97 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…

Bloch's conjecture. For every $\gamma\in A^2(Y\times X)$, the induced action $Gr^*_F\gamma$ depends only on $cl(\gamma)$.

Beilinson's conjecture. For every $X\in\mathcal V_k$ there exists a decreasing filtration $F^\bullet$ on $A^i(X)$ such that: (a) $F^0A^j(X)=A^j(X)$ and $F^1A^j(X)=(A^j(X))_{hom}$;…

Cancellation conjecture for effective Chow motives. If

Nori's conjecture. If $n$ is sufficiently large, then the natural map

Let $A$ be an abelian surface, and let $F$ be a simple vector bundle on $A$ with Mukai vector $v:=v(F)$. Suppose that $c_1(F)$ is symmetric. For a positive integer $d$, define … wh…

Chow tautological-generation conjecture. The Chow group

Martin--Vial's conjecture.

Let $\mathcal{F}$ be the moduli stack of a locally complete family of polarised hyperkähler varieties, and let $\mathcal{X}\to\mathcal{F}$ be its universal family. For each…

Virtual modularity conjecture. The series $\Theta^{2,\mathrm{vir}}_{\mathcal A_n}$ is a holomorphic Siegel modular form of weight $2n$ for $\operatorname{Sp}_4(\mathbb Z)$, valued…

Let $G$ be a split semisimple linear algebraic group over a field $F$, let $B\subset G$ be a Borel subgroup, and let $E$ be a generic $G$-torsor. Write $X=E/B$ and let … be the can…

Green–Griffiths implication. The induced second filtration step vanishes:

For each $g\ge 3$, let $\mathcal F_g'$ be the smooth dense open moduli space of primitively polarized K3 surfaces of degree $2g-2$ with trivial automorphism groups, and let … be th…

Let $X$ be a $K3$ surface, and let ${\operatorname{Aut}}_s(X)$ be the group of symplectic automorphisms of $X$, meaning automorphisms that act trivially on a nonzero holomorphic…

Let $K$ be a global field, let $Y$ be a smooth projective variety over $K$, and let $i\in\mathbb{Z}$. Write $\operatorname{CH}^i(Y)_0$ for the kernel of the cycle class map and let…

Let $Y$ be a smooth projective variety over a global field, and let $i\in\mathbb{Z}$. Write $\operatorname{CH}^i(Y)_0$ for the kernel of the cycle class map on…

Let $S$ be a smooth projective surface, and let $G\subset\mathrm{Aut}(S)$ be a group of automorphisms. Assume that every element of $G$ acts trivially on $H^{2,0}(S)$. Trivial-acti…

Let $S$ and $T$ be smooth projective surfaces. Let $\Gamma\in\mathrm{CH}^2(S\times T)$ be a codimension-$2$ cycle, and let … be the induced map. Define…

Stable Chow groups conjecture. The $i$th rational Chow groups of $\operatorname{CHur}^{G,c_1,\ldots,c_k}_{n_1,\ldots,n_k}$ stabilize, in the sense that they are independent of…

Let $r$ be a positive integer and let $(\kappa,\lambda)\in\mathcal A_r^{(2)}$. Write $X_r=X_1(Np^r)$, $f=\mathbf f_\kappa$, and $g=\mathbf g_\lambda$. Assume that $f$ and $g$ are n…

Let $X=F_r(Y)$ be as above, with Voisin rational self-map $\Psi:X\dashrightarrow X$, and let $F\subset X$ be the closure of its fixed locus. A closed subvariety is constant-cycle w…

Let $Y$ be a cubic hypersurface and let $X=F_r(Y)$ be the variety of $r$-planes in $Y$, equipped with the Voisin rational self-map $\Psi:X\dashrightarrow X$. For $r\geq 2$ and…

Let $X$ be a projective hyper-Kähler manifold. Consider the cycle class map from the Chow ring $CH^*(X)$ to cohomology. Beauville's weak splitting conjecture. The cycle class map i…

Beilinson–Bloch conjecture. The Beilinson–Bloch conjecture for the motive $\mathfrak h^3(J)^{\mu_6}$ predicts

Finiteness conjecture for isotropic motivic Homs. Homs between compact objects of $DM(k/k;\mathbb{F}_p)$ are finite groups.