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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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    $\begingroup$ 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. $\endgroup$
    – J.C. Ottem
    Mar 11, 2013 at 19:23
  • $\begingroup$ Yes, you are exactly right. Thanks for pointing that out! $\endgroup$
    – rfauffar
    Mar 11, 2013 at 19:25
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    $\begingroup$ @J.C.Ottem It seems that algebraic equivalence coincides with numerical equivalence up to torsion for divisors holds for any smooth variety. See this answer and its reference: mathoverflow.net/questions/15001/… $\endgroup$
    – Li Yutong
    Dec 30, 2020 at 3:11

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