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 $\overlin...
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Zusammenfassung: | 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. |
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DOI: | 10.48550/arxiv.2205.15148 |