55 problems

Beilinson–Hodge conjecture. The fundamental class map $\mathrm{cl}^{p,m}_{\pi}$ is surjective. This predicts that every Hodge class in $\mathrm{Hg}(H^1(\mathcal{U},\mathbf{H}(p)))$…

Let $X$ be a $K3$ surface. The group ${\operatorname{Aut}}_s(X)$ of symplectic automorphisms, namely automorphisms acting trivially on a nonzero holomorphic $2$-form, acts on Bloch…

Let $X$ be an equidimensional compact complex algebraic manifold. Write $K_n(X)_{coperatorname{\mathbb{Q}}}$ for rational $K$-theory, $CH^{p}_{s}(X,n)_{coperatorname{\mathbb{Q}}}$…

Graph isomorphism conjecture. For every such $A$, the graph homomorphism is an isomorphism.

Let $K$ be a global field, let $X/K$ be a smooth projective geometrically integral variety of dimension $d$, let $n$ be prime to the characteristic of $K$, and let $i>0$. Set…

Hodge-$\mathcal D$-Conjecture. The map $r_{\mathcal{D},\mathbb{R}}^{p,n}$ is surjective. This is a higher-dimensional and higher-cycle analogue of the Hodge conjecture, formulated…

Voisin–Beilinson conjectures. For $p=2$, $\text{CH}_{\text{ind}}^{2}(X,1)_{{\Bbb Q}}$ should be countable, and the map in (4.5.2) should be surjective.

Let $X$ and $S$ be as in (0.6), and suppose $\d7eta \in \text{CH}^{2}(X,1)_{{\Bbb Q}}$ restricts to zero on the relevant fibers. The higher Abel–Jacobi map … is injective for any f…

Let $X$ be a projective variety over ${\bold C}$, and let $F^\bullet{\mathrm{CH}}^r(X,m)$ be the decreasing filtration on its higher Chow groups constructed in the paper. Discreten…

Termination conjecture for the motivic filtration. The filtration terminates, meaning

Higher Hodge conjecture. The map $cl_{U}^{p,m}$ is surjective. This is the arithmetic quasiprojective extension of the Hodge conjecture attributed in the source to predictions of B…

Topological consequence. For every positive integer $b$ there is an integer $D$ depending only on $b$, $Y$ and ${\mathcal O}(1)$ such that for all $d\geq D$ and…

The algebraicity conjecture. For all integers $b\in\mathbb N^*$ and $p\in\mathbb N$ there is an integer $D$ depending only on $b$, $p$, $Y$ and ${\mathcal O}(1)$, such that for all…

Let $K$ be a number field, let $v$ be a finite place, let $\Lambda$ be the completion of the ring of integers at $v$, and consider the local case with $q=1$ and $a=0$. Let the regu…

Let $X$ be in the function-field setting, let $q$ be the cardinality of the constant field, and suppose the integral regulator maps have finite kernel and cokernel with orders…

Let $X$ be a variety over a number field $K$ with at worst semistable reduction at all places. Let $S$ be a finite set of places containing all Archimedean places, let…

Let $Y$ be a nonsingular variety and suppose $q-2a>1$. The higher Chow group $CH^{q-a-1}(Y,q-2a-1)$ is the group occurring in the definition of the $v$-adic Deligne cohomology. Par…

Beilinson's Tate conjecture. In case $k$ is finitely generated over $\mathbb Q$, $\operatorname{reg}_{U,\mathrm{et}}^q$ is surjective.

Let $F$ be a field. Let $B_2(F)$ be the Bloch group, let $R_3(F)$ be the subgroup generated by the stated relations in ${\mathbb Z}[{\mathbb P}^1_F]$, and define … Let…

Countability conjecture. The group $\operatorname{coker}(\tau)$ is countable.

Let $X$ be a smooth projective variety, and let $CH^2_{\mathrm{ind}}(X,1)$ denote the quotient of the higher Chow group $CH^2(X,1)$ by its subgroup of decomposable cycles. Voisin's…

Let $S$ be a smooth projective surface. Write ${{\rm CH}}^2(S,1)_{{\rm ind}}$ for the indecomposable part of the higher Chow group and $J^2(S)_{{\rm tr}}$ for the transcendental in…

Let $X$ be a smooth projective curve over a number field $F$, and let … where $f\colon X\to \operatorname{Spec}(F)$ is the structure morphism and $SK_1(X)=CH^{2}(X,1)$. Bloch's con…

Toric–Sreekantan regulator conjecture. For each prime $\ell$, the valuation of the toric regulator at $\ell$ is the Sreekantan regulator tensored with ${\mathbb Q}_\ell$. This conj…

Let $X$ be a smooth projective variety, and let $\underline{CH}^{k}(X,m)$ be the relevant higher Chow group, $\underline{r}_{k,m}$ its regulator, and $D(k,m)$ the indicated denomin…