142 problems

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…

Let $X$ be a terminal $\mathbb Q$-factorial variety which is not minimal. Termination conjecture. After finitely many flips, there is an extremal ray whose exceptional locus has co…

Canonical ${\bf Q}$-Fano contraction conjecture. There exist a resolution of singularities

Rationality conjecture. If

Existence conjecture for log canonical models. The graded ring of $\cal O_S$-modules

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

Existence of log terminal models. The pair $(X,B+\boldsymbol{\beta})$ has a log terminal model.

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

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 $(X,B+\boldsymbol{\beta})$ be a generalized klt Kähler pair, where $X$ is a normal compact Kähler variety, and let $\alpha\in H_{\mathrm{BC}}^{1,1}(X,\mathbb{R})$ be a nef clas…

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…

Deformation invariance conjecture. For every $t\in\Delta$, $K_{X_t}$ is semiample and $K_{\mathcal{X}}$ is $\pi$-semiample over $\pi^{-1}(U_t)$, where $U_t$ is a Zariski neighborho…

Finiteness conjecture for wlc klt models. The number of projective wlc klt models in a fixed 0-class is finite up to log isomorphism.