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The starting point of this note is our recent paper with Laza and Sacc\`a onthe construction of deformations of O'Grady's $10$-dimensional manifolds ascompactifications of intermediate Jacobian fibrations associated to cubicfourfolds. The note provides a complement to that paper consisting in theanalogous construction in the twisted case, leading to isogenous but presumablynot isomorphic or birational hyper-K\"ahler manifolds.

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