On numerical dimensions of Calabi--Yau varieties

TL;DR

This paper explores the numerical dimensions of extremal rays in the movable cone of Calabi--Yau varieties, revealing a specific dimension relation for varieties with Picard number two and extending the analysis to hyperk"ahler manifolds.

Contribution

It establishes the numerical dimension of extremal rays for certain Calabi--Yau varieties and investigates the relationship between different numerical dimensions, including explicit computations for hyperk"ahler manifolds.

Findings

Numerical dimension of extremal rays is half the dimension of the variety for Picard number two.

Relation between $ppa^{R}_{ps}$ and $ppa^{R}_{ ext{vol}}$ is analyzed.

Explicit computation of $ppa^{R}_{ps}$ for non-big divisors in hyperk"ahler manifolds.

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.