37 problems

Fargues–Scholze categorical conjecture. This setting enables a version of the Fargues–Scholze categorical conjecture that includes $\textnormal{Kal}_v$-inner forms.

Let $\mathcal{Y}$ be a gerbe with dual $\widehat{\mathcal{Y}}$ and associated $\mathbb{C}^*$-gerbe $c$. Let $D\subset \mathcal{B}$ be a smooth irreducible divisor, and define its i…

Let $G$ be a finite group and let $\pi:\mathcal{Y}\to\mathcal{B}$ be a $G$-gerbe over a smooth Deligne–Mumford stack with trivial generic stabilizer. Let…

Let $X$ be a smooth projective variety over $\mathbb{C}$, let $G$ be a finite abelian group, and let $\epsilon:\mathcal{G}\to X$ be a $G$-banded gerbe. Let $\lvert\hat{G}\rvert$ de…

The obstruction conjecture. The obstruction to the gerbe $R$ over $D$ being trivial equals the obstruction to the normal bundle $N_{D/X}$ having an $n$-th root as a line bundle on…

Let $B_{1}=F_{1}=B_{2}=F_{2}=S^{1}$, let $E_{i}=B_{i}\times F_{i}$ be elliptic curves, and let $S_{i}=B_{i}\times F_{i}^{\vee}$. Set $X=E_{1}\times E_{2}$ and $Y=S_{1}\times S_{2}$…

Let $X$ be a complex manifold elliptically fibered with at worst $I_{1}$ fibers over a normal analytic variety $B$ such that … Let $\alpha,\beta\in\mathop{\mathcal{S}h}_{an}(X)$ be…

Let $\mathcal{G}_{r}$ be the nontrivial $\mu_{r}$-gerbe over $\mathbb{P}^{d}$ and let $\mathbb{P}^{d}\times\mathcal{B}\mu_{r}$ be the trivial $\mu_{r}$-gerbe over $\mathbb{P}^{d}$.…

Let $X\to\mathbb P^n$ be an abelian fibration, let $P\to\mathbb P^n$ be its relative Picard, and let $X_0$ be the relative Picard of $P$. Let…

Main Conjecture. There exists an equivalence

Let $M$ be a manifold, let $\Omega(M,*)$ denote the based loop space, and let $C(M,*)$ be the groupoid of based loops and homotopies between them. Let $N(M,*)$ be the normal monoid…

Let $W$ be the coarse projective variety in the KSBA moduli problem, let ${\mathfrak W}\to W$ be a $\mu_N$-gerbe, and let $M$ be the associated KSBA moduli space. Suppose that the…

Let $M$ be a smooth manifold, let $n$ be a positive integer, let $[g,\mathcal{A}]$ be an $(n{-}1)$-gerbe with full connection, and let $H\in\Omega^{n+1}_{\rm cl}(M)$ be the field s…

Gerbe decomposition conjecture. For the Ruan–Zhang/Chern–Simons level corresponding to ordinary quantum $K$-theory,

Drinfeld's gerbe isomorphism conjecture. These two gerbes are isomorphic.

Let $\mathbb{F}$ be the algebraic closure of a finite field, and let $X$ be a variety over $\mathbb{F}$. Write $\operatorname{Rep}(\operatorname{Kt}_{\mathbb{Q}})$ for the Tannakia…

Essential dimension conjecture. One has

Let $M$ be a smooth manifold. Write $4\pi_0(\mathrm{FFT}_2^{sf}(M)^\times)4$ for the set of isomorphism classes of two-dimensional superficial, invertible smooth functorial field t…

Let $X$ be a surface and let ${\mathfrak X}\to X$ range over all $\mu_r$-gerbes. For each such gerbe, consider the twisted Vafa–Witten invariants defined using the moduli space of…

Let $G$ be a finite group, let $Q$ act on the finite set of isomorphism classes of irreducible $G$-representations, and let $\mathcal{Y}$ be a gerbe with dual…

Let $S$ be a smooth projective surface and let $r\in{\mathbb Z}_{>0}$. For $L\in\operatorname{Pic}(S)$, define the $\operatorname{SU}(r)$ Vafa–Witten partition function by … Define…

Let $S$ be a smooth projective surface and let ${\mathfrak S}\to S$ be a $\mu_r$-gerbe. The twisted Vafa–Witten invariants are defined using twisted stable sheaves and Higgs sheave…

Let $X$ be the smooth scheme considered above, let $G$ be the reductive group with maximal torus $T$, and let $\ell$ be the chosen prime. Consider the diagram … The horizontal and…

A string datum consists of local data whose underlying bundle is a principal $\operatorname{Spin}(n)$-bundle $\bar{P}$. Gluing these local data is expected to produce an $S^1$-gerb…

Let $(X,varphi)$ be a $G_2$-manifold equipped with a monopole gerbe $(\mathcal{G},F)$, and consider the analogue for gerbes of the condition that a connection on a complex line bun…