59 problems

Let $X$ be a nonsingular projective variety with $\kappa(X)=0$. Ueno's Conjecture K asserts that the Albanese map is birational to an étale fiber bundle over…

Let $X$ be a smooth projective variety. A variety is uniruled if there are a variety $B$ of dimension $\dim X-1$ and a dominant rational map…

Let $\overline{\mathcal{M}}_g$ be the moduli space of stable curves of genus $g$, and let $\kappa(\overline{\mathcal{M}}_g)$ denote its Kodaira dimension. Low-genus Kodaira-dimensi…

Let $F$ be a projective manifold of dimension $d$ with Kodaira dimension $\kappa(F)=0$, and let $L$ be an almost strictly nef line bundle, meaning that $L$ is the pullback of a str…

Let $X$ be a projective manifold. Suppose that, for some positive integer $N$, … where $A$ is an effective divisor and $B$ is a pseudo-effective line bundle. Campana's abundance-ty…

Self-$K$-isocorrespondence conjecture. If there exists a self-$K$-isocorrespondence $\Sigma\subset X\times X$ passing through an arbitrary pair $(x,y)$, then

Let $Y^\partial$ be a smooth or e-admissible effective orbifold, let $g^\partial:Y^\partial\rightarrow Z$ be a fibration, and let $Y_z^\partial$ be its orbifold general fiber. Orbi…

Let $X$ be a projective variety with at most terminal singularities. Minimal model conjecture. There is a minimal model $X'$ birational to $X$ if and only if … The source presents…

Let $g:(Y/H)\to Z$ be a prepared and high holomorphic fibration between manifolds, with $Y\in\mathcal C$. Let $\Delta(g,H)$ be its base orbifold, let $z\in Z$ be general, and write…

Let $(M,J)$ be a compact complex surface, where $M$ is its underlying 4-manifold and $Y(M)$ is its Yamabe invariant. Negative Yamabe invariant conjecture. If … then $M$ is of gener…

Let $M$ be the underlying 4-manifold of a compact complex manifold of Kodaira dimension $0$ or $1$, and let $b_1(M)$ denote its first Betti number. Yamabe invariant conjecture for…

Let $M$ be the underlying 4-manifold of a complex algebraic surface of Kodaira dimension $\geq 0$. A sequence of unit-volume constant-scalar-curvature metrics on $M$ has uniformly…

Let $(X,\Delta)$ be a Klt pair, and let $f:X\to Y$ be an algebraic fiber space, where $X$ and $Y$ are smooth projective varieties of dimensions $n$ and $m$, respectively. Let $F$ b…

Kollár's conjecture. If

Let $X$ be a smooth projective variety with big fundamental group and Kodaira dimension satisfying … A fundamental group is big in the sense defined in the source. Kollár's structu…

Let $X$ be a smooth projective variety over $\mathbb{C}_p$, and let $X^{\mathrm{an}}$ denote its analytification. Kodaira-dimension conjecture. If there is a $p$-adic analytic map…

Let $f\colon X\to Y$ be an algebraic fiber space between smooth projective varieties, and let $F$ be its general fiber with $\kappa(F)=0$. Suppose that there are an ample divisor…

Let $f:X\to Y$ be an algebraic fiber space between smooth projective varieties, and suppose that the general fiber $F$ satisfies $\kappa(F)=0$. Let $H$ be an ample divisor on $Y$.…

Let $f:X\to Y$ be an algebraic fiber space, meaning that $X$ and $Y$ are smooth projective varieties and $f$ is surjective with connected fibers. Let $F$ be a generic fiber satisfy…

Let $X$ be a smooth projective variety and let $E$ be a simple normal crossing divisor on $X$. Let $\kappa_\sigma$ denote Nakayama's numerical dimension. Generalized abundance conj…

Infinite fundamental-group conjecture. If

Let $X$ be a smooth projective variety with $\kappa(X)=0$, and let $\alpha:X\rightarrow A$ be its Albanese map, with general fibre $F$. Ueno Conjecture K. The map $\alpha$ is surje…

Let $f\colon X \to Y$ be a fibration between smooth projective varieties with general fiber $F$ and $\kappa(Y)\ge0$. For a sufficiently divisible positive integer $m$ such that…

Let $f\colon X \to Y$ be a fibration between smooth projective varieties with general fiber $F$ and $\kappa(Y)\ge0$. Define $\operatorname{Var}(f)$ to be the variation of the fibra…

Let $f\colon X\to A$ be a fibration from a smooth projective variety $X$ to an abelian variety $A$. For a coherent sheaf $\mathcal{F}$ on $A$, write … Let $\operatorname{Var}(f)$ d…