Hyper-Kaehler compactification of the intermediate Jacobian fibration of a cubic fourfold : the twisted case

The starting point of this note is our recent paper with Laza and Sacc`a on the construction of deformations of O’Grady’s $10$-dimensional manifolds as compactifications of intermediate Jacobian fibrations associated to cubic fourfolds. The note provides a complement to that paper consisting in the analogous construction in the twisted case, leading to isogenous but presumably not isomorphic or birational hyper-K"ahler manifolds.