On a conjecture of Oguiso about rational curves on Calabi-Yau threefolds

Abstract

Let X be a Calabi-Yau threefold. We show that if there exists on X a non-zero nef non-ample divisor then X contains a rational curve, provided its second Betti number is greater than 4.

13 pages, 1 figure, final version. To appear on Comment. Math. Helv