51 problems

Adjunction and effective adjunction conjectures. One can choose an effective representative $M_Z\geq 0$ in its $\mathbb{R}$-linear equivalence class such that $(Z,D_Z+M_Z)$ is log…

Fix $n\in\mathbb{N}$, $\varepsilon>0$, and let $I\subseteq [0,1]$ be a finite set. For a klt singularity $x\in (X,\Delta)$, let $\operatorname{mld}^{\mathrm{K}}(x,X,\Delta)$ denote…

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…

Borisov–Alexeev boundedness conjecture. Log varieties $X$ with $-K_X$ ample, $a(X)\geq \epsilon$, and $\dim(X)=d$ form a bounded family.

Let $\delta>0$ be a real number. A delta-log-canonical weak log Fano pair is a pair $(X/\operatorname{pt.},B)$ of dimension $d$ that is $\delta$-log canonical and weak log Fano, wh…

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.

Strong $b5$-lc complement conjecture. For every $0<b5$ and $d$, there exists a finite set $a9_{b5,d}$ of positive integers such that every such pair has an $(b5,n)$-complement over…

Weak $b5$-lc complement conjecture. For every $0<b4$ and $d$, there exist a finite set $a9_{b4,d,91b}$ of positive integers and $0<b5$ such that every such pair is $(b5,n)$-complem…

Index boundedness conjecture. The family of $n$-dimensional log terminal Fano varieties of index $I$ is bounded.

Boundedness-of-index conjecture. The set of possible values of $r(W)$ for canonical ${\bf Q}$-Fano varieties $W$ of dimension $n$ is finite.

Jiao's conjecture. There exists a positive integer $m$ with the following property: there is an effective $\mathbb{Q}$-Weil divisor $\Delta$ on $X$ such that $(X,\Delta)$ is klt an…

Let $T$ be a torus and let $c>0$ be a real number. Consider isomorphism classes of $T$-equivariant $(\mathbb{Q})$-Fano fibrations $T\curvearrowright(X\xrightarrow{\pi}Y)$, with wei…

BAB conjecture. The set of $\epsilon$-lc Fano varieties of dimension $d$ forms a bounded family.

For a positive integer $n$, let $\mathcal{F}_n$ be the family of $n$-dimensional projective varieties $X$ such that there exists a log Calabi--Yau pair $(X,B)$ with … and…

Let $\mathcal{T}(X,B)$ denote the torus exceptional degree of a log Calabi--Yau pair $(X,B)$. Torus exceptional degree boundedness conjecture. For every positive integer $n$, there…

Fix an algebraically closed field $k$, an integer $d>0$, and a rational number $\varepsilon>0$. Let $X$ be a $d$-dimensional normal projective variety over $k$, and let $B$ be a…

Bounded-volume boundedness conjecture. If

Boundedness conjecture. The family $\mathcal{S}_{klt}(d,\Phi,\leq u,v)$ is bounded.

Let $n$ be a positive integer, let $\varepsilon>0$, and let $\operatorname{mult}_xX$ denote the multiplicity and the embedded dimension of $x\in X$ its embedding dimension. Multipl…

Let $n$ be a positive integer. A set of klt singularities is bounded if all its members occur as fibers of one $\mathbb{Q}$-Gorenstein family; for Fano cone singularities, the fami…

Let $d$ be a positive integer and let $\boldsymbol{\mathcal{X}_{d,\varepsilon}}$ denote the class of $\varepsilon$-klt Fano varieties of dimension $d$, where $\varepsilon>0$ is rat…

Multisection-index conjecture. One has

Let $f:Y\to X$ be a finite morphism between smooth projective varieties of dimension $n\ge 2$, each having second Betti number $1$. Suppose that $X\not\cong\mathbb{P}^n$. Amerik's…

Let $d$ be a positive integer and let $\Lambda$ be a set of rational numbers satisfying the descending chain condition. A projective $d$-dimensional klt log Calabi–Yau pair is a pr…

Let $d$ be a positive integer. Consider all $d$-dimensional klt Calabi–Yau rationally connected projective varieties. Rationally connected Calabi–Yau boundedness conjecture modulo…