4 problems

Hodge realization conjecture. The functor $\iota_{X,\sigma}^{\textup{Hdg}}$ is fully faithful, with essential image stable under subquotients. This is a Hodge-type conjecture for N…

Fullness conjecture. The functor $\iota_{\mathbb{C}}^{\textup{Hdg}}$ is fully faithful, and its essential image is stable under subquotients. This is known for the Tannakian subcat…

Let $X$ be an algebraic variety over $\mathbb{Q}$. An object $M$ of the abelian category $\mathcal{M}_\mathrm{univ}(X,\mathbb{Z})$ is torsion-free if multiplication by every nonzer…

Let $X$ be a smooth $k$-variety, and let $\mathscr N(X)$ be the category of perverse motives. Let $\mathscr M(X)$ be the motivic category introduced earlier in the paper, and let t…