A Fourier transform for sheaves on real tori: Part II. Relative theory

If X is a symplectic family of Lagrangian tori, the dual family X̂ has a natural complex structure. We define, for any dimension of X, a Fourier transform which yields a bijective correspondence between local systems supported on Lagrangian submanifolds of X and holomorphic vector bundles supported on complex subvarieties of X̂ (suitable conditions being verified on both sides).