142 problems

ACC conjecture for log canonical thresholds. The set

Termination of flips. Any sequence of $K_X+B+\boldsymbol{\beta}_X$ flips is finite.

Minimal model conjecture. Then $(X/Z,B,\mathbf{M})$ has a minimal model.

Let $V$ be a normal affine variety of complex dimension $n$ with an isolated singularity at $0$ that is numerically $\mathbb{Q}$-Gorenstein. Write $\operatorname{min.disc}(V,0)$ fo…

Bounded-volume boundedness conjecture. If

Extension conjecture for dlt pairs. The restriction map

Let $(X,\Delta)$ be a $\mathbb Q$-factorial projective dlt pair, and assume that $K_X+\Delta$ is pseudo-effective. Good minimal model conjecture. The pair $(X,\Delta)$ has a good m…

Finiteness of minimal models. The number of (log) isomorphism classes of projective wlc klt models in a fixed $0$-class is finite.

Let $(X,\Delta)$ be a klt pair on a normal projective variety $X$ such that $K_X+\Delta$ is pseudo-effective. Let $L$ be a nef $\mathbb{Q}$-divisor on $X$. Generalized Nonvanishing…

Generalised Nonvanishing Conjecture. The numerical class of $K_X+\Delta+tL$ belongs to the effective cone for every $t\geq0$.

Generalised Abundance Conjecture. There exists a semiample $\mathbb{Q}$-divisor $M$ such that

Abundance Conjecture. For every such $X$,

Log abundance conjecture. If $K_X+B$ is nef, then it is semi-ample.

Non-vanishing conjecture. There exists an effective $\mathbb{R}$-divisor $E\geq 0$ such that

Huybrechts' projectivity and bigness conjecture.

Flip conjecture. There exists a unique birational map $\phi:X\dashrightarrow X^+$ to a $\mathbb{Q}$-factorial projective variety $X^+$ with only terminal singularities, together wi…

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…

Existence of flips. The flip of $f$ exists.

Kawamata's fully faithful embedding conjecture. Under these assumptions, there exists a fully faithful exact functor

Let $X$ be a minimal model, meaning a complex projective variety with at most terminal singularities and nef canonical divisor $K$. For a nef divisor $K$, its nef dimension…

Let $X$ be a projective variety with at most terminal singularities. Minimal model conjecture. There is a minimal model $X'$ birational to $X$ if and only if … The source presents…

Let $X$ be a projective variety with at most terminal $\mathbb Q$-factorial singularities. Minimal model conjecture. There exists a minimal model $X'$ birational to $X$ if and only…