  The classical Fourier-Mukai duality establishes an equivalence of categories
between the derived categories of sheaves on dual complex tori. In this article
we show that this equivalence extends to an equivalence between two dual
objects. Both of these are generalized deformations of the complex tori. In one
case, a complex torus is deformed formally in a non-commutative direction
specified by a holomorphic Poisson structure. In the other, the dual complex
torus is deformed in a B-field direction to a formal gerbe. We show these two
deformations are Fourier-Mukai equivalent.
