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-K\"ahler compactification $J(X)$ with regular Lagrangian $J \to \mathbb P^5$. This builds upon arXiv:1602.05534, where result is proved for general $X$, as well on degeneration techniques arXiv:1704.02731 and from minimal model program. then study some aspects of birational geometry $J(X)$: very we compute movable nef cones $J(X)$, showing not twisted version arXiv:1611.06679, nor an OG$10$-type moduli space objects in Kuznetsov component $X$; show, using normal functions, Mordell-Weil group $MW(\pi)$ abelian $\pi: J P^5$ isomorphic integral degree $4$ primitive algebraic cohomology i.e., $MW(\pi) = H^{2,2}(X, Z)_0$.