Singularities on vertical $\epsilon$-log canonical Fano fibrations

TL;DR

This paper extends previous results on the singularities of Fano fibrations by weakening the assumptions from global to vertical $ ext{lc}$ conditions, confirming conjectures about the nature of the base's singularities.

Contribution

It proves that the generalized pair on the base remains generalized $ ext{lc}$ under weaker vertical $ ext{lc}$ assumptions on the total space.

Findings

Generalized pair on the base is generalized $ ext{lc}$ under weaker assumptions.

Confirms a conjecture of Shokurov in a broader setting.

Extends previous results to vertically $ ext{lc}$ 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 $Z$.