Abstract
Given a Fano type log Calabi–Yau fibration \((X,B)\to Z\) with \((X,B)\) being \(\epsilon\) -lc, the first author in 2023 proved that the generalised pair \((Z,B_Z+M_Z)\) given by the canonical bundle formula is generalised \(\delta\) -lc, where \(\delta>0\) depends only on \(\epsilon\) and \(\dim X-\dim Z\) , which confirmed a conjecture of Shokurov. In this paper, we prove the above result under a weaker assumption. Instead of requiring \((X,B)\) to be \(\epsilon\) -lc, we assume that \((X,B)\) is \(\epsilon\) -lc vertically over \(Z\) , that is, the log discrepancy of \(E\) with respect to \((X,B)\) is \({\geq}\,\epsilon\) for any prime divisor \(E\) over \(X\) whose centre on \(X\) is vertical over \(Z\) .
