I have recently been wondering about the existence of a canonical bundle formula in the following situation and am not sure how to proceed.
Suppose $(X,B) \xrightarrow{f} Y$ is a fibration where $ (X,B)$ is a sub-klt pair such that rank $f_* \mathcal{O}_X(\left\lceil{A(X,B)} \right\rceil) =1$ and $K_X+B \equiv_{f, \mathbb{Q}} 0$ (i.e. we are replacing $\mathbb{Q}$-linear equivalence with numerical equivalence).
In the above setup, I wonder if we can still write $K_X+B \equiv_{\mathbb{Q}} f^*(K_Y+B_Y+M_Y)$ as usual such that $M_Y$ is b-nef. Here $B_Y= \Sigma (1-b_D).D$ where $b_D= sup\{t>0| K_X+B+tf^*D$ is sub-lc over the generic point of $D\}$ as usual.
Please let me know your thoughts on this.
