176 problems

Let $X$ be a hyper-Kähler variety, and let $\mathrm{R}^{\bullet}(X)\subset \operatorname{CH}^{\bullet}(X)_{\mathbb{Q}}$ be the subalgebra generated by divisor classes. Beauville's…

Voisin's conjecture. If $\phi$ is symplectic, then

Let $Y$ be a double EPW quartic. Assume that the multiplication on holomorphic forms … is surjective, and let $F^\bullet$ denote the conjectural Bloch–Beilinson filtration. Bloch–B…

Let $X$ be a hyper-Kähler variety. Write $A^\ast(X)$ for its Chow ring with rational coefficients, and let $c_j(T_X)$ denote the Chern classes of its tangent bundle. Beauville–Vois…

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…

Beilinson–Bloch conjecture. The Abel–Jacobi map is injective for varieties defined over number fields; consequently, every such cycle that is trivial under the Abel–Jacobi map is t…

Let $X$ be a smooth projective variety over a field $k$ of pure dimension $d$. A Chow–Künneth decomposition of $X$ consists of pairwise orthogonal projectors…

Let $F=f_1\boxtimes f_2\boxtimes f_3$ be the triple product motive associated with the three forms in the setting above, let $L(F,s)$ be its finite triple product $L$-function, let…

Franchetta conjecture. Every locally complete family of hyperkähler varieties satisfies the Franchetta property. The property is known for some Fano varieties, but the source prese…

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 $M$ be a projective hyper-Kähler manifold. A filtration on the Chow ring is called a Beauville–Voisin filtration when it is analogous to the filtration on the Chow group of zer…

Anti-symplectic involution conjecture. Then

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 $X$ be a smooth projective variety. Write $A^2_{AJ}(X)$ for the subgroup of codimension-two cycles Abel–Jacobi trivial, $A^2_{alg}(X)$ for the subgroup algebraically trivial, a…

Let $K$ be a number field, let $X$ be a smooth projective variety over $K$, and let ${\mathrm{CH}}^m(X)$ denote the Chow group of codimension-$m$ algebraic cycles on $X$ modulo rat…

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 $F^1A^j(X)=(A^j(X))_{hom}$;…

Separatedness and motivic identification conjecture. The filtration $F_{\mathcal{M}}$ is separated and gives the conjectural motivic filtration of Beilinson and Bloch.

Transcendence-degree realization conjecture. If

Independence and Murre compatibility conjecture. The filtration $F_{\mathcal{M}}$ should not depend on the choice of $\mathcal{M}$ and should coincide with Murre's conjectural filt…

Cancellation conjecture for effective Chow motives. If

Let $X$ be smooth and projective of dimension $n$ over a finite field, and let $\operatorname{CH}^q(X)$ denote its codimension-$q$ Chow group. Finite-generation conjecture. The gro…

Beauville's conjecture. The eigenspace is nonzero precisely in the expected range:

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…