12 problems

Let ${\EuScript{N}}_d({\Phi})$ denote the set of positive integers occurring as complement indices in the setting of the assumptions referred to in the source. Boundedness conjectu…

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…

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…

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…

Shokurov's boundedness of complements conjecture. There exists a constant $n$ depending only on $d$, $p$, and $R$ such that $K_{X'}+B'+M'$ has an $n$-complement…

Existence of Complements Conjecture. There exists a positive integer $n$, depending only on $d$ and $\Gamma$, such that every such pair $(X/Z\ni z,B)$ has an $n$-complement…

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…

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…

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…

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

Yoshihara's conjecture. Then $C$ and $D$ are projectively equivalent.

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…