Singularities on Vertical $$\epsilon$$-Log Canonical Fano Fibrations

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 $$Z$$ .