457 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 $X$ be a smooth projective variety defined over a finitely generated subfield $F\subset\mathbb{C}$. Under the Artin comparison identification … let…

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 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…

Moore's attractor conjecture. Attractor points are defined over $\overline{\mathbb{Q}}$. Moore verified this in several cases. The conjecture asks for algebraicity of the special p…

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

Let $X\subset\mathbb{P}^5$ be a smooth sextic defined by a $\sigma$-invariant equation, where $\sigma$ is the involution under consideration, and write … The invariant part…

Let $(U,f)$ be a Landau–Ginzburg model, where $U$ is a smooth quasi-projective variety and $f\colon U\to \mathbb{A}^1$ is a regular function with reduced pole divisor. The Landau–G…

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

Let $({\mathbb V}^{\sigma})_{\sigma}$ be a de Rham motivic variation of Hodge structure on a smooth connected complex quasi-projective variety $S$. Let ${\mathbf G}$ be the group d…

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 $f:\mathfrak{X}\to S$ be an algebraic family of polarized algebraic manifolds over a quasi-projective manifold $S$, with period map $\Phi:S\to\Gamma\backslash D$ and lifted per…

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…

Integral cohomology and non-isomorphism conjecture. One has an isomorphism of integral cohomology groups

Hodge conjecture for the Kuga–Satake correspondence. There is a cycle $\Gamma\in {\rm CH}^{2n}(X\times KS(X))_{\mathbb{Q}}$ such that $[\Gamma]=\alpha$, and hence

Grothendieck's standard conjecture of Hodge type. For any $i \le d/2$, the symmetric bilinear form

Cappell–Shaneson conjecture.

Let $Y_{[3]}^\Gamma$ be a smooth complex threefold, and let an algebraic $2k$-cycle on it be a $2k$-cycle that is holomorphically embedded in $Y_{[3]}^\Gamma$. The Poincaré dual of…

Let $S$ be an irreducible algebraic variety with a non-trivial bi-algebraic structure. Let $U\subset \widetilde{S^{\mathrm{an}}}\times S^{\mathrm{an}}$ be an algebraic subvariety f…

Let $X$ be a smooth connected projective variety over ${\mathbb C}$, and let $k$ be a nonnegative integer. A Hodge class is a class in $H^{2k}(X, {\mathbb Q})(k)$ of Hodge type…

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…

Hodge conjecture. The group $Z^{2i}(X)$ is finite.

Let $X$ be a smooth projective variety over $\mathbb{C}$. A Hodge cycle on $X$ is a cohomology class of Hodge type, and an algebraic cycle is a formal rational linear combination o…