n},n\geq 1$. As a result, the spaces of morphisms are not complexes: $m_{1}^{2}\neq 0$. On the other hand, the derived category of coherent sheaves arises from an $A_{\infty}$-category without $m_{0}$ and with the condition $m_{1}^{2}=0$. Hence one should explain in which sense two $A_{\infty}$-categories in question are equivalent.

The above-mentioned problems can be resolved by an appropriate generalization of the notion of $A_{\infty}$-category. This generalization involves numerous preparations and will be given elsewhere (see [KoS]). On the other hand, the problem with $m_{0}$ does not appear in the case of abelian varieties, which is the main application of the approach offered in this paper. Hence, for the purposes of present paper it is sufficient to work with $A_{\infty}$-pre-categories (or $A_{\infty}$-categories with transversal structure, cf. [P1]). This gives a partial solution to the transversality problem, and provides a solution to the problem with the identity morphisms.

Using $A_{\infty}$-pre-categories we formulate and prove a variant of the homological mirror symmetry conjecture. It can be applied to the case of abelian varieties. In particular, one can obtain certain formulas for Massey products for abelian varieties in terms of partial theta-sums similar to those considered in [P1].