33 problems

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…

Let $f\colon X\to Z$ be the fibre space and let $D$, $\Theta$, $D_{\operatorname{div}}$, and $D_{\operatorname{mod}}$ be as in the paper's adjunction construction. Assume that $D$…

Let $(X,B)$ be a klt pair. Write $u(KX+B)$ for the numerical dimension and $\kappa{\iota}(KX+B)$ for the invariant Iitaka dimension of $KX+B$. Abundance conjecture. One has … This i…

A primitive symplectic variety is a normal compact Kähler variety with symplectic singularities whose smooth locus carries a symplectic form and whose reflexive powers of the canon…

Generalized abundance for symplectic varieties. Then $L$ is semiample.

Abundance conjecture. The manifold $X$ admits a good minimal model $X'$ such that

Let $X$ be a smooth projective variety and let $E$ be a simple normal crossing divisor on $X$. Let $\kappa_\sigma$ denote Nakayama's numerical dimension. Generalized abundance conj…

Generalised abundance conjecture. If $K_X+B+M$ is nef, then $K_X+B+M$ is num-semiample.

Generalized abundance conjecture. For every $X$ and every integer $p>0$,

Work over $\mathbb{C}$. Let $(X,\Delta)$ be a projective klt log pair of dimension $n\geq 2$. Let $\kappa(X,\Delta)$ denote the Kodaira dimension of $K_X+\Delta$, and let…

Work over $\mathbb{C}$. Let $(X,\Delta)$ be a projective klt log pair of dimension $n\geq 2$, and let $\kappa(X,\Delta)$ denote the Kodaira dimension of $K_X+\Delta$. A divisor is…

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

Let $X$ be a compact complex manifold in class $\mathcal C$, meaning that $X$ is bimeromorphic to a compact Kähler manifold. Let $K_X$ denote its canonical bundle. A line bundle is…

Let $X$ be a compact hyperkähler manifold, and let $L$ be a nef line bundle. Let $q$ denote the Beauville–Bogomolov–Fujiki form; $L$ is parabolic when $q(L,L)=0$. SYZ conjecture. A…

Let $X$ be a normal projective klt variety with $H^1(X,\mathcal O_X)=0$ and $K_X\sim_{\mathbb Q}0$, and let $L$ be a nef divisor. Singular abundance conjecture. Then $L$ is semiamp…

Let $X$ be a projective manifold with $H^1(X,\mathcal O_X)=0$ and $K_X\sim 0$, and let $L$ be a nef divisor. Abundance conjecture for varieties with trivial canonical class. Then…

Non-vanishing conjecture for smooth varieties. The canonical divisor $K_{X}$ has an effective $\mathbb{Q}$-linearly equivalent representative.

Abundance conjecture. $\mathcal L$ is semiample, i.e. there exists a positive integer $m$ such that $\mathcal L^{\otimes m}$ is generated by global sections.

Generalized abundance conjecture.

Let $X$ be a smooth variety with nef canonical divisor. Write $\kappa(X)$ for the Kodaira dimension of $X$ and $\nu(X)$ for the numerical dimension of its canonical divisor. Abunda…

Generalized abundance conjecture.

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…

Let $(X,\Delta)$ be a $\mathbb Q$-factorial projective dlt pair. The numerical Kodaira dimension $\kappa_\sigma(X,K_X+\Delta)$ and the invariant Kodaira dimension…