26 problems

Let $(X,\Delta)$ be a projective log canonical pair of dimension $n$. Log canonical model conjecture. If $K_X+\Delta$ is big, then $(X,\Delta)$ has a log canonical model. The sourc…

Let $(X,\Delta)$ be a projective kawamata log terminal pair. Abundance conjecture. If $K_X+\Delta$ is nef, then it is semiample. The source describes this as one of the main outsta…

Let $(X,\Delta)$ be a projective kawamata log terminal pair. If $K_X+\Delta$ is pseudo-effective and, for some ample divisor $H$, … is not a bounded function of $m$, then…

Let $(X,\Delta)$ be a projective kawamata log terminal pair. Nonvanishing conjecture. If $K_X+\Delta$ is pseudo-effective, then $\kappa(X,K_X+\Delta)\geq 0$. This is identified as…

Log crepant derived equivalence conjecture. Then there exists an equivalence of triangulated categories

Let $(X,B)$ and $(Y,C)$ be birationally equivalent pairs of varieties with effective $\mathbb{Q}$-divisors satisfying the smooth-local-covering condition: each pair admits a quasi-…

Let $(X/Z,B)$ be a log pair and let $\pi(X/Z)$ be its local or global algebraic fundamental group. A trivial complement is a complement with unchanged boundary, $B^+=B$; such compl…

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 $(X/Z,B)$ 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…

Decomposable Iitaka fibration conjecture. There exist a positive integer $m$, a finite set $\Gamma_0\subset(0,1]$, and a DCC set $\Gamma'\subset[0,1]$, depending only on $d$ and…

Non-vanishing conjecture. The relative real linear system is nonempty:

Decomposable Iitaka fibration conjecture. There exist a positive integer $m$, a finite set $\Gamma_0\subset(0,1]$, and a DCC set $\Gamma'\subset[0,1]$, depending only on $d$ and…

Effective log Iitaka fibration conjecture. There exists a positive integer $m$ depending only on $d$ and $\tau$ such that, whenever $(X,B)$ is an lc pair of dimension $d$ with $B$…

sRC quotient structure conjecture. Such an orbifold morphism with all the stated properties exists.

Let $(X,\Delta;L)$ be a polarized log pair, and let K-stability and uniform K-stability be the corresponding notions defined using test configurations. Folklore conjecture. Uniform…

Uniform boundedness conjecture. There is a positive integer $N=N(n,I)$ depending only on $n$ and $I$ such that the linear system

Let $(X,\Delta)$ be a klt pair. Suppose that $X$ is rationally connected, that $\Delta$ is a nef $\mathbb{Q}$-divisor whose support is a prime divisor, that $\kappa(X,\Delta)=0$, a…

Let $n\in\mathbb N$ be positive and let $I\subset [0,1]\cap\mathbb Q$ be a DCC set, meaning that $I$ satisfies the descending chain condition. Let $(X,B)$ be an $n$-dimensional com…

Let $(X,\Delta)$ be a projective divisorial log terminal pair, with $\Delta$ a $\mathbb{Q}$-divisor, and write $\lfloor\Delta\rfloor=S$. Suppose that $K_X+\Delta$ is nef and … wher…

Let $(X/Z,B)$ be a log canonical (lc) pair such that $K_X+B$ is a $\mathbb Q$-divisor, and let $f\colon X\to Z$ be the given morphism. Define the log canonical algebra by … Finite…

Let $(X/Z,B)$ be a log canonical (lc) pair, and suppose that $K_X+B$ is pseudo-effective over $Z$. A divisor $M$ is effective, written $M\ge 0$, when all its coefficients are nonne…

Let $(Y/Z,B_Y)$ be a log canonical (lc) pair, and suppose that $K_Y+B_Y$ is nef over $Z$. A divisor is semi-ample over $Z$ if it defines a contraction over $Z$ after a suitable pos…

Let $(X/Z,B)$ be a log canonical (lc) pair, meaning that $(X/Z,B)$ is a pair with log canonical singularities over $Z$. A log minimal model is a birational model on which $K_Y+B_Y$…

Global ACC conjecture. There is a finite subset $I_0\subset I$ such that every coefficient $\delta_i$ belongs to $I_0$.

Non-vanishing conjecture. If $K_X+\Delta$ is pseudo-effective, then