Asymptotic base loci on hyper-Kähler manifolds

Abstract

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

26 pages. Final version. The exposition improved, thanks to the referee's comments. Proposition 3.9 was deleted. To appear in Commun. Contemp. Math