12 problems

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.

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.

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…

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…

Let $X$ be a smooth projective variety and let $D$ be a smooth divisor such that $(X,D)$ is asymptotically log Fano. For $\beta\in(0,1]$, write $(X,(1-\beta)D)$ for the correspondi…

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

Weak BAB conjecture. 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 $X$ is an $n$-dimensiona…

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

Reid's base point free conjecture. The linear system