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In this paper, we first generalize to any hyper-K\"ahler manifold withnonzero third Betti number results proved by O'Grady for hyper-K\"ahlermanifolds of generalized Kummer type. In the second part, we restrict tohyper-K\"ahler manifolds of generalized Kummer type and prove, using results ofMarkman, that their Kuga-Satake correspondence is algebraic.

Footnotes to papers of O'Grady and Markman