Is there any characterization of rational nef classes that don't come from effective $\mathbb{Q}$divisors on abelian varieties? Is there any result along the lines of "Any nef $\mathbb{Q}$divisor is the sum of an effective $\mathbb{Q}$divisor with a numerically trivial divisor"? A useful observation is that the nef cone coincides with the pseudoeffective cone on an abelian variety.
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