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.
Francesco Antonio Denisi (Thu,) studied this question.