13 problems

Let $X$ be a projective nonsingular surface over ${\bold C}$, and let $T(X)$ be the kernel of the Albanese map on degree-zero zero-cycles. Let $\underline{{\mathrm{M}}}({\bold C})$…

Let $r\geq 2$ be an integer. For a nonsingular projective variety $X$ over $\overline{\bold Q}$, let … be the Abel-Jacobi map. If $X_{\bold C}=X\otimes_{\overline{\bold Q}}\bold C$…

Termination conjecture for the motivic filtration. The filtration terminates, meaning

Let $X$ be a smooth projective variety of dimension $g$, and let $V\subset H^m(X,\mathbb{Q})$ be a $\mathbb{Q}$-Hodge substructure of level at most $m-2p$. A nonsingular projective…

Griffiths's conjecture on incidentally trivial cycles. For every $p\in\{1,\ldots,d\}$ and every $z\in I^p(X)$, there is a positive integer $N$ such that

Let $X\subseteq \mathbb P^N$ be a smooth projective variety of dimension $n$, let $Y=X\cap H$ be a smooth hyperplane section, let $U\subseteq (\mathbb P^N)^*$ be the open set of sm…

Let $V/\overline{\mathbb Q}$ be a smooth quasiprojective variety. Let … be the Abel–Jacobi map. Bloch–Beilinson conjecture on Abel–Jacobi maps. The map $\Phi_{k,\mathbb Q}$ is inje…

Jacobi-type theorem. There is some positive integer $k$ such that the topological Abel--Jacobi mapping

Strong Tube Theorem. The image of the topological Abel--Jacobi homomorphism

Beilinson–Bloch conjecture. The $m$-th higher Abel–Jacobi map $AJ^m$ is injective. This is presented as another part of the Beilinson–Bloch conjecture for smooth projective varieti…

Let $W$ be a smooth projective variety over $\overline{\mathbb{Q}}$, and let $r$ and $m$ be integers for which the displayed higher Chow group is defined. Let … be the generalized…

Let $W$ be a smooth projective variety over $\overline{\mathbb{Q}}$, and let $r\geq 0$ be an integer. Let $J(H^{2r-1}(W(\mathbb{C}),\mathbb{Q}(r)))$ denote the relevant intermediat…

For generic $V\subset\Lambda^2W^*$, let $X$ and $Y$ be the associated Calabi–Yau varieties, and let $F(Y)$ and $H(X)$ be the birational moduli spaces described in the source. Since…