Singularities on vertical $ε$-log canonical Fano fibrations
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916520014118912 |
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| author | Birkar, Caucher Chen, Bingyi |
| author_facet | Birkar, Caucher Chen, Bingyi |
| contents | Given a Fano type log Calabi-Yau fibration $(X,B)\to Z$ with $(X,B)$ being $ε$-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 $δ$-lc where $δ>0$ depends only on $ε$ 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 $ε$-lc, we assume that $(X,B)$ is $ε$-lc vertically over $Z$, that is, the log discrepancy of $E$ with respect to $(X,B)$ is $\geq ε$ for any prime divisor $E$ over $X$ whose center on $X$ is vertical over $Z$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_09382 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Singularities on vertical $ε$-log canonical Fano fibrations Birkar, Caucher Chen, Bingyi Algebraic Geometry Given a Fano type log Calabi-Yau fibration $(X,B)\to Z$ with $(X,B)$ being $ε$-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 $δ$-lc where $δ>0$ depends only on $ε$ 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 $ε$-lc, we assume that $(X,B)$ is $ε$-lc vertically over $Z$, that is, the log discrepancy of $E$ with respect to $(X,B)$ is $\geq ε$ for any prime divisor $E$ over $X$ whose center on $X$ is vertical over $Z$. |
| title | Singularities on vertical $ε$-log canonical Fano fibrations |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2412.09382 |