26 problems

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…

Kuga--Satake Hodge conjecture. The embedding $\iota_V$ is induced by a correspondence in $A^\ast(X\times KS(V)\times KS(V))$. This is a special case of the Hodge conjecture; the pa…

Hodge conjecture. The Hodge classes are $$ -linear combinations of algebraic classes.

General Hodge conjecture. This embedding is fully faithful.

Higher Hodge conjecture. The map $cl_{U}^{p,m}$ is surjective. This is the arithmetic quasiprojective extension of the Hodge conjecture attributed in the source to predictions of B…

Let $X$ be a smooth hypersurface of dimension $m=2n$ in ${\mathbb P}^{2n+1}$. Let $H^{n,n}(X;\Q)=H^{n,n}(X)\cap H^{2n}(X;\Q)$, and let $h_n^{n,n}$ be the dimension of the subspace…

The Hodge conjecture for $A_v$. Every Hodge class on $A_v$ is algebraic. This is the specialization of the Hodge conjecture to the abelian sixfolds arising from non-CM points of…

Let $Y_t$ be a smooth fibre of a semistable degeneration with nodal central fibre $Y_0$, let $\beta\in H^{2,2}(Y_t,\mathbb{Q})$ be rational, and let…

Let $\pi:\mathcal{Y}\to\Delta$ be a $G$-equivariant semistable degeneration of Calabi–Yau threefolds with ordinary double points, and let $\widetilde{Y}_0$ be a $G$-equivariant cre…

Let $(X,h)$ be a $2n$-dimensional polarized hyper-Kähler variety, and let $H^2(X,\mathbb{Q})_{\mathrm{prim}}$ denote the orthogonal complement of the polarization class in…

Let $X$ be an abelian threefold, let $\hat X$ be its dual, and let ${{{{{{{{\mathcal G}}}}}}}}_{F_1,F_2}$ be the sheaf constructed from non-zero coherent sheaves $F_1,F_2$ whose Ch…

Let $\mathfrak X$ be a smooth projective tropical variety. For an integer $p$, consider the tropical cohomology groups … and let $N$ be the tropical monodromy operator between them…

Let $\mathscr{X}_{-1/7}$ be the Calabi–Yau threefold considered in the paper, let $h^3(\mathscr{X}_{-1/7})$ be its third cohomology motive, and let $K$ be a number field. Let…

Let $\mathcal{M}_{\mathbb{C}}$ be the category of pure homological motives over $\mathbb{C}$ with rational coefficients, and let … be the enriched Betti realization functor to rati…

Hodge conjecture. The two $\mathbb{Q}$-structures coincide:

Let $f:\mathcal{X}\longrightarrow S$ be a smooth proper morphism and let ${\mathbb V}=R^kf_*{\mathbb Q}$. For a point $s\in S$, exceptional Hodge tensors in ${\mathbb V}_s$ are Hod…

Hodge conjecture for log mixed motives. The realization map is an isomorphism

Demailly's Hodge conjecture for strongly positive currents. The current $\mathscr{T}$ is a weak limit

Let $(G,X)$ be a Shimura pair of Hodge type with an integral canonical model ${\scr N}$, let $y:\operatorname{Spec}\,\db F\to{\scr N}$ be a point, and let $A$ be the abelian variet…

Let $S$ be a projective $K3$ surface, let $B$ be a connected quasi-projective variety, let $\mathcal{S}\rightarrow B$ be a smooth projective family of projective $K3$ surfaces, let…

The big-class coniveau conjecture. The sub-Hodge structure $L$ is supported on a divisor of $X$, namely, there exists a closed algebraic subset $Z$ of codimension $1$ such that

Let a supermanifold be described by a matrix-factorization category, and let its Jacobian ideal be viewed inside Hochschild cohomology. A $(p,p)$ class is a Hodge-type class of bid…

Let $X$ be a smooth projective variety of dimension $n$, let $k\leq n$, and let $H'$ be a sub-Hodge structure of $H^k(X,\mathbb{Q})$ with no component of type $(k,0)$. Generalized…

Let $U$ be a smooth quasiprojective variety over $\mathbb{C}$, let $\CH^{r}(U,m;\mathbb{Q})=\CH^{r}(U,m)\otimes\mathbb{Q}$ be Bloch's higher Chow group with rational coefficients,…

Let $A$ be an abelian variety over an algebraic closure of $\mathbb{Q}$ with good reduction to an abelian variety $A_{0}$ over an algebraic closure of a finite field. A Hodge class…