Is the Grothendieck's Standard Conjecture D (stating that the numerical equivalence relation for algebraic cycles with rational coefficients coincides with the homological one) known to be true for dimension 1 cycles? I have found a book saying that this result was proved by Lieberman in 1968, so the question is whether the conjecture is known to be true in the positive characteristic case.
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asked Mar 3, 2016 at 19:13
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$\begingroup$ For "homological equivalence", are you looking at the kernel of the cycle class map to $\ell$-adic 'etale cohomology? $\endgroup$– Jason StarrMar 3, 2016 at 19:15
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$\begingroup$ Yes, I do. However, I would be interested in a result of this sort for "any reasonable" (Weil?) cohomology theory. $\endgroup$– Mikhail BondarkoMar 3, 2016 at 19:17
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$\begingroup$ Lieberman showed that homological and numerical equivalence coincide for cycles of dimension 0,1,2,m-1,m on a Hodge manifold of complex dimension m, and also that the conjecture is true for abelian varieties. Sebastian (2013) has shown that numerical and smash-nilpotent equivalence coincide for 1-cycles on varieties dominated by products of curves. This result has no assumption on characteristic of the base field. Later, he generalized his results a bit. $\endgroup$– Anandam BanerjeeMar 15, 2016 at 14:40
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$\begingroup$ Thanl you! So, the general case of the statement is not known at the moment. $\endgroup$– Mikhail BondarkoMar 16, 2016 at 9:07
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