Let $C$ be a (smooth projective algebraic) curve. It is well known that the Jacobian $J$ of $C$ is a principally polarized abelian variety. In other words, $J$ is self-dual in the sense that $J$ is identified with the space of topologically trivial line bundles on itself. Suppose now that $C$ is singular. The Jacobian $J$ of $C$ parametrizes topologically trivial line bundles on $C$; it is an algebraic group that is no longer compact. By considering torsion-free sheaves instead of line bundles, one obtains a natural singular compactification $J'$ of $J$.
In this talk, I consider (projective) curves $C$ with planar singularities. The main result is that $J'$ is self-dual: $J'$ is identified with a space of torsion-free sheaves on itself. This identification also provides an auto-equivalence of the coherent derived category of $J'$ (the Fourier-Mukai transform). The autoduality naturally fits into the framework of the geometric Langlands conjecture; I hope to sketch this relation in my talk.