The pseudo-effective cone of the known irreducible holomorphic symplectic manifolds

Abstract

Thanks to Kovács' work, it is known that the pseudo-effective cone Eff(S) of a smooth projective K3 surface S of Picard number ρ(S) ≥ 3 is either circular or equalsOn a higher dimensional (projective) irreducible holomorphic symplectic (IHS) manifold, the structure of the pseudo-effective cone is quite similar to that of a smooth projective surface, due to the existence of the Beauville-Bogomolov-Fujiki form, and the smooth rational curves are naturally replaced by the prime exceptional divisors. In this note we show that, in some sense, Kovacs' result still holds true if X is a projective IHS manifold belonging to one of the 4 known deformation classes. In particular, we show that if ρ(X) ≥ 3, Eff(X) is either circular or equals E R ≥0 [E], where the sum runs over the prime exceptional divisors of X. The proof goes through the description of Mon 2 (X) (the monodromy group of X) and of the stably exceptional classes. Moreover, we provide some consequences of this fact.