Timeline for Why is the Hodge conjecture equivalent to the assertion that $ \mathcal{R}_{ \mathrm{Hodge} } $ is fully faithfull?

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Apr 21, 2017 at 9:52	history	edited	David Roberts♦	CC BY-SA 3.0	Formatting
Apr 21, 2017 at 0:54	review	Close votes
Apr 21, 2017 at 8:31
Apr 21, 2017 at 0:37	comment	added	R. van Dobben de Bruyn		The definition is, as @WillSawin suggested, straightforward. First define the functor on smooth projective varieties (classical Hodge theory), and then extend to motives (e.g. modulo rational equivalence) in the obvious way (see Will's comment for the steps involved). By definition of homological equivalence, this functor factors through motives modulo homological equivalence (which these notes assume coincides with numerical equivalence; see the second paragraph of section 5). This finishes the construction of the functor, and the other statement is just an unwinding of the definitions.
Apr 20, 2017 at 22:48	comment	added	Will Sawin		What perspective are you asking this perspective from? Do you understand how to define a contravariant functor from the category of smooth projective varieties $\mathcal V_k$ to $\mathbb QHS$? If so, have you tried checking that this functor is preserved under the linearization, pseudoabelianization, and inversion procedures defined in the notes? If not, then from where do you know pure Hodge structures?
Apr 20, 2017 at 21:12	history	asked	YoYo	CC BY-SA 3.0