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
Ampk(X), for any 1≤k≤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
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.Comment: 26 pages, v2. Fixed some typos. Updated bibliography. Improved the
exposition of some part