35 problems

Let $k$ be a global field and let $X$ be a smooth, projective, geometrically integral variety over $k$. Write $CH_0(X)$ for the Chow group of zero-cycles modulo rational equivalenc…

Let $X$ be a smooth, proper, geometrically irreducible variety over a number field $k$. A zero-cycle of degree one on $X$ is a formal integer linear combination of closed points wh…

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…

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

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…

Martin--Vial's conjecture.

Colliot-Thélène's conjecture. The complex above is exact.

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.

Kollár–Szabó conjecture. The group

Divisible-plus-finite conjecture. The group $F^2(X)$ is the direct sum of a divisible group and a finite group.

Let $X$ be a smooth projective variety, let $i$ be an integer, and let $w\in\operatorname{CH}^i(X)$ be a cycle. Nori's conjecture. If $w|_T=0\in\operatorname{CH}_0(T)$ for every…

Let $X=C_1\times\cdots\times C_d$ be a product of smooth projective curves over a number field $k$, and let $F^2(X)$ be the kernel of the Albanese map on degree-zero zero-cycles. D…

Let $k$ be a number field and let $V$ be a proper smooth geometrically integral variety over $k$. Consider the complex … where the modified Chow groups are the usual Chow groups at…

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 $A$ 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…

Let $K$ be a number field, let $X$ be a smooth projective geometrically integral variety over $K$, and let $P_K$ be the set of places of $K$. Write $X_{K_v}$ for the base change to…

Let $X \subset \mathbb{P}^n$ be a very general complete intersection of multidegree $(d_1, \dots, d_k)$. For $p \in X$, let $R_{X,p}$ denote the space of points of $X$ rationally e…

Let $k$ be a number field, and let $\{X_\omega\}_\omega$ be the family of all smooth, proper, geometrically integral varieties over $k$. The Brauer–Manin obstruction to weak approx…

Let $X$ be a smooth proper variety over a number field $k$. Let $\Omega_f$ be the set of finite places of $k$ and $\Omega_\infty$ the set of infinite places. For each infinite plac…

Normalized-orbit dimension conjecture. If $A$ is a very general abelian variety, then

Let $X$ be a smooth projective geometrically connected variety over a number field $K$ with a $K$-rational point. For a finite place $v$ of $K$, let … be the base change to the com…

Bloch–Quillen formula. The cycle class map $\operatorname{cyc}_X^M$ is an isomorphism. Together with the known surjectivity, this would identify the Levine–Weibel Chow group of zer…

Let $K$ be a global field with separable closure $K^s$, and let $X^s$ be a smooth projective variety over $K^s$. Let $\operatorname{CH}_0(X^s)^0$ denote the degree-zero Chow group…