A hyper-kähler compactification of the intermediate jacobian fibration associated with a cubic 4-fold

Let X be a general cubic 4-fold. It was observed by Donagi and Markman that the relative intermediate Jacobian fibration JU/U (with U=(P⁵)^v\X^v) associated with the family of smooth hyperplane sections of X carries a natural holomorphic symplectic form making the fibration Lagrangian. In this paper, we obtain a smooth projective compactification J of JU with the property that the holomorphic symplectic form on JU extends to a holomorphic symplectic form on J. In particular, J is a 10-dimensional compact hyper-Kähler manifold, which we show to be deformation equivalent to the exceptional example of O’Grady. This proves a conjecture by Kuznetsov and Markushevich.