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

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?

• 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? – Will Sawin Apr 20 '17 at 22:48