Let $X$ be a smooth projective variety and $K_X$ the canonical line bundle. If $K_X$ is nef, then the abundance conjecture predicts that it is semiample, so in particular a multiple $mK_X$ has many sections. If $-K_X$ is nef, then it does not have to be semiample (see examples in reference below). I have the following weaker questions:

Question 1: If $-K_X$ nef, does there exist an integer $m>0$ such that $-mK_X$ effective?

Question 2: If $-K_X$ nef, does there exist $m$ as above, and $H$ ample divisor, such that $-mK_X|_H$ effective?

Of course, if $-K_X$ is big - in the interior of the effective cone, both questions are true, so the issue is really on the boundary. From the paper ''Nef Reduction and Anticanonical Bundles'' I can extract that Bauer-Peternell claim that the result is true in dimensions $\leq3$: see 1.5-1.7 for surfaces, and the various theorems in the case of threefolds. An idea for a counterexample for Q1 would be to cook up a semistable projective bundle, perhaps over something $K$-trivial, and follow something like the Hartshorne-Mumford construction for non-closed effective cone, but it does not seem clear how to make this work in higher dimension or for $-K_X$. Of course, in higher dimension there is more space for a complicated base locus for $-K_X$, which an ample divisor $H$ could perhaps not avoid. Note that one could also ask Q2 for $K_X$ as a weak version of abundance, but even here I am not sure what is known.

  • $\begingroup$ I just mention related to Q1: even under the much stronger hypothesis that $-K_X \cdot C >0$ for all curves $C$, the answer is not known to be positive (as far as I am aware). $\endgroup$ – potentially dense Feb 16 '16 at 13:37
  • $\begingroup$ That's a pretty good point, thanks, and it seems that this paper of Serrano deals with the situation you mention for threefolds. However Serrano and later Campana-Chen-Peternell seem to be interested in getting -K_X ample, rather than just getting a section of a multiple. Also Q2 does not seem to appear in the literature at all. $\endgroup$ – Frank Feb 16 '16 at 14:16

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