The non-vanishing conjecture for log canonical pairs

Equivalent formulations 2

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. Non-vanishing conjecture for log canonical pairs

   Let $(X,\Delta)$ be a projective log canonical pair. Non-vanishing conjecture. If $K_X+\Delta$ is pseudo-effective, then there exists an effective $\mathbb{R}$-divisor $D$ such that

   $$D\sim_{\mathbb{R}}K_X+\Delta.$$

   The conjecture asks for an effective representative of a pseudo-effective log canonical divisor. It is a basic existence statement in the minimal model program and is used in the paper as a step toward abundance and good minimal models.

   source: Osamu Fujino and Yoshinori Gongyo, “On log canonical rings”, arXiv:1302.5194 (2013).

2. Non-vanishing conjecture for log canonical pairs

   Let $(X/Z,B)$ be an lc pair. The relative real linear system $|(K_X+B)/Z|_{\mathbb{R}}$ is nonempty whenever $K_X+B$ is pseudo-effective over $Z$.

   Non-vanishing conjecture. If $K_X+B$ is pseudo-effective over $Z$, then

   $$|(K_X+B)/Z|_{\mathbb{R}}\not=\emptyset.$$

   This conjecture concerns the existence of effective representatives of pseudo-effective log canonical divisors in the relative setting. The supplied text uses it as an assumption in a main theorem, but gives no evidence of a resolution, so its status is left open.

   source: Guodu Chen, Jingjun Han and Jihao Liu, “Uniform rational polytopes for Iitaka dimensions”, arXiv:2208.04663 (2022).