63 problems

O'Grady's conjecture. For every ${\mathcal{E}}\in D^b(S)$,

Let $X$ be a smooth projective variety whose algebra of holomorphic forms is generated in degree at most $2$, and let $x,y$ be closed points of $X$. Let $F_{\mathrm{BB}}^\bullet$ d…

Let $A$ be a henselian discrete valuation ring of characteristic zero with residue field of characteristic $p$ and function field $K$. Let $X$ be smooth and projective of relative…

Colliot-Thélène–Sansuc–Kato–Saito conjecture. The complex

Raskind–Spiess conjecture. The group $T(Y)$ is the direct sum of its maximal divisible subgroup and a finite group.

Kerz–Esnault–Wittenberg conjecture. For every prime power $p^r$, the map

Let $X$ be a very general hypersurface of degree $d$ in $\mathbb{P}^n$, and let $p\in X$. Denote by $R_{\mathbb{P}^1,X,p}$ the space of points of $X$, other than $p$, that are rati…

Let $E$ be an Enriques surface, and let $C\subset E$ be an irreducible nonsingular curve of genus $g\geq 2$. Its moduli point in $\overline{\mathcal{M}}_g$ determines a $0$-cycle.…

Let $U$ be a smooth algebraic surface, let $p_g(U)$ denote its geometric genus, and let $T(U)$ be the Albanese kernel, namely the kernel of the Albanese morphism…

Let $K$ be a global field, and let $V/K$ be a smooth projective variety such that its base change $V_{\overline K}=V\operatorname*{\otimes}\overline K$ is rational and has Picard g…

Let $V$ be a smooth projective geometrically integral variety over a $p$-adic field $k$. Let $A_0(V)$ be the degree-zero subgroup of the Chow group of zero-cycles, and let … be the…

Let $V$ be a smooth projective geometrically integral variety over a number field. Let $A_0(V)$ be the degree-zero part of the Chow group of zero-cycles, and let … be the Albanese…

Let $X$ be a complex surface, and let the group of $0$-cycles modulo rational equivalence on $X$ be the corresponding quotient group. Severi's conjecture. The group of $0$-cycles m…

Let $K$ be a field finitely generated over ${\mathbb Q}$, and let $X$ be a $K$-variety. Finite generation conjecture for zero-cycles. The group $\operatorname{CH}_0(X)$ is finitely…

Brauer–Manin zero-cycle conjecture. If $X({\bf A}_k)^{\operatorname{Br}}$ is non-empty, then $I(X)=1$.

Let $A$ be an abelian surface, and let $A^{(k)}$ denote its $k$th symmetric product. Let $z\subset A^{(k)}$ be a cycle in the kernel of the summation map…

Let $X$ be an abelian surface, let $K_n(X)$ be the generalized Kummer variety, and let $z=\{z_0,\ldots,z_n\}\in K_n(X)$ with $z_m\in X$. For a point $z$, let $O_z$ denote its ratio…

Let $(X,\alpha)$ be a twisted abelian surface, let $K=K_\sigma(X,\alpha,\mathbf{v})$ be the kernel of the Albanese map, and suppose that $\dim K=2n$. For $E\in K$, let $O_E$ denote…

Let $Y$ be a double EPW quartic, and let $S$ be the associated $K3$ surface. The construction as a moduli space gives a map … Let $F^\bullet$ denote the Bloch--Beilinson filtration…

Martin--Vial's conjecture.

The twisted Beauville--Voisin class conjecture.

Let $U$ be a smooth quasi-projective surface over a finite extension $k$ of the $p$-adic field $\mathbb{Q}_p$. Suslin's singular homology $H_0^{\operatorname{sus}}(U)$ has a degree…

Gazaki–Hiranouchi conjecture. If $v$ is an unramified place of good reduction, then

La conjecture d'exactitude du complexe des zéro-cycles. Le complexe ci-dessus est exact pour toute variété $X$ propre, lisse et géométriquement intègre. Cette conjecture généralise…

Soit $k$ un corps de nombres et $X$ une variété propre, lisse et géométriquement intègre sur $k$. Soit $Z_0^1(X)$ l'ensemble des zéro-cycles de degré $1$ sur $X$, et soit…