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Nov 5, 2010 at 15:15 comment added Donu Arapura Your first question sounds a lot like Grothendieck's standard conjecture "Hdg", which would still be open in general. Regarding your last question, it seems that such statements are proved in Jannsen's book in chapter 1, sections 1 and 4.
Nov 5, 2010 at 11:04 history asked Mikhail Bondarko CC BY-SA 2.5