487 problems

Tate conjecture. The representation $\rho_{\ell}$ is semisimple, and the cycle class map

Let $X\subset\mathbb{P}^{n+1}(\mathbb{C})$ be a smooth hypersurface, and let $h\in A^1(X)$ denote the hyperplane class. Hartshorne's conjecture. … Apart from some easy results when…

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…

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…

Let $X$ be a smooth projective variety, and let $z$ be an algebraic cycle on $X$. Two cycles are numerically equivalent if they have the same intersection numbers with all cycles o…

Hodge conjecture. The map $cl^{p,0}$ should be surjective. This is the classical Hodge conjecture, asserting that the relevant rational Hodge classes arise from algebraic cycles; t…

Lefschetz standard conjecture. The inverse of this hard Lefschetz isomorphism is induced by an algebraic correspondence; assuming this conjecture, the Hodge conjecture for…

Let $K$ be a number field and let $Z/K$ be a smooth projective irreducible variety of dimension $d$. Define the Albanese kernel by … Here $\mathrm{CH}^d(Z)_0$ is the group of homol…

Let $X$ be a smooth projective complex variety, and let a rational $(p,p)$ cohomology class mean a class in $H^{2p}(X,\mathbb{Q})\cap H^{p,p}(X)$. The Hodge conjecture. Every ratio…

Let $X$ be a smooth projective variety defined over $\overline{\mathbb{Q}}$, and let $\mathrm{H}^{n}_{\mathrm{dRB}}(X,\mathbb{Q})$ be its de Rham–Betti structure. Write…

Let $S$ be a smooth projective surface defined over $\bar{\mathbb Q}$. Denote by $\operatorname{CH}_0(S)^0$ the Chow group of zero-cycles of degree zero, by $\operatorname{Alb}(S)$…

Let $X$ be a smooth projective variety of dimension $m$. Assume that $H^{p,q}(X)=0$ for $p\ne q$ and $p<c$ (or $q<c$). Generalized Bloch conjecture. The cycle class map … is inject…

Friedlander–Mazur conjecture. For any smooth projective variety $X$ and $k\geq 2p\geq 0$, one has

Let $\mathcal{C}\to U$ be a universal curve over a Zariski open subset $U$ of the moduli space of curves of genus $g\geq 2$. For a point $b\in U$, write $\mathcal{C}_b$ for the fib…

Hodge conjecture. Every Hodge class on $X$ is a linear combination with rational coefficients of the cohomology classes of complex subvarieties (algebraic cycles) of $X$.

Grothendieck's general Hodge conjecture. The inclusion above is an equality. The conjecture strengthens the classical Hodge conjecture by comparing coniveau with the largest Hodge…

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 $Y$ be a smooth projective algebraic variety over $\mathbb{C}$, let $j \geq 0$ be an integer, and let $c \in H^{2j}_{B}(Y, \mathbb{Q}(j))$ lie in the Hodge summand…

Let $X$ and $Y$ be smooth projective varieties of dimension $n$, and let $Z\in CH^n(X\times Y)_{\mathbb Q}$ be a correspondence. Let $F^\bullet$ be the Bloch–Beilinson filtration a…

Beauville's conjecture. The cycle class map is injective on the subalgebra of $A^{*}(X)$ generated by divisors.

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 complex projective variety and let $Y\subset X$ be a closed algebraic subset. Suppose $Z\subset X$ is a codimension-$k$ algebraic cycle whose class…

Rost nilpotence principle. For every $\Gamma\in \operatorname{End}_k(X)$ such that $\Gamma_E=0$ for some field extension $E/k$, $\Gamma$ is nilpotent as a correspondence.

Let $X$ be a smooth projective variety whose algebra of holomorphic forms is generated in degree at most $2$, and let $x,y$ be closed points of $X$. Let $F_{\mathrm{BB}}^\bullet$ d…

Grothendieck's generalized Hodge conjecture. The subspace $N^{d-2}H^{2d-3}(X,\mathbb{Q})$ should be the largest rational Hodge substructure contained in