Quantum tori, mirror symmetry and deformation theory

We suggest to compactify the universal covering of the moduli space of comp lex structures by noncommutative spaces. The latter are described by certai n categories of sheaves with connections which are flat along foliations. I n the case of Abelian varieties, this approach gives quantum tori as a nonc ommutative boundary of the moduli space. Relations to mirror symmetry, modu lar forms and deformation theory are discussed.