Birational geometry of the intermediate Jacobian fibration of a cubic fourfold

We show that the intermediate Jacobian fibration associated to any smooth cubic fourfold X admits a hyper-Kähler compactification J.X / with a regular Lagrangian fibration W J ! P 5 .This builds upon work of Laza, Saccà and Voisin (2017), where the result is proved for general X , as well as on the degeneration techniques introduced in the work of Kollár, Laza, Saccà and Voisin, and the minimal model program.We then study some aspects of the birational geometry of J.X /: for very general X we compute the movable and nef cones of J.X /, showing that J.X / is not birational to the twisted version of the intermediate Jacobian fibration, nor to an OG10-type moduli space of objects in the Kuznetsov component of X ; for any smooth X we show, using normal functions, that the Mordell-Weil group MW./ of the fibration is isomorphic to the integral degree-4 primitive algebraic cohomology of X , ie MW. / Š H 2;2 .X; Z/ 0 .