98 problems

For every genus $g\ge 2$, let $\pi:\mathcal{X}_g\to\mathcal{M}_g$ be the universal family of polarized K3 surfaces of genus $g$, and let $o_S\in\mathrm{CH}^2(S)$ denote the Beauvil…

Let $X$ be a holomorphic symplectic variety, let $\operatorname{CH}(X)_{(0)}$ denote the degree-zero part of the Chow grading associated with a Chow--Künneth decomposition, and let…

Murre's conjecture. There exists a Chow–Künneth decomposition $\{\pi^0,\ldots,\pi^{2d}\}$, and the induced filtration satisfies $F^0\operatorname{CH}^i(X)=\operatorname{CH}^i(X)$,…

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…

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…

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…

Assume the setup of the paper: $\Pi_0$ and $\Pi_1$ are the relevant representations under consideration, $F$ is the CM field, $E$ is their coefficient field, and the notation…

Let $A$ be an abelian variety over $\mathbb C$ of dimension $g$. Its Beauville decomposition is … where … Here $n^*$ is induced by multiplication by $n$ on $A$. Beauville's conject…

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 $S$ 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…

Let $S$ be a smooth projective surface. A symplectic automorphism is an automorphism acting trivially on the space of holomorphic two-forms $H^{2,0}(S)$. It acts on the Albanese ke…

Let $X$ be a smooth projective variety over the complex numbers, and let $D\subset X$ be a very general smooth ample divisor of sufficiently large degree. Write…

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.

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:

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…