Asymptotic base loci on hyper-Kähler manifolds

Given a projective hyper-Kähler 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 ${\mathrm{A}\mathrm{m}\mathrm{p}}_k(X)$, for any $1\le k\le \mathrm{d}\mathrm{i}\mathrm{m}(X)$, answering affirmatively (in the case of projective hyper-Kähler manifolds) a question asked by Sam Payne. We provide a decomposition for the effective cone $\mathrm{E}\mathrm{f}\mathrm{f}(X)$ into chambers of Mori-type, analogous to that for Mori dream spaces into Mori chambers. To conclude, we illustrate our results with several examples.