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

C. Birkar^a, Bingyi Chen<sup>b

a</sup> Yau Mathematical Sciences Center, Jingzhai, Tsinghua University, Haidian District, Beijing, 100084, China
^b Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

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 $Z$.

Keywords: Fano type fibrations, singularities of pairs, canonical bundle formula.

Received: December 12, 2024
Revised: February 20, 2025
Accepted: March 12, 2025

DOI: 10.4213/tm4461

 English version: Proceedings of the Steklov Institute of Mathematics, 2025, 329, 55–64

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