10 problems

Let $X$ be a smooth projective complex variety. Write $CH^2(X)_{hom}$ for codimension-$2$ cycles homologous to $0$, $CH^2(X)_{alg}$ for those algebraically equivalent to $0$, and l…

Let $X$ be a projective nonsingular variety, and define its Griffiths group by … The source identifies this group conjecturally with a motivic extension group involving…

Let $X$ be a smooth projective variety, let $B^2_{\mathrm{hom}}(X)$ denote the codimension-two Griffiths group with rational coefficients modulo algebraic equivalence, and suppose…

Griffiths-group rank prediction. One expects

Let $T$ be the smooth projective threefold considered under Hypothesis, and let $t_3(T)$ denote its transcendental degree-three motive. Write $\mathcal{M}(\mathbb{L},t_3(T))$ for m…

Let $X$ be a smooth complete variety of dimension $n=3$, and let $\operatorname{Griff}^3(X\times X)$ denote the relevant Griffiths group. Griffiths-group vanishing conjecture for t…

Let $X$ be a smooth projective variety of dimension $n=3$, and let $\operatorname{Griff}^3(X\times X)_{\mathbb{Q}}$ denote the Griffiths group of codimension-$3$ cycles on…

Let $X_3$ and $X_4$ be general members of the families of cubic sevenfolds and smooth hypersurfaces in $\mathbb{P}(1^6;2)$ described above, respectively. Write…

Nori's conjecture. The map $\operatorname{AJ}^2$ is injective for every $t$, or at least for very general $t$. This is the injectivity assumption used to identify Griffiths-group c…

Schoen's conjecture. The class of $3(C-C^{-})\in CH^2(JC)$ is algebraically equivalent to zero. Equivalently, $C-C^{-}$ is $3$-torsion in $\operatorname{Griff}^2(JC)$. This conject…