Asymptotic base loci on hyper-Kahler manifolds

Given a projective hyper-Kähler manifold [Formula: see text], we study the asymptotic base loci of big divisors on [Formula: see text]. 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 [Formula: see text]-ample divisors [Formula: see text], for any [Formula: see text], answering affirmatively (in the case of projective hyper-Kähler manifolds) a question asked by Sam Payne. We provide a decomposition for the effective cone [Formula: see text] into chambers of Mori-type, analogous to that for Mori dream spaces into Mori chambers. To conclude, we illustrate our results with several examples.