53 problems

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

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

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.

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 $X$ be a four-dimensional projective variety with only terminal singularities. Recall that a $\mathbb{Q}$-Cartier divisor is quasi-numerically positive if it is nef and has pos…

SYZ Conjecture. The line bundle $L$ is semiample.

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…

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

Abundance conjecture. Then $K_X+\Delta$ is semiample.

Let $X$ be a smooth projective variety. Denote its Kodaira dimension by $\kappa(X)$ and its numerical dimension by $\nu(X)$. Non-vanishing conjecture. If … then … This conjecture a…

Let $X$ be a smooth projective variety. Its Kodaira dimension is denoted by $\kappa(X)$ and its numerical dimension by $\nu(X)$. Abundance conjecture. … The conjecture predicts equ…

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…

Abundance conjecture. The divisor $K_X$ is semi-ample.

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

Abundance conjecture. The canonical divisor $K_{X'}$ is semi-ample; equivalently, there exists an $m\in\mathbb{N}$ such that $\mathcal{O}_{X'}(mK_{X'})$ is generated by its global…

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

Abundance conjecture. For every $X$,

Let $(X,B)$ be a klt pair with $K_X+B$ pseudoeffective, and let $L$ be a nef divisor on $X$ such that $K_X+B+L$ is also nef. Generalized abundance conjecture. The divisor $K_X+B+L$…

Let $(X,B)$ be a klt pair such that $K_X+B$ is pseudoeffective, and let $L$ be a nef divisor on $X$ with $K_X+B+L$ nef and abundant. Non-vanishing conjecture. The divisor $K_X+B+L$…

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…