Birational geometry of the intermediate Jacobian fibration

A few year ago with Laza and Voisin we constructed a hyperkähler compactification of the intermediate Jacobian fibration associated to a general cubic fourfold. In this talk I will first show how such a compactification J(X) exists for any smooth cubic fourfold X and then discuss how the birational geometry of the fibration J(X) is governed by any extra algebraic cohomology classes on J(X).