294 problems

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…

Abundance. The divisor $K_X+B$ is semiample in the following senses:

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

Bounded-volume boundedness conjecture. If

Good Minimal Model Conjecture. The pair $(X/Z,B)$ has a good minimal model $(Y/Z,B_Y)$ over $Z$, meaning that $(Y,B_Y)$ is a minimal model of $(X,B)$ over $Z$ and $K_Y+B_Y$ is semi…

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

Let $X$ be a smooth projective variety. Minimal model conjecture. Either $X$ is uniruled, or there is a birational modification $X\dashrightarrow X'$ such that $X'$ has only termin…

Work over an algebraically closed field $k$ of characteristic zero, and let varieties over $k$ mean separated integral schemes of finite type over $k$. Let $\varphi:X\to X$ be an e…

Extension conjecture for dlt pairs. The restriction map

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

Let $G$ be an algebraic group, let $X$ be a smooth projective variety with a $G$-action, and let $\pi:X\to Y$ be a $G$-contraction. Let $\mathrm{Stab}(X_G)$ denote the space of sta…

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

Minimal-model conjecture. If $Y$ has at most terminal singularities and $K_Y$ is a nef $\mathbb{Q}$-Cartier divisor, then $Y$ is smooth and the rational map $f$ is a morphism. More…

Minimal model and abundance conjectures. Either $X$ is uniruled, or $X$ has a minimal model $X'$; moreover, some multiple $mK_{X'}$ is spanned, so $X'$ is a good minimal model.

Abundance Conjecture. For every such $X$,

Minimal-model conjecture. Every compact Kähler manifold of non-negative Kodaira dimension should have a minimal model.

Huybrechts' projectivity and bigness conjecture.

p-Quantum Minimal Model Conjecture. Any two smooth minimal models have p-isomorphic quantum cohomology.

Quantum Minimal Model Conjecture. Any two smooth minimal models in any dimension have isomorphic quantum cohomology.

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…