17 problems

Borisov–Alexeev boundedness conjecture. The collection of $b5$-klt log Fano varieties is bounded.

Birkar–Shokurov conjecture. There is $n\in\mathbb N$ such that, whenever $X$ is $\epsilon$-lc of dimension $d$, $f\colon X\to Z$ is a contraction, and $-K_X$ is ample over $Z$, the…

Boundedness conjecture for log Fano varieties. Then $(X,\operatorname{Supp}(D))$ belongs to a finite number of algebraic families. This conjecture is known to be true when…

AlexeevbBorisov boundedness conjecture. For every $b4>0$ and $91bb[0,1]$, the varieties $X$ for which such a pair $(X,B)$ exists are elements of an algebraic family.

The examples above associate to a two-dimensional klt or strictly log canonical singularity a log Fano variety $(E,\operatorname{Diff}_E(0))$, where $E$ is the exceptional divisor…

Let $(\mathfrak{X},\mathfrak{D})$ be a weakly log Fano scheme over the ring of integers $\mathfrak{o}$, with generic fiber $(X,D)$, and write…

Let $\mathcal{X}=(\mathfrak{X},D,L)$ be a polarized log Fano variety, with integral-point scheme $\mathfrak{U}=\mathfrak{X}\setminus\overline{D}$, height function $H_L$, and counti…

Let $M$ be a smooth complex projective manifold and let $D=D_1+\ldots+D_r$ be a simple normal crossing divisor. Assume that $(M,D)$ is asymptotically log Fano and admits Kähler–Ein…

Finite-generation conjecture. The associated graded ring $\operatorname{gr}_v R$ is finitely generated.

Let $(X,\Delta)$ be a log Fano pair. K-polystability conjecture. If $(X,\Delta)$ is K-polystable, then it is reduced uniformly K-stable. This is the K-polystable analogue of the co…

Generalized Ambro conjecture. Fix $0<\epsilon<1$ and an integer $n>0$. Then there exists a number $\mu(n,\epsilon)>0$ depending only on $n$ and $\epsilon$ such that

Ambro's conjecture. Fix $0<\epsilon<1$ and an integer $n>0$. Then there exists a number $\mu(n,\epsilon)>0$ depending only on $n$ and $\epsilon$ such that, for every $\epsilon$-klt…

Higher-dimensional uniformization conjecture. (i) There exist no Kähler–Einstein edge metrics with small $\beta$ if $\kappa=\infty$. (ii) If $(K_X+D)^{\dim X}=0$, then there exist…

Positivity classification conjecture. The positivity classification completely determines the existence problem for Kähler–Einstein edge metrics on $(S,C)$.

Karzhemanov's conjecture. Suppose that $X$ is $\mathbb{Q}$-factorial and

Reid's base point free conjecture. The linear system

A log-Fano variety is a projective variety with mild singularities for which there exists an effective boundary divisor making the pair log Fano. Hu–Keel's conjecture. Every log-Fa…