Twisted Intermediate Jacobian Fibrations

In this talk, I will report on a joint work in progress with D. Mattei and E. Shinder, where we construct, using Hodge modules, a group scheme that can be thought of as the intermediate Jacobian of a certain complete family of cubic threefolds. We show that the group scheme acts on a well-known abelian fibration. The action gives rise to twisted versions of the abelian fibration. This is similar to twisting genus 1 fibrations with irreducible fibres via its Tate-Shafarevich group.