Take the 2-minute tour ×  Suppose that over an algebraically
closed field $K$ of finite characteristic the numerical
equivalence of cycles relation (for algebraic cycles of smooth
projective varieties) coincides with the homological one (with
respect to etale $\mathbb{Q}_l$-adic cohomology). Does the Hodge
standard conjecture follow? Moreover, does it follow from the
existence of a 'nice' motivic $t$-structure for Voevodsky's
motives over $K$ (i.e. we assume that this $t$-structure
'splits' Chow motives into 'numerical pieces', and that mixed
motives have filtration 'by these pieces'). If the answer (even)
to the second question is 'no', then why does one usually
believe in the HSC?  share|improve this question I would be
surprised if the Hodge standard conjecture would follow from the
others (which is not to say that it doesn't). This conjecture is
true in char $0$ by Hodge theory. Perhaps that and a general
sense of optimism, at the time, were sufficient, but I'm just
speculating. –  Donu Arapura Jan 23 '11