33 problems

Let $A$ be a geometric $n$-stack, and let $D=\operatorname{Spec}(k[\epsilon]/(\epsilon^2))$. Let $TDef_A$ denote the $(n+1)$-category of deformations of $A$ parametrized by $D$. Le…

Let $T$ be a very presentable geometric $n$-stack, and let $X$ be a smooth projective variety. Write $X_{DR}$ for its de Rham formal stack. The de Rham mapping-stack conjecture. Th…

Let $T$ be any geometric, locally geometric, or slightly geometric $n$-stack, and let $X\to S$ be a flat projective morphism. The slightly geometric mapping-stack conjecture. The m…

Let $T$ be a geometric $n$-stack whose homotopy sheaves $\pi_i$ are vector sheaves for every $i\geq 3$, and let $X\to S$ be the morphism under consideration. The higher-vector-homo…

Let $S$ be a base scheme, let $X\to S$ be a flat projective morphism, and let $K({\bf G}_m,2)$ denote the Eilenberg–Mac Lane stack. For any group scheme $G$ of finite type, let…

Let $T$ be a geometric $n$-stack that is very presentable in the sense that its fundamental groups over artinian bases are affine and its higher homotopy groups are vector sheaves.…

Reductive local structure conjecture. Under these hypotheses, such an étale morphism exists.

The stack of all curves conjecture. 1. $\mathcal{M}_g^{\text{all}}$ satisfies Vakil's Murphy's law. 2. $\operatorname{Pic}(\mathcal{M}_g^{\text{all}})=0$. 3. The structure morphism

Let $X$ be a quasi-separated Deligne–Mumford stack with generically trivial stabilizer. Let $\mathcal{M}\in \operatorname{QCoh}(X)$ and let $\alpha\in \mathrm{Br}(X)$. Write…

Coherent completeness and formal functions conjecture. The stack $[X/G]$ is cohomologically proper over $A^G$, and the pair $([X/G],\mathcal{G}_x)$ is coherently complete and satis…

Let $G$ be a smooth affine group scheme over ${\mathbb{Z}}/p^n{\mathbb{Z}}$ and let $\mu:{\mathbb{G}}_m\to G$ be 1-bounded. Let $\operatorname{BT}_n^{G,\mu,?}$ be the quotient 2-st…

Drinfeld's algebraicity conjecture. This restriction is a quasicompact smooth algebraic stack over $\mathbb Z/p^m\mathbb Z$ with affine diagonal.

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

Drinfeld's extension conjecture. The isomorphism between these restrictions extends to an isomorphism over the whole $\operatorname{Disp}_1^G$.

Persistence Conjecture. If $X$ is arithmetically hyperbolic over $k$, then $X$ is absolutely arithmetically hyperbolic.

Finite-dimensionality conjecture. Its cohomology

Let $X$ be a smooth proper curve over $\mathbb{F}_p$, let $G\to X$ be a smooth affine group scheme with semisimple simply connected generic fiber and connected fibers, and let…

Let $\mathfrak{X}$ be a “nice” algebraic stack defined over a number field $K$, and let $\mathcal{E}$ be a “nice” vector bundle on $\mathfrak{X}$. Let…

Let $H_\Sigma$ be the formal group over the stack $\Sigma$ introduced above. An algebraization of $H_\Sigma$ is an isomorphism class of a smooth affine group scheme over $\Sigma$ w…

Let $(g,n)\neq(1,0)$, and let $d\geq 0$ be fixed. For each $m\geq 0$, consider the forgetful morphism … It forgets the last $m$ markings without stabilizing the curve. Pullback inj…

Height-multiple nonexistence conjecture. There is no vector bundle $\mathscr W$ on $\overline{\mathscr M}_{1,1}$ for which there exists an integer $n$ such that

Let $K$ be a finite extension of $\mathbb Q_p$, let $\mathcal O_K$ be its ring of integers, and let $\mathpzc X$ be a smooth formally proper stack over $\mathcal O_K$. Let…

Let $K$ be a finite extension of $\mathbb Q_p$, let $\mathcal O_K$ be its ring of integers, and let $\mathpzc X$ be a smooth Hodge-proper stack over $\mathcal O_K$. Let…

Minimal skeleton conjecture. The homeomorphism type of the dual complex of $(\mathcal{X},\sum_{1\le i\le s}\mathcal{D}_{i})$ does not depend on the choice of such compactification.

Criterion for quasi-affineness. Under these hypotheses, $\mathcal{X}$ is a quasi-affine scheme.