Let $X$ be a Calabi--Yau variety of Picard number two with infinite birational automorphism group. We show that the numerical dimension $\kappa^{ \mathbb{ R}}_{ \sigma}$ of the extremal rays of the closed movable cone of $X$ is $\dim X/2$. More generally, we investigate the relation between the two numerical dimensions $\kappa^{ \mathbb{ R}}_{ \sigma}$ and $\kappa^{ \mathbb{ R}}_{ \mathrm{ vol}}$ for Calabi--Yau varieties. We also compute $\kappa^{ \mathbb{ R}}_{ \sigma}$ for non-big divisors in the closed movable cone of a projective hyperk\"{ a}hler manifold.
Categories: math.AG