Suppose (X,B)/Z is a contraction with relative dimension >0, where (X,B) is a generically lc projective pair and K_X+B sim_ Q 0/Z. Let eta=operatornameSpec(k(Z)) be the generic point of Z. For each prime divisor D on Z, define the discriminant coefficient using the log canonical threshold operatornamelct_eta(X,B,f^D). Canonical bundle formula conjecture. There is B_Z:=∑_D prime divisor on Z(1-operatornamelct_eta(X,B,f^D))D nand some pseudo-effective Q-b-divisor M_Z such that K_X+B_X sim_ Q f^(K_Z+B_Z+M_Z). Moreover, if (X,B) is lc, then M_X is a b-nef Q-b-divisor, where M_X=K_X+B_X-f^(K_Z+B_Z)sim_ Q f^M_Z. The formula decomposes the relative canonical class into discriminant and moduli contributions and is a central tool in the birational geometry of fiber spaces. In characteristic 0, the conjecture was completely settled by a series of works; the positive-characteristic situation is part of the motivation for this paper.

The canonical bundle formula conjecture

Suppose $(X,B)/Z$ is a contraction with relative dimension $>0$ , where $(X,B)$ is a generically lc projective pair and $K_X+B\sim_\mathbb Q 0/Z$ . Let $\eta=\operatorname{Spec}(k(Z))$ be the generic point of $Z$ . For each prime divisor $D$ on $Z$ , define the discriminant coefficient using the log canonical threshold $\operatorname{lct}_\eta(X,B,f^*D)$ .

Canonical bundle formula conjecture. There is

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

\nand some 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.

The formula decomposes the relative canonical class into discriminant and moduli contributions and is a central tool in the birational geometry of fiber spaces. In characteristic $0$ , the conjecture was completely settled by a series of works; the positive-characteristic situation is part of the motivation for this paper.

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).

The canonical bundle formula conjecture

The canonical bundle formula conjecture

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