67 problems

Let $X$ be a smooth projective variety over $\bC$, let $A=\operatorname{Spec}R$ where $R$ is a finitely generated $\bZ$-algebra containing $\bZ$, and let $X_A\to A$ be a spreading…

Integral-Tango characterization. If there is a counterexample to the Kodaira vanishing theorem on $Y$, then there exists a fibration

Ruled-surface characterization. There exists a counterexample to the Kawamata–Viehweg vanishing theorem on some geometrically ruled surface over $C$ if and only if $C$ is a Tango c…

Tango-curve characterization. If there is a counterexample to the Kawamata–Viehweg vanishing theorem on $X$, then there exists a dominant rational map

Let $A$ be the coefficient ring in the reduction-modulo-$p$ setup, let $\frak{a}\subset A[x_1,\ldots,x_n]$ be an ideal, let $c$ be its log canonical threshold, and let $c_p$ denote…

Let $(R,\mathfrak{m})$ be a KLT singularity essentially of finite type over $\mathbb{C}$. For every sufficiently large prime $p$, let $\mathscr{s}_p(R)$ denote the $F$-signature of…

Let $S$ be a scheme smooth and dominant over $\operatorname{Spec} \mathbb{Z}$, and let $Y_S\to S$ be a smooth proper morphism whose fibers are Calabi–Yau varieties. Weak ordinarity…

Let $\Pi$ be the discretization map from the paper, and let $f_0,\ldots,f_{25}$ be the intrinsic invariants used to construct a five-component raw rank function…

Let $g$ and $h$ be the base polynomials in the monic Weierstrass family of hypersurfaces … and let the clean lex tuple be … A bounded-delay descent condition means that improvement…

Semistability–cohomological semistability conjecture. $E$ is semistable if and only if $E$ is cohomologically semistable.

Let $K$ be an absolutely finitely generated field, meaning that it is finitely generated over its prime field. Let … be the localization of the category of finite type $K$-schemes…

Let $X$ be a fake projective plane over an algebraically closed field of characteristic $p>0$, meaning a smooth projective surface with the same Betti numbers as the projective pla…

Let $X$ be a normal complex quasi-projective variety, and let $\mathcal{F}\subset T_X$ be a coherent subsheaf closed under the Lie bracket. The distribution $\mathcal{F}$ is algebr…

Achinger–Witaszek–Zdanowicz conjecture. There exists a finite étale Galois cover $f:Y\to X$ such that the Albanese morphism

Achinger–Zdanowicz conjecture. There is a flat deformation of $X$ over the Witt vectors $W(k)$, not only over $W_2(k)$.

Let $M_{Kah,reg}$ be the specialized quantum $D$-module, let $x\in H^2_{T\times\mathbb{G}_m}(X)$, and let $s$ be the Artin--Schreier map defining $p$-curvature. Quantum-side p-curv…

Let $M_{eq,reg}$ be the $D$-module of twisted traces, let $\mathcal{A}_0^2$ be the degree-$2$ part of the relevant quantization algebra, and let $s$ be the Artin--Schreier map defi…

Let $\lambda$ be a parameter, written in characteristic $p$ as a lifting $(\lambda_0,\lambda_1)$, and let $M_p(\lambda_0,\lambda_1)$ denote the matrix associated with the periodici…

Canonical bundle formula conjecture. There is

Let $X$ be a smooth projective scheme defined over a field of characteristic zero. After spreading out $X$, for a prime $p$, write $X_{p=0}$ for its reduction modulo $p$, and let…

Let $k$ be a finitely generated field of characteristic $p>0$ with absolute Galois group $G_k$, let $m$ be a non-negative integer, and let $U_i$ be hyperbolic curves over $k$ for…

Structural formula conjecture. For $d\geq e>0$,

END-tile parametrization conjecture. The END tiles are precisely

Duality and stabilization conjecture. (a) If a tile is valid for $n$, then appending a zero to its vector produces a valid tile for $n+1$. (b) If $c_{\mathrm{in}}\in\mathbb{Z}$, th…

Tile-structure conjecture. For fixed $n$ and $p$, there is a finite list of valid tiles such that $\mathrm{Prim}(d)$ is the sum