28 problems

Huybrechts's conjecture. For every hyper-Kähler variety $X$, the divisibility

Lang's conjecture. The function field $\mathbb R(X)$ is $C_i$.

Period–Index Conjecture for Function Fields. For every $\alpha\in\operatorname{Br}(k(X))$,

Let $k$ be a field, let $C/k$ be a smooth projective curve with a rational point, let $\mathscr L\in\operatorname{Pic}(C)$ be an invertible sheaf, and let $n$ be an integer prime t…

Let $K$ be a $C_d$ field, meaning that every homogeneous form of degree $e$ in more than $e^d$ variables over $K$ has a nontrivial zero, and let $\alpha\in\operatorname{Br}(K)$. Th…

Let $X$ be a smooth, connected, projective variety over an algebraically closed field $k$, and let $lpha \in \operatorname{Br}(X)$. Write $\operatorname{per}(\alpha)$ for the order…

Let $K$ be a $C_d$ field, and let $\alpha\in \operatorname{Br}(K)$ be a Brauer class. The period $\operatorname{per}(\alpha)$ is its order in the Brauer group, and the index…

Period-index conjecture. The index should divide the period raised to the power one less than the dimension of $B_0$:

Let $\mathcal{K}$ be a complete discrete valued field of characteristic zero with residual characteristic $p$, and let $\kappa$ be its residue field. For a field $E$ of characteris…

Generalized Artin conjecture. If $K$ is a $C_d$-field, then

Generalized Artin conjecture. If $K$ is a $C_d$-field, then

Let $K$ be a field and let $A$ be a central simple algebra over $K$. Write $\operatorname{per}(A)$ and $\operatorname{ind}(A)$ for the period and index of $A$, respectively. A fiel…

Let $k$ be a $C_2$ field, meaning that every homogeneous polynomial of degree $d$ in more than $d^2$ variables over $k$ has a nontrivial zero. Write $\operatorname{Br.dim}(k)$ for…

Unramified period-index conjecture. For every $\alpha \in \operatorname{Br}(X)$,

Period-index conjecture. For every $\alpha \in \operatorname{Br}(K)$,

Let $F$ be a field, let $d$ be an integer, and let $\ell$ be the prime appearing in the cohomology group. The diophantine dimension $\operatorname{dd}(F)$ is the smallest integer…

Let $K$ be a field with $\operatorname{cd}_\ell(K)=n$, and fix a prime $\ell$ coprime to the characteristic of $K$. Let $F$ be a function field of a curve over $K$. The indices of…

Let $X$ be a variety and let $\alpha$ be an unramified Brauer class on $X$. Write $e_\alpha$ for the exponent in a period–index bound for $\alpha$. de Jong–Perry's conjecture. Ther…

Let $k$ be a totally imaginary number field, let $C/k$ be a smooth projective geometrically integral curve, and let $\mathscr{C}/\mathscr{O}$ be a regular proper model, where…

For a finite $2d$-dimensional CW complex $X$ and a Brauer class $\alpha\in\operatorname{Br}(X)$, write $\mathrm{per}(\alpha)$ for its period and $\mathrm{ind}(\alpha)$ for its inde…

Antieau-Williams' conjecture.

Antieau-Williams' conjecture.

The standard period-index conjecture. For every $\beta\in\operatorname{Br}(\mathbb{Q}_p(S))$, one should have

Let $p$ be prime, let $e<f$ be positive integers, and set $m=p^e$, $n=p^f$ as above. With $n=km$, consider the symmetric-function expression … where the sum runs over partitions fi…

Effective-index conjecture. One should have