On K3 correspondences

We consider relationships between families of K3 surfaces, in the context of string theory. An important ingredient of string theory also of interest in algebraic geometry is T-duality. Donagi and Pantev have extended the original duality on genus one fibred K3 surfaces with a section to the case of any genus one fibration, via a Fourier-Mukai transform. We investigate possibilities of extending this result to the more general case of non fibred K3s, in the context of Caldararu's conjecture. We also provide an explicit construction of a quasi-universal sheaf associated to the classical Mukai correspondence.