230 problems

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

Bloch–Beilinson filtration conjecture. There exists a filtration

Let $X$ be a smooth projective variety defined over $\overline{\mathbb{Q}}$, and let $0\leq j\leq\dim(X)$ be an integer. Consider the de Rham-Betti structure … If…

Let $X$ be a hyper-Kähler manifold of dimension $2n$, and let $L,L'\subset X$ be $n$-dimensional constant-cycle subvarieties, meaning that all their points are rationally equivalen…

The Lefschetz standard conjecture. For each $k\leq n$, there exists an algebraic self-correspondence

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…

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 $X$ 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…

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…

Let $X$ be a smooth projective variety over a field $k$, and let a cycle on $X$ be smash-nilpotent if some positive external power of it is rationally equivalent to zero on a corre…

Hodge standard conjecture. The quadratic form $q_Z$ is positive definite. This is Grothendieck’s Hodge-type standard conjecture; it is known in characteristic zero through Hodge–Ri…

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…

Let $X\subset\mathbb{P}^4$ be a very general hypersurface of degree $d\ge6$, and let $C\subset X$ be a curve. Griffiths–Harris conjecture. The degree of $C$ is divisible by $d$. Th…

Griffiths–Green conjecture. For every $\alpha\in Hg^p(X)$, the associated admissible normal function $\nu_\alpha$ is singular on $\bar{S}=|L^{\otimes d}|$ for some $d>0$. This conj…

Let $k\in\mathbb{Z}_{>0}$, let $A$ be a very general abelian variety, and let $\mathscr{G}(k)$ be the least dimension such that a very general abelian variety of that dimension has…

Kimura's conjecture. All smooth projective varieties have finite-dimensional motive.

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…

Let $X$ be a smooth projective complex variety. Write … for the integral cycle class map, and let … be the group of integral Hodge classes. Integral Hodge conjecture. For every suc…

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

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 $A$ 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…

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

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…

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…