A Fourier transform for sheaves on Lagrangian families of real tori
Abstract
We systematically develop a transform of the Fourier-Mukai type for sheaves on symplectic manifolds X of any dimension fibred in Lagrangian tori. One obtains a bijective correspondence between unitary local systems supported on Lagrangian submanifolds of X and holomorphic vector bundles with compatible unitary connections supported on complex submanifolds of the relative Jacobian of X (suitable conditions being verified on both sides).