44 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…

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

Higher direct-image vanishing conjecture. One has

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 $\operatorname{Hilb^n\mathbb{C}^2}$ be the Hilbert scheme of $n$ points in $\mathbb{C}^2$, and let $\mathcal{B}_1$ denote its tautological rank $n$ bundle and $\mathcal{P}_1$ i…

Generic-amplitude conjecture. The vanishing in Corollary should continue to hold when $\alpha({\mathcal E})$ is replaced by $\alpha_{gen}$.

Vanishing conjecture. The same vanishing should hold without the semistability hypothesis on $f$.

Let $X$ be a smooth algebraic manifold of dimension $n$ over an algebraically closed field $k$ of characteristic $0$, equipped with a birational projective map $\pi:X\to Y$ to a no…

A GIT quotient is a quotient constructed from a linearized action of a reductive group on a variety, and a smooth Fano variety is a smooth projective variety with ample anticanonic…

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

Let $X$ be a smooth projective curve over a ground field $k$, let $E$ be an $m$-dimensional local system on $X$, and let $\Bun_n$ be the moduli stack of rank $n$ vector bundles on…

Let $H_n=\operatorname{Hilb}^n(\mathbb{C}^2)$ be the Hilbert scheme of $n$ points in the plane, let $Z_n$ be the zero fiber of its Chow morphism, and let $P$ and $B$ be the bundles…

Let $X$ be a smooth projective variety of dimension $n$, let $E$ be an ample vector bundle of rank $e$ on $X$, and let $L$ satisfy the preceding positivity hypotheses. For an integ…

Vanishing conjecture. The higher cohomology groups vanish:

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…

Buch–Thomsen–Lauritzen–Mehta conjecture. Partial flag varieties that are not products of projective spaces do not satisfy Bott vanishing.

Intersection-complex vanishing conjecture.

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 … and the associated range $[,]$. Small-slope vanishing conjecture. Strongly small slope can be replaced by small slope in the second and third points of the vanishing theorem…