62 problems

Bounded-volume boundedness conjecture. If

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…

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

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

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

Let $X$ be a projective rationally connected variety of dimension $d$ with $\epsilon$-lc singularities, where $\epsilon>0$, and suppose that $-K_X$ is nef. McKernan–Prokhorov's bou…

Let $(X,\Delta;x)$ be a klt singularity, and let $\operatorname{\widehat{vol}}(X,Delta;x)$ denote its normalized volume. Say that a collection of singularities is bounded up to spe…

Multisection-index conjecture. One has

Weak BAB conjecture for Mori fiber spaces. Fix $0<\epsilon<1$ and an integer $n>0$. Then there exists a number $M(n,\epsilon)$ depending only on $n$ and $\epsilon$ such that, if…

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…

Let $(X/Z\ni o,D)$ be a log variety admitting at least one $\mathbb{Q}$-complement of $K_X+D$ near the fiber over $o$. It is exceptional over $o$ if every such complement has at mo…

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…

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

Let $(X/Z,B)$ be a log pair with standard boundary coefficients, meaning each coefficient of $B$ is $(m-1)/m$ for a natural number $m$, or is $1$. Assume that $(X/Z,B)$ is weak log…

Boundedness conjecture. For every positive rational number $C$ there exist a positive integer $N>0$ such that, for every stable $n$-dimensional stable pair $(X,B)$ with $(K+B)^n=C$…

Higher-dimensional boundedness conjecture. The conditions for boundedness formulated in the paper should be the most natural ones, and direct analogues of all four main theorems sh…

Geometric Shafarevich conjecture. The set

Birkar's conjecture. The Stein degree

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 $d\in\mathbb{N}$ and let $t\in\mathbb{R}^{>0}$. Let $(X, B)\to Z$ be a log Calabi-Yau fibration of dimension $d = \dim X$, and let $S$ be a horizontal$/Z$ irreducible component…

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…

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…