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 ${\kappa }_{\sigma }^{\mathbb{R}}$ 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 ${\kappa }_{\sigma }^{\mathbb{R}}$ and ${\kappa }_{\mathrm{vol}}^{\mathbb{R}}$ 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.

arXiv commentv2. 19 pages, reorganized and more details added, final version

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Cited by in corpus (1)

• The volume of a divisor and cusp excursions of geodesics in hyperbolic manifolds