6 problems

Bogomolov's conjecture.

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 $k$ 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 $\mathbb{F}$ 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 unirational variety over an algebraically closed field of characteristic zero, and let $H^i_{nr}$ denote unramified cohomology. Unramified-cohomology finiteness conjec…

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…