On Numerical Dimensions of Calabi–Yau Varieties

Abstract

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 $\kappa ^{\mathbb{R}}_{\textrm{vol}}$ for Calabi–Yau varieties. We also compute $\kappa ^{\mathbb{R}}_{\sigma }$ for non-big divisors in the closed movable cone of a projective hyperkähler manifold.