Does a conservativity conjecture (e.g. Conjecture 2.1 of http://user.math.uzh.ch/ayoub/PDF-Files/Article-for-Steven.pdf) imply the standard conjectures? Specifically I am confused with Beilinson's article arXiV:1006.1116 "Remarks on Grothendieck’s standard conjectures", where it seems to show that the conservativity of a realization functor implies the standard conjectures on algebraic cycles.
-
2$\begingroup$ What part of Beilinson's article are you looking at? As far as I can tell most of that article relies on the stronger hypothesis of existence of a motivic t-structure, which to my knowledge does not easily follow from conservativity. $\endgroup$– dhyMay 6, 2019 at 4:06
1 Answer
Here is a partial answer: In characteristic 0 it is known that conservativity + algebraicity (edit: modulo rational equivalence) of the Künneth projectors implies the remaining standard conjectures.
Indeed, if the Künneth projectors are algebraic, then one may use conservativity to show that the inverse of the Lefschetz operator is algebraic (i.e. Lefschetz standard conjecture). Once that is known, the hom=num standard conjecture also follows. For example, André's category of “motivated” motives would be equivalent to the categories of homological/numerical motives.
-
1$\begingroup$ I think there is a slight complication, coming from the difference between rational and homological equivalence: to do this argument, one needs a lift of the Künneth projectors to the category of Chow motives, i.e. a Chow-Künneth decomposition, rather than just the algebraicity of Künneth projectors in the sense of the standard conjecture of Künneth type. $\endgroup$ Jun 6, 2019 at 13:08
-
$\begingroup$ @SimonPepinLehalleur — Yes, that's completely correct. I should have been more explicit with what I meant. I'll edit accordingly. $\endgroup$– jmcJun 7, 2019 at 17:30