15 problems

Let $k$ be a flexible field and let $X$ be a smooth projective variety over $k$. Write $\operatorname{CH}_{k/k}(X,\mathbb{Z}/p^m)$ for the isotropic Chow group with…

Let $k$ be a field, and let $\operatorname{Ch}^*_{k/k}$ denote the isotropic Chow groups with $\mathbb{F}_p$-coefficients, obtained from Chow groups by quotienting by classes comin…

Let $X$ be a smooth projective variety of dimension $n$ over $\mathbf{K}$, let $k$ be a codimension, and let $Z\in\mathsf{Z}^k(X)_\mathbf{Q}$ be a rational algebraic cycle. Standar…

Let $X$ be a smooth projective variety of dimension $n$ over $\mathbf{K}$. Let $\|\cdot\|_{\iota}$ denote the chosen operator norm on $\ell$-adic étale cohomology and $\|\cdot\|$ t…

Let $X$ be as above, and let $CH^i(X)^{(0)}$ be the subgroup defined in Proposition 7. Let $CH^i_{\operatorname{num}}(X)$ denote the subgroup of cycles numerically equivalent to…

Let $X$ be a smooth projective variety of dimension $d$ over a finitely generated field $K$, and let $B$ be a smooth $k$-scheme of finite type with function field $K$. For…

Let $k$ be a flexible field. The local Chow groups are the groups obtained from the local Chow theory over $k$, while $\operatorname{Ch}^*_{Num}$ denotes Chow groups with…

Numerical triviality conjecture. In $\mathrm{LM}(S)$, $f\sim g$ implies $f=g$; equivalently,

Let $X$ be a smooth projective holomorphic symplectic variety. Write $\operatorname{CH}(X)$ for its Chow ring with rational coefficients and $\overline{\operatorname{CH}}(X)$ for t…

Numerical-equivalence conjecture. The functor $\operatorname{num}$ is an equivalence of categories. If true, both functors $p$ and $\pi$ considered in the paper are equivalences of…

Strong Lefschetz conjecture. This map is an isomorphism for all $k\leq\lfloor n/2\rfloor$.

The standard conjecture (D). Homological equivalence and numerical equivalence for algebraic cycles with $\mathbb{Q}$-coefficients on $X$ coincide.

Let $X$ be the variety and let $p$ and $V$ be as in Setup. Write $\mathcal{Z}^p({X^{\operatorname{sep}}})$ for the group of codimension-$p$ cycles on $X^{\operatorname{sep}}$. For…

Numerical Chow-group conjecture.

Codimension-one-nef formulation. If