Pseudo-effective classes on projective irreducible holomorphic symplectic manifolds

We show that Kovács’ result on the cone of curves of a K3 surface generalizes to any projective irreducible holomorphic symplectic manifold X . In particular, we show that if ρ ( X ) ≥ 3 , the pseudo-effective cone Eff ( X ) ¯ is either circular or equal to ∑ E R ≥ 0 [ E ] ¯ , where the sum runs over the prime exceptional divisors of X . The proof goes through hyperbolic geometry and the fact that (the image of) the Hodge monodromy group Mon Hdg 2 ( X ) in O + ( N 1 ( X ) ) is of finite index. If X belongs to one of the known deformation classes, carries a prime exceptional divisor E , and ρ ( X ) ≥ 3 , we explicitly construct an additional integral effective divisor, not numerically equivalent to E , with the same monodromy orbit as that of E . To conclude, we provide some consequences of the main result of the paper, for instance, we obtain the existence of uniruled divisors on certain primitive symplectic varieties.