4 problems

Let $X$ be a proper scheme over $\mathbb C$, let $\pi:X_\bullet\to X$ be a smooth proper hypercovering, and let $H^{2p+i}(X)$ be Deligne's mixed $\mathbb Q$-Hodge structure on…

Let $X$ be smooth and proper over $\mathbb C$. Let $\operatorname{gr}^0_{F_m}CH^p(X)$ denote the degree-zero graded piece of the motivic filtration on codimension-$p$ Chow groups,…

Let $M$ be a 1-motive defined over a sub-field $K$ of $\mathbb{C}$. Its periods are the coefficients of the comparison isomorphism between its De Rham and Hodge realizations, and l…

Étale 1-motive realization conjecture. These functors should satisfy, functorially in $X$, for every prime $\ell\ne p$,