9 problems

Martin--Vial's conjecture.

Green–Griffiths implication. The induced second filtration step vanishes:

Let $X$ and $Y$ be smooth projective varieties of dimension $n$, and let $Z\in CH^n(X\times Y)_{\mathbb Q}$ be a correspondence. Let $F^\bullet$ be the Bloch–Beilinson filtration a…

Let $X$ be a smooth complex projective variety. Let $(V,\nabla)$ be a flat vector bundle on $X$, and let … be the conjectural functorial Bloch–Beilinson filtration. Bloch's conject…

Let $X$ be a smooth projective variety and let $i$ be a positive integer. Assume that $\mathrm{H}^{*}_{\mathrm{tr}}(X)$ is generated by $\mathrm{H}^i_{\mathrm{tr}}(X)$. Bloch–Beili…

Let $X$ be a hyper-Kähler variety of dimension $2n$. For an integer $i$, let $C_i(X)\subset \operatorname{CH}_i(X)=\operatorname{CH}^{2n-i}(X)$ be the $\mathbb{Q}$-vector space gen…

Let $X$ be a hyper-Kähler variety of dimension $2n$, and let $S_i\operatorname{CH}_0(X)$ denote the subgroup generated by classes of points whose rational-equivalence orbits have d…

Let $X$ be a projective hyper-Kähler manifold of dimension $2n$. Let $S_\bullet$ be the filtration on zero-cycles induced by the loci $S_iX$, and let $F'_{BB}$ denote the even leve…

Let $Y$ be a smooth projective variety and let $F$ be a Bloch–Beilinson filtration with rational coefficients on its Chow groups. Assume that $F$ also induces a filtration on Chow…