105 problems

Let $X_{0}$ be a Calabi–Yau threefold over an algebraically closed field $k$ of characteristic $p$. Write $h^{1,0}$ for its Hodge number and $h_{W}^{1,2}$ for the corresponding cry…

Blickle–Mustaţă–Smith's equality conjecture.

Artin's conjecture. Every Artin-supersingular K3 surface $X$ is supersingular.

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…

Kodaira-type vanishing conjecture. Kodaira-type vanishing theorems should hold for $X$ when the characteristic is sufficiently large, namely when $p\geq p(n)$.

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 $(E,\theta)$ be a nilpotent semistable Higgs bundle of exponent at most $p$. Strong Higgs semistability conjecture. The Higgs bundle $(E,\theta)$ is strongly Higgs semistable.…

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…

Large-characteristic quasi-$F^{\infty}$-splitting conjecture. There exists an integer $p_d>0$ such that, if $p>p_d$, every smooth proper quasi-$F$-split $K$-trivial variety of dime…

Let $(X,\Delta)$ be a KLT log Fano pair over $\mathbb{C}$, and suppose $(X_A,\Delta_A)$ is a family of characteristic-$p$ models of $(X,\Delta)$ for sufficiently large $p$. Uniform…

Let $F$ be a local field, and let $\phi:\bfM\to\bfN$ be a flat morphism of smooth algebraic varieties defined over $F$, whose fibers are reduced and normal. Say that $\phi$ is geo-…

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 $K$ be a field finitely generated over $\mathbb{F}_p(t)$. A perfection of a finite type $K$-scheme is obtained by inverting the universal homeomorphisms arising from Frobenius.…

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 $W_2(k)$.

Canonical bundle formula conjecture. The stated discriminant and moduli decomposition, together with the b-nefness assertion for the moduli b-divisor in the lc case, holds.