Autoduality and Fourier-Mukai for compactified Jacobians

Margarida Melo, October 22nd, 2013

Among Abelian varieties, Jacobians of smooth curves C have the important property of
being autodual, i.e., they are canonically isomorphic to their dual abelian variety. This
is equivalent to the existence of a Poincare line bundle P on J(C)xJ(C) which is universal
as a family of algebraically trivial line bundles on J(C). Another 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, where we try to generalize both the autoduality
result and Mukai's equivalence result for singular reducible curves X with locally planar
singularities. Our results generalize previous results of Arinkin, Esteves, Gagne and
Kleiman and can be seen as an instance of the geometric Langlands duality for the
Hitchin fibration.