Acta Mathematica

Let $X$ be a general cubic $4$-fold. It was observed by Donagi and Markman that the relative intermediate Jacobian fibration $\mathcal{J}_U/U$ (with $U=(\mathbb{P}^5)^\vee\setminus X^\vee$) 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 $\overline{\mathcal{J}}$ of $\mathcal{J}_U$ with the property that the holomorphic symplectic form on $\mathcal{J}_U$ extends to a holomorphic symplectic form on $\overline{\mathcal{J}}$. In particular, $\overline{\mathcal{J}}$ 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.