Asymptotic base loci on hyper-Kähler manifolds

Given a projective hyper-K\"ahler manifold \(X\), we study the asymptotic base loci of big divisors on \(X\). We provide a numerical characterization of these loci and study how they vary while moving a big divisor class in the big cone, using the divisorial Zariski decomposition, and the Beauville-Bogomolov-Fujiki form. We determine the dual of the cones of \(k\)-ample divisors \(\mathrm{Amp}_k(X)\), for any \(1\leq k \leq \mathrm{dim}(X)\), answering affirmatively (in the case of projective hyper-K\"ahler manifolds) a question asked by Sam Payne. We provide a decomposition for the effective cone \(\mathrm{Eff}(X)\) into chambers of Mori-type, analogous to that for Mori dream spaces into Mori chambers. To conclude, we illustrate our results with several examples.