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)(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)(R) and kappa(R)(vol) for Calabi-Yau varieties. We also compute kappa(sigma)(R) for non-big divisors in the closed movable cone of a projective hyperahler manifold.