On numerical dimensions of Calabi--Yau varieties

Abstract

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.