213 problems

For the Simpson stratification of the Hodge moduli space of Higgs/de Rham objects over a smooth complex projective curve of genus $g\geq 2$, Simpson's nestedness conjecture asserts…

For every smooth and proper differential $\mathbb{Z}/2$-graded category $\mathcal{C}$ over $\mathbb{C}$, the Getzler–Gauss–Manin connection $\nabla^{\mathcal{C}}_{\partial_t}$ in t…

For integers $m\ge 1$ and $n\ge 1$, let $X_m^n\subset \mathbb{P}^{n+1}_{\mathbb{C}}$ be the Fermat hypersurface $x_0^m+x_1^m+\cdots+x_{n+1}^m=0$. The Hodge conjecture for Fermat va…

For every integer $m\ge 1$, let $X_m^4=\{x_0^m+x_1^m+x_2^m+x_3^m+x_4^m+x_5^m=0\}\subset\mathbb{P}^5$ be the Fermat fourfold. The Hodge conjecture asserts that every rational Hodge…

Nonnegativity conjecture. All stringy Hodge numbers $h^{p,q}_{\rm st}(X)$ are nonnegative.

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

Hodge-theoretic Hopf conjecture. For every $p$ with $0\leq p\leq n$,

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 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…

Grothendieck's conjecture. There exists a dense open $U\subseteq Y$ such that this restriction map vanishes.

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…

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 $({\mathbb V}^{\sigma})_{\sigma}$ 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…

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

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 realization conjecture. The functor $\iota_{X,\sigma}^{\textup{Hdg}}$ is fully faithful, with essential image stable under subquotients. This is a Hodge-type conjecture for N…

Cappell–Shaneson conjecture.

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…

Let $X$ be a complex projective manifold, let $\mathbb{M}$ be a Mixed Hodge Module on $X$, and let $H^k_2(\widetilde{X},\mathbb{M})$ denote the associated $L_2$ cohomology, equippe…

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 $\mathbb{Q}^{\mathrm{al}}$ with good reduction at $w$, and let $\gamma$ be a Hodge class on $A$. Call $\gamma$ $w$-Lefschetz if its specializatio…

Let $X$ be a smooth complex projective variety, and set … Let $\mathrm{B}_2(X)$ denote the cone of universally pseudoeffective $2$-dimensional classes, and let a class be realizabl…