| Among abelian varieties, Jacobians of smooth projective curves C have the important property of being autodual, i.e., they are canonically isomorphic to their dual abelian varieties. This is equivalent to the existence of a Poincare' line bundle P on J(C)×J(C) which is universal as a family of algebraically trivial line bundles on J(C). A yet other instance of this fact was discovered by S. Mukai who proved that the Fourier-Mukai transform with kernel P is an auto-equivalence of the bounded derived category of J(C). I will talk on joint work with Filippo Viviani and Antonio Rapagnetta (partially still in progress), where we try to generalize Mukai's results for singular reducible curves X with locally planar singularities. We prove that the Fourier-Mukai tranform from the derived category of the generalized Jacobian of X to the derived category of any fine compactified Jacobian of X (following Esteves's construction), induced by the Poincarč sheaf, is fully faithful, generalizing a previous work of Arinkin in the case of integral curves. As a consequence, we prove that there is a canonical isomorphism (called autoduality) between the generalized Jacobian of X and the connected component of the Picard scheme of any fine compactified Jacobian of X, generalizing previous results of Arinkin, Esteves, Gagnč, Kleiman. We show further that there is a Poincare' sheaf on the product of any two (possibly equal) fine compactified Jacobians of X whose Fourier-Mukai transform is an equivalence of categories. These results can be seen as an instance of the geometric Langlands duality for the Hitchin fibration. |