Semi-ampleness conjecture for nef integral divisors on Calabi–Yau threefolds

A SCY3 is a smooth Calabi–Yau threefold. A divisor is nef if its intersection with every curve is nonnegative, and an integral divisor is one whose class is integral. A divisor is semi-ample if some positive multiple is base-point-free.

Semi-ampleness conjecture. Any nef integral divisor on a SCY3 is semi-ample.

This is described as a notoriously difficult open conjecture. It concerns the relationship between numerical positivity and the existence of morphisms on Calabi–Yau threefolds.