Assume $X$ be a normal projective variety with $\mathbb Q$-Cartier divisor $D$, then can we extend adjunction formula on pair $(X,D)$?



Adjunction formula on pair
If the pair $(X,D)$ is log canonical and $S$ be a component of $D$ with coefficient 1, then we have adjunction type formula as $$K\_S+D\_S=(K\_X+D)|\_S$$


[edited]: If $f:Z\to X$ be a finite Galois map, then there exists a branch divisor $B$ on $X$ s.t, $K\_X+B$ is $\mathbb Q$-Cartier and $K\_Z=f^\*(K\_X+B)$.


As an example on surfaces: Let $(X,D)$ be a surface with Kawamata log terminal singularities and $\pi:Y\to X$ be a minimal resolution of $X$. Then we can write $\pi^\*(K\_X+D)=K\_Y+D\_Y$ where $D\_Y\geq 0$ and each exceptional curve of $\pi$ is a smooth rational curve.


Note: If $K\_X+B$ is not $\mathbb Q$-Cartier, it is not clear what the adjunction formula should mean, but even then one can have a sort of adjunction formula involving $Ext$'s which is Grothendieck Duality.


See Kawamata-Kodaira canonical bundle formula also


Example: Let $f:V\to X$ is a contraction but not birational such that $K\_V\sim\_\mathbb Q 0$ (like Calabi-Yau model)then it often happen that $K\_V\sim\_\mathbb Q f^\*(K\_X+B)$ for some divisor $B$


