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On pages 17 and 18 of the following document: https://www.math.tifr.res.in/~sujatha/ihes.pdf, we find the following paragraph:

Let $ \mathbb{Q} \mathrm{HS}$ be the category of pure Hodge structures over $\mathbb{Q}$. There is a functor: $$ \mathcal{R}_{ \mathrm{Hodge} } \colon \mathop{\mathrm{Mot}}^{ \bullet }_{ \mathrm{num} } ( k, \mathbb{Q} ) \to \mathbb{Q} \mathrm{HS} $$ and the Hodge conjecture is equivalent to the assertion that $\mathcal{R}_{ \mathrm{Hodge} }$ is fully faithful.

My questions are:

How is the Hodge realization functor $ \mathcal{R}_{ \mathrm{Hodge} }$ explicitly defined? And how to prove explicitly that the Hodge conjecture is equivalent to the assertion that $\mathcal{R}_{\mathrm{Hodge}}$ is fully faithful?

Thanks in advance for your help.

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    $\begingroup$ 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? $\endgroup$
    – Will Sawin
    Commented Apr 20, 2017 at 22:48
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    $\begingroup$ 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. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Apr 21, 2017 at 0:37

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