The canonical bundle formula conjecture

Equivalent formulations 1

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

1. The canonical bundle formula conjecture

   Suppose $f\colon (X,B)\to Z$ is a contraction, where $(X,B)$ is a generically lc projective pair and $K_X+B\sim_{\mathbb Q}0/Z$. For the generic point $\eta=\operatorname{Spec}(k(D))$ of a prime divisor $D$ on $Z$, define the discriminant divisor

   $$B_Z:=\sum_{D\text{ prime divisor on }Z}(1-\operatorname{lct}_\eta(X,B,f^*D))D.$$

   There is also a pseudo-effective $\mathbb Q$-b-divisor $M_Z$ such that

   $$K_X+B_X\sim_{\mathbb Q}f^*(K_Z+B_Z+M_Z).$$

   Moreover, if $(X,B)$ is lc, then $M_X$ is a b-nef $\mathbb Q$-b-divisor, where $M_X=K_X+B_X-f^*(K_Z+B_Z)\sim_{\mathbb Q}f^*M_Z$.

   Canonical bundle formula conjecture. The stated discriminant and moduli decomposition, together with the b-nefness assertion for the moduli b-divisor in the lc case, holds.

   The canonical bundle formula is a central form of fiber-space adjunction. It was completely resolved in characteristic zero, whereas the source states that it fails in general in positive characteristic.

   source: Xintong Jiang, “Boundedness of complements for fibered Fano threefolds in positive characteristic”, arXiv:2506.00553 (2025).