18 problems

Let $d$ be a positive integer, and let $\Gamma\subset [0,1]$ be a DCC set of rational numbers whose closure is contained in $\mathbb Q$. A projective $\mathbb R$-complementary norm…

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…

Let $f\colon X\to Z\ni o$ be a contraction from a threefold with only terminal singularities such that $-K_X$ is $f$-nef and $f$-big. An $n$-complement of $K_X$ is a complement wit…

A log surface is a log pair whose boundary has standard coefficients, and a minimal complement index is the least index of a complement in the relevant complement theory. In the ex…

For each dimension $d$, let $N_d$ denote the bounded set of complement indices in the complements conjecture, and define the exceptional indices by $EN_d=N_2\setminus N_{d-1}$. Let…

Bounded relative-global klt complement conjecture. For every positive integer $d$ and positive real number $\epsilon$, there exists a positive integer $N$, depending only on $d$ an…

Let $f\colon X\to Z$ be a Fano fibration, with $X$ $\epsilon$-lc of dimension $d$, and let $Z$ be a curve. Shokurov's bounded-complements conjecture. The existence of a bounded klt…

Foliated complement conjecture. For every positive real number $\epsilon$ and positive integer $d$, there exists a positive real number $n$ depending only on $\epsilon$, $d$ and…

A log Calabi–Yau pair $(X,B)$ is a projective log canonical pair such that $K_X+B\equiv 0$, and its index is the smallest positive integer $I$ such that $I(K_X+B)\sim 0$. Index bou…

Moraga's complement conjecture. Every Fano type variety of dimension $n$ has an $N$-complement for some

Let $c$ be a nonnegative integer, and let $s(c)$ be the $c$-th Sylvester's number. Let $X$ be a Fano type variety of coregularity at most $c$. Bounded-complement conjecture. If…

Plt-type lc place conjecture. There exists a constant $A=A(n,N,\varepsilon)$ such that there is a Kollár component $E$ over $\eta\in X$ satisfying

Bounded complement approximation conjecture. There is a natural number $N$ depending only on $(X,\Delta)$ and a sequence of prime divisors $\{E_i\}$ over $X$ such that

Let $d$ be a positive integer, let $\epsilon$ be a positive real number, and let $\Gamma\subseteq[0,1]$ be a DCC set. A generalized pair $(X,B+M)$ has data…

Let $P(X/Z\ni z,\Delta;D_1,\ldots,D_s)$ denote the set of vectors $(t_1,\ldots,t_s)\in\mathbb{R}_{\ge0}^s$ for which $(X/Z\ni z,\Delta+t_1D_1+\cdots+t_sD_s)$ is…

Let $n$ and $d$ be positive integers, and let $(X,B)$ be a connected projective log canonical pair of dimension $d$ such that … For a positive integer $r$, let … be the natural rep…

Let $d$ be a natural number and let $\mathfrak{R}\subset [0,1]$ be a finite set of rational numbers. For a finite set $\mathfrak{R}$, write … An $n$ complement of $K_X+B$ is a divi…

Strong $(\delta,n)$-complement conjecture. There exist a natural number $n$ and a non-negative real number $\delta$, depending only on $d,p,\epsilon$ and $\Lambda$, such that for e…