Pseudo-effective classes on projective irreducible holomorphic symplectic manifolds

We show that Kov\'acs' 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 $\rho(X)\geq 3$, the pseudo-effective cone $\overline{\mathrm{Eff}(X)}$ is either circular or equal to $\overline{\sum_{E}\mathbf{R}^{\geq 0} [E]}$, 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 $\mathrm{Mon}^2_{\mathrm{Hdg}}(X)$ in $\text{O}^+(N^1(X))$ is of finite index. If $X$ belongs to one of the known deformation classes, carries a prime exceptional divisor $E$, and $\rho(X)\geq 3$, 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.