Shokurov–Kawamata adjunction and effective adjunction conjectures

Let $(T/Z,B)$ be a log canonical pair of dimension $d$ such that

$$K_T+B\sim_{\mathbb{R}}0/Z.$$

Define the discriminant divisor $D_Z$ by

$$1-d_i=\sup\{c\mid K_T+B+cf^*D_i\text{ is log canonical over the generic point of }D_i\},$$

where $D_Z=\sum_i d_iD_i$, and define the moduli class $M_Z$, uniquely up to $\mathbb{R}$-linear equivalence, by

$$K_T+B\sim_{\mathbb{R}}f^*(K_Z+D_Z+M_Z).$$

Adjunction and effective adjunction conjectures. One can choose an effective representative $M_Z\geq 0$ in its $\mathbb{R}$-linear equivalence class such that $(Z,D_Z+M_Z)$ is log canonical. Moreover, after fixing $\Gamma_f$, there is a constant $I\in\mathbb{N}$ depending only on $d$ and $\Gamma_f$ such that, for an appropriate choice of $M_Z$, the linear system $|IM_Z|$ is free and

$$I(K_T+B)\sim f^*I(K_Z+D_Z+M_Z).$$

These assertions are general forms of the canonical bundle formula: the first gives log canonicity of the base pair, while the second predicts bounded effective representatives and an index depending only on the dimension and the specified data. They are attributed in the source to Shokurov and Kawamata and are used there as an input for boundedness arguments; no resolution status is supplied in the paper.