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Let $X$ be a Hyperk\"ahler manifold, and let $H$ be an ample divisor on $X$.We give a lower bound in terms of the Beauville-Bogomolov form $q(H)$ for thetwisted cotangent bundle $\Omega_X \otimes H$ to be pseudoeffective. If $X$ isdeformation equivalent to the Hilbert scheme of a K3 surface the lower boundcan be written down explicitly and we study its optimality.

Twisted cotangent bundles of Hyperk\"ahler manifolds