Pseudo-effective classes on projective irreducible holomorphic symplectic manifolds

TL;DR

This paper generalizes Kovács' result on the cone of curves from K3 surfaces to projective irreducible holomorphic symplectic manifolds, analyzing the structure of their pseudo-effective cones using hyperbolic geometry and monodromy groups.

Contribution

It extends the understanding of the pseudo-effective cone structure to higher-dimensional symplectic manifolds, providing explicit constructions and geometric consequences.

Findings

Pseudo-effective cone is either circular or generated by prime exceptional divisors.

The monodromy group acts with finite index on the Hodge lattice.

Existence of uniruled divisors on certain primitive symplectic varieties.

Abstract

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.