10 problems

Grothendieck's general Hodge conjecture. The inclusion above is an equality. The conjecture strengthens the classical Hodge conjecture by comparing coniveau with the largest Hodge…

Let $X$ be a smooth complex projective variety. Define the coniveau filtration by … where $Z\subset X$ ranges over subvarieties, and let $H^{m,i}_{\operatorname{Hdg}}(X)$ be the la…

Grothendieck's generalized Hodge conjecture. The coniveau $1$ condition on the étale cohomology of $V\times_K\bar K$ and the vanishing of $H^m(V,\mathcal O_V)$ for all $m\geq 1$ sh…

Let $Y$ be a smooth proper variety of dimension $n$ over $C$. Consider the following properties: $Y$ is uniruled; $\mathrm{CH}_0(Y)$ is supported in codimension at least $1$;…

Let $X$ be a smooth projective $n$-fold, and let $N_H^rH^\bullet(X,\mathbb{Q})$ denote the Hodge coniveau filtration. Hodge-theoretic polynomial decomposition conjecture. $X$ admit…

Generalized Hodge conjecture. There exists a closed algebraic subset $Y\subset X$ of codimension $c$ such that

Let $X$ be a smooth projective variety of dimension $n$, and let $L$ be an ample line bundle. Let $N^1H^i(X,\mathbb{Q})$ denote the first coniveau piece of cohomology. Hard Lefsche…

The big-class coniveau conjecture. The sub-Hodge structure $L$ is supported on a divisor of $X$, namely, there exists a closed algebraic subset $Z$ of codimension $1$ such that

Let $X\subset\mathbb{P}^n$ be a smooth complete intersection of $r$ hypersurfaces of degrees $d_1\leq\cdots\leq d_r$, and let $H^{n-r}(X,\mathbb{Q})_{\mathrm{prim}}$ denote its pri…

Let $X$ be a smooth complex projective variety, and let $L\subset H^k(X,\mathbb{Q})$ be a sub-Hodge structure of coniveau $c$. The coniveau of a rational Hodge structure of weight…