230 problems

Let $Y$ be the variety in the preceding construction and let $X=F_r(Y)$, with $r\geq 2$, where $\Psi:X\dashrightarrow X$ is the Voisin rational map. For $z\in CH_0(X)_{hom}$, the s…

ℓ-adic integral Tate conjecture in codimension $c$. The cycle class map

ℓ-independent Lefschetz standard conjecture in degree $i$. For $X$, $h$, and $i$ as above, there exists a codimension $i$ cycle…

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 $X_{\bold C}=X\otimes_{\overline{\bold Q}}\bold C$…

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…

Polynomial-cycle injectivity conjecture. If the cohomology class of $P$ vanishes, then $P$ vanishes in the Chow group:

Let $X_1$ and $X_2$ be varieties defined over $\mathbb{Q}$, and let $\rho_1$ and $\rho_2$ be isomorphic two-dimensional Galois representations occurring in their étale cohomology.…

Tate–Beilinson conjecture. For every prime $l$: (1) the cycle map above is surjective; (2) the $\hat G$-module $H^{2n}_{cont}(\bar X,\mathbb{Q}_l(n))$ is semisimple at the eigenval…

The algebraicity conjecture. For all integers $b\in\mathbb N^*$ and $p\in\mathbb N$ there is an integer $D$ depending only on $b$, $p$, $Y$ and ${\mathcal O}(1)$, such that for all…

Let $X=T^*(G/B)$ be the Springer resolution of the nilpotent cone $Y={\mathcal N}\subset{\mathfrak g}^*$ in the coadjoint representation of a semisimple algebraic group $G$, let…

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 $F^1A^j(X)=(A^j(X))_{hom}$;…

Hard Lefschetz conjecture. For all $k$, the Lefschetz operator induces an isomorphism

Let $T\mathcal M(n)$ denote the additive part of the degree-$n$ component of the indecomposable space, and let $Z_n(a)$ be a codimension-$n$ cycle on … For $a\in\mathbb A^1-\{0\}$,…

Let $S^{[3]}$ and $S^{[4]}$ be the strata appearing in the semistable degeneration, and let $[N]$ denote the cohomology class of the monodromy operator. The relevant restriction ma…

Let $X$ be a complex algebraic manifold and let $d$ be a positive integer. Write $QH^k(X)$ for the quotient cohomology groups appearing in the exact sequence associated with iterat…

Let $X$ be smooth and proper over $\mathbb C$. Let $\operatorname{gr}^0_{F_m}CH^p(X)$ denote the degree-zero graded piece of the motivic filtration on codimension-$p$ Chow groups,…

Let $X$ be a proper smooth scheme over $\mathbb C$. For integers $i,j$, define the coniveau filtration by … The inclusion … identifies the left-hand side with a subspace of rationa…

Let $k\subset F$ be the fields in the paper, let $n=\infty$, and let $X$ be a smooth irreducible proper variety over $k$. Write $CH_0(X_F)_{\mathbb Q}$ for the rationalized group o…

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

Let $X$ be a smooth complex projective variety. Define the coniveau filtration by … where $Z\subset X$ ranges over subvarieties, and let $H^{m,i}_{\operatorname{Hdg}}(X)$ be the la…

Let $X$ be a smooth, projective variety over $\mathbb{C}$, and write $h^{p,0}(X)$ for the Hodge number measuring global holomorphic $p$-forms. Bloch's conjecture. If … for every…

Generalized Tate conjecture. There is a closed algebraic subset $Z\subset X$ of codimension $k$ such that

Hu's rational cycle class conjecture. For all $d\geq 0$ and $0\leq p\leq n$,