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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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I think your statement follows from Lemma 1.1 in Bauer's paper 'On the cone of curves of an abelian variety', since algebraic equivalence coincides with numerical equivalence on an abelian variety. – J.C. Ottem Mar 11 '13 at 19:23
Yes, you are exactly right. Thanks for pointing that out! – rfauffar Mar 11 '13 at 19:25

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