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 \cite{Bi23} 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 center on $X$ is vertical over $Z$.