Logarithmic base change theorem and smooth descent of positivity of log canonical divisor
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866910069420982272 |
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| author | Park, Sung Gi |
| author_facet | Park, Sung Gi |
| contents | We prove a logarithmic base change theorem for pushforwards of pluri-canonical bundles and use it to deduce that positivity properties of log canonical divisors descend via smooth projective morphisms. As an application, for a surjective morphism $f:X\to Y$ with $κ(X)\ge 0$ and $-K_Y$ big, we prove $Y\setminus Δ(f)$ is of log general type, where $Δ(f)$ is the discriminant locus. In particular, when $Y=\mathbb{P}^n$ we have $\dim Δ(f)=n-1$ and $\mathrm{deg}\,Δ(f)\ge n+2$, generalizing the case $n=1$ proved by Viehweg-Zuo. In addition, we prove Popa's conjecture on the superadditivity of the logarithmic Kodaira dimension of smooth algebraic fiber spaces over bases of dimension at most three and analyze related problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_02825 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Logarithmic base change theorem and smooth descent of positivity of log canonical divisor Park, Sung Gi Algebraic Geometry 14D06, 14J10, 14F10, 14E30 We prove a logarithmic base change theorem for pushforwards of pluri-canonical bundles and use it to deduce that positivity properties of log canonical divisors descend via smooth projective morphisms. As an application, for a surjective morphism $f:X\to Y$ with $κ(X)\ge 0$ and $-K_Y$ big, we prove $Y\setminus Δ(f)$ is of log general type, where $Δ(f)$ is the discriminant locus. In particular, when $Y=\mathbb{P}^n$ we have $\dim Δ(f)=n-1$ and $\mathrm{deg}\,Δ(f)\ge n+2$, generalizing the case $n=1$ proved by Viehweg-Zuo. In addition, we prove Popa's conjecture on the superadditivity of the logarithmic Kodaira dimension of smooth algebraic fiber spaces over bases of dimension at most three and analyze related problems. |
| title | Logarithmic base change theorem and smooth descent of positivity of log canonical divisor |
| topic | Algebraic Geometry 14D06, 14J10, 14F10, 14E30 |
| url | https://arxiv.org/abs/2210.02825 |