25 problems

Let $X$ be an $n$-dimensional compact complex manifold, and let $\alpha$ be a real $d$-closed $C^\infty$ $(1,1)$-form on $X$. Define … The BottChern cohomology class…

Let $\operatorname{SL}_n\mathbb{Z}$ be the special linear group over the integers, and let $H^*(-;\mathbb{Q})$ denote rational group cohomology. Vanishing conjecture.…

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

The averaging-functor vanishing conjecture. Assume that $E$ is irreducible and has rank greater than $n$. If

Let $X$ be a klt projective variety over $\C$. Suppose that … for $i>0$ and every semi-ample line bundle $L$ on $X$. The vanishing criterion conjecture asserts that $K_X$ is not ps…

Suppose $X$ is embedded in a smooth variety $Y$ of dimension $d$. Choose a log resolution … of the pair $(Y,X)$ that is an isomorphism away from $X$, and let … Thus $E$ is a simple…

Let $X$ be a weakly pseudoconvex manifold, let $L\longrightarrow X$ be a holomorphic line bundle, and let $\mathscr{J}$ be an ideal sheaf on $X$. Assume that $L$ is exhaustion sing…

Let $\mathbb F_{e,r}$ be the blow-up of the Hirzebruch surface $\mathbb F_e$ at $r$ very general points, and let $\widetilde{C}_e$ be the strict transform of the negative section…

Intersection-complex injectivity conjecture. The natural morphism

Let $C$ be an algebraically closed non-archimedean field, let $A$ be a $C$-affinoid algebra, and let $n$ be an integer invertible in $\mathcal{O}_C$. Let…

Let $S_K$ be the special fiber of a Hodge-type Shimura variety at a prime $p$ of good reduction which splits in the reflex field $\mathbf E$. Let $G$ be the attached reductive grou…

Assumption-free discrepancy and vanishing conjecture. The assumptions that $\mathbb{B}l_f$ is smooth and that $\pi_0(Y)-\pi_0(X)$ is nonempty can be removed from the discrepancy fo…

Let $f:Y\to X$ be a proper morphism between smooth projective algebraic varieties. Assume that the degenerate locus of $f$ is contained in a normal crossing divisor $D\subset X$. L…

Grauert–Riemenschneider vanishing conjecture. For every $i>0$,

Fujita-transform conjecture. The derived pushforward

Let $S$ be a fake projective plane, let $G$ be an ample generator of $\operatorname{Pic}(S)$, and let $\sigma$ be a torsion line bundle on $S$. Vanishing conjecture. One has … for…

Vanishing conjecture. The higher cohomology of the $m$-th reflexive log-pluricanonical sheaf vanishes:

Let $X$ be an affinoid rigid space over a complete algebraically closed nonarchimedean field $C$. Let $n$ be prime to the residue characteristic of $C$, and let $\mathscr{G}$ be a…

Broad vanishing conjecture. For every $i\in\{1,\ldots,n\}$,

Kähler vanishing conjecture. If $q>n- u(L)$, then

Fujino's logarithmic injectivity conjecture. The multiplication homomorphism

Let $\operatorname{Mod}_g$ be the mapping class group of a closed, oriented surface of genus $g\geq 2$, and let $H^*(-;\mathbb{Q})$ denote rational group cohomology. Vanishing conj…

Let $X$ and $Y$ be proper and smooth $S$-schemes, let $f:X\rightarrow Y$ be an $E$-semistable $S$-morphism, and let $D$ be a simple normal crossing divisor on $X$ containing…

Let $(Z,\Delta)$ be a logarithmic pair, with $Z$ a variety and $\Delta$ a boundary divisor, and assume that $(Z,\Delta)$ is log canonical. Let $\Omega_Z^{[p]}(\log\Delta)$ denote t…