A Fourier transform for sheaves on real tori : Part I. The equivalence Sky (T)≃ Loc (T̂)

As a first step toward a theory of a real Fourier transform for sheaves on Calabi–Yau manifolds fibred in special Lagrangian tori, we explicitly construct the functors which establish the equivalence between the category of skyscraper sheaves of finite-dimensional vector spaces on a real torus T, and the category of local systems (locally free sheaves of C-modules of finite rank) on the dual torus ˆT .