Campana's orbifold conjecture for numerically equivalent divisors
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913869846282240 |
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| author | Ru, Min Wang, Julie Tzu-Yueh |
| author_facet | Ru, Min Wang, Julie Tzu-Yueh |
| contents | We prove the following version of the Campana's orbifold conjecture: Let $X$ be a complex non-singular projective variety of dimension $n$. Let $D_1,\ldots,D_{n+1}$ be $\mathbb Z$-linearly independent effective divisors in ${\rm Div}(X)$ and $D:=D_1+\cdots+D_{n+1}$ be a normal crossing divisor of $X$. Assume furthermore that they are numerically parallel. Let $Δ=\sum_{i=1}^{n+1} (1-m_i^{-1}) D_i$ and let $f:\mathbb C\to (X,Δ) $ be an orbifold entire curve. Then, there exists a positive integer $\ell$ such that, the orbifold $ (X,Δ_{\ell}) $ is of general type, where $Δ_{\ell}=\sum_{i=1}^{n+1} (1-\frac1{\ell})D_i$, and if $f$ has multiplicity at least $\ell$ along $D_i$, $1\le i\le n+1$, then $f$ must be algebraically degenerate. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_00873 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Campana's orbifold conjecture for numerically equivalent divisors Ru, Min Wang, Julie Tzu-Yueh Complex Variables Algebraic Geometry 32H30 We prove the following version of the Campana's orbifold conjecture: Let $X$ be a complex non-singular projective variety of dimension $n$. Let $D_1,\ldots,D_{n+1}$ be $\mathbb Z$-linearly independent effective divisors in ${\rm Div}(X)$ and $D:=D_1+\cdots+D_{n+1}$ be a normal crossing divisor of $X$. Assume furthermore that they are numerically parallel. Let $Δ=\sum_{i=1}^{n+1} (1-m_i^{-1}) D_i$ and let $f:\mathbb C\to (X,Δ) $ be an orbifold entire curve. Then, there exists a positive integer $\ell$ such that, the orbifold $ (X,Δ_{\ell}) $ is of general type, where $Δ_{\ell}=\sum_{i=1}^{n+1} (1-\frac1{\ell})D_i$, and if $f$ has multiplicity at least $\ell$ along $D_i$, $1\le i\le n+1$, then $f$ must be algebraically degenerate. |
| title | Campana's orbifold conjecture for numerically equivalent divisors |
| topic | Complex Variables Algebraic Geometry 32H30 |
| url | https://arxiv.org/abs/2506.00873 |