On fibrations and measures of irrationality of hyper-K\"ahler manifolds

We prove some results on the fibers and images of rational maps from a hyper-K"ahler manifold. We study in particular the minimal genus of fibers of a fibration into curves. The last section of this paper is devoted to the study of the rational map defined by a linear system on a hyper-K"ahler fourfold satisfying numerical conditions similar to those considered by O’Grady in his study of fourfolds numerically equivalent to $K{3}^{[2]}$. We extend his results to this more general context.