On numerical dimensions of Calabi--Yau varieties

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