20 problems

Let $\mathbb{F}$ be the finite field under consideration, let $\ell$ be a prime, and let $X$ be a smooth projective geometrically uniruled $\mathbb{F}$-variety of dimension $3$. Th…

Let $X$ be a smooth projective variety over $\mathbb C$. The integral Hodge conjecture concerns whether every integral Hodge class … is algebraic. Integral Hodge conjecture. Every…

Let $X$ be a smooth projective variety of dimension $d$ over a finite field ${\mathbb{F}}$, and let $\ell$ be a prime different from the characteristic of ${\mathbb{F}}$. Write…

Let $q$ be an anisotropic quadratic form over $k$, let $K=k(q)$, and let $r$ be a quadratic form defined over $K$. Let $W_{\mathrm{nr}}(K/k)$ denote the unramified Witt ring. Weak…

Let $q$ be an anisotropic non-degenerate quadratic form over $k$, let $K=k(q)$, and let $r$ be a quadratic form defined over $K$. Let $W_{\mathrm{nr}}(K/k)$ denote the unramified W…

Let $k$ be the base field, let $\mathcal{X}=(X,D)$ be an object of $\category{mSm}$, and let $q\geq 0$. Write $\mathcal{X}^{\circ}$ for the complement of the boundary,…

Let $X$ be a smooth variety over a field $k$, and let $A(n)$ be a twisted locally constant torsion étale sheaf for which the exponential characteristic of $k$ is invertible. Let ……

Let $Y$ be a smooth proper variety of dimension $n$ over $C$. Consider the following properties: $Y$ is uniruled; $\mathrm{CH}_0(Y)$ is supported in codimension at least $1$;…

Let $A$ be an abelian torsion group and $X$ be a smooth quasi-projective variety defined over a field $k$. Consider the identity continuous morphism of sites … and let…

Refined unramified homology and cohomology are expected to arise as a hypercohomology theory over a suitable motivic category. Hypercohomologicality conjecture. The refined unramif…

Let $\dvr$ be the ring of integers of a non-archimedean local field, let $\my'$ be a regular scheme proper over $\dvr$, and let $\fy'$ be its function field. Kato conjecture. The u…

Let $X$ be a unirational variety over an algebraically closed field of characteristic zero, and let $H^i_{nr}$ denote unramified cohomology. Unramified-cohomology finiteness conjec…

Let $K$ be a finitely generated field of transcendence degree $n$ over an algebraically closed field $k$ of characteristic $\ne 2$. Write $H_{\mathrm{ur}}^n(K/k,\mu_2^{\otimes n})$…

Conjecture 5.2. The set of $K$-isomorphism classes of $K$-forms of $G$ having good reduction at every $v\in V_0$ is finite.

Let $K$ be a local field with ring of integers $\mathcal{O}_K$ and finite residue field $k$. Let $X$ be a regular proper flat scheme over $\mathcal{O}_K$, with smooth generic fiber…

Let $k$ be a field, let $X\subset \mathbb{P}^n$ be a smooth hypersurface of degree at most $n$ with $X(k)\neq\emptyset$, and let $K_0(\operatorname{Var}/k)$ be the Grothendieck rin…

Conjecture de finitude de l'obstruction $K$-théorique. Le groupe $\cyr{X}\mathcal N(V)$ est fini. Cette conjecture est un principe local-global en $K$-théorie algébrique; son statu…

Soit ${\mathbb F}$ un corps fini, soit $l\neq p=\operatorname{car}({\mathbb F})$ un nombre premier, et soit $X$ une variété projective et lisse de dimension $3$ sur ${\mathbb F}$,…

Let $X$ be a smooth projective complex variety, and let $i$ be an integer with $i\geq 4$. For $j\geq i$, consider the Hodge-cohomology groups $H^j(X,\mathcal O_X)$ and the integral…

Vanishing conjecture.