${\mathbb Z}_{p}^{m}$-actions of type $(d;p,n)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912784638279680 |
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| author | HIdalgo, Ruben A. Leyton-Alvarez, Maximiliano |
| author_facet | HIdalgo, Ruben A. Leyton-Alvarez, Maximiliano |
| contents | A ${\mathbb Z}_{p}^{m}$-action of type $(d;p,n)$, where $2 \leq d \leq m \leq n$ are integers, is a pair $(S,N)$ where $S$ is a $d$-dimensional compact complex manifold, $N \cong {\mathbb Z}_{p}^{m}$ is a group of holomorphic automorphisms of $S$ such that the quotient orbifold $S/N$ is the $d$-dimensional projective space ${\mathbb P}^{d}$ whose branch locus consists of $n+1$ hyperplanes in general position, each one of branch order $p$.
If $(d;p,n) \notin \{(2;2,5),(2;4,3)\}$ and $d+1 \leq n$, then we prove that: (i) $N$ is a normal subgroup of ${\rm Aut}(S)$ and (ii) if $(S,M)$ is a ${\mathbb Z}_{\hat{p}}^{\hat{m}}$-action of type $(d;\hat{p},\hat{n})$, then $M=N$. If, moreover, $d+1 \leq n \leq 2d-1$, then we observe that $S$ is not algebraically hyperbolic |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_04462 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | ${\mathbb Z}_{p}^{m}$-actions of type $(d;p,n)$ HIdalgo, Ruben A. Leyton-Alvarez, Maximiliano Algebraic Geometry 14J50, 32Q40, 53C15 A ${\mathbb Z}_{p}^{m}$-action of type $(d;p,n)$, where $2 \leq d \leq m \leq n$ are integers, is a pair $(S,N)$ where $S$ is a $d$-dimensional compact complex manifold, $N \cong {\mathbb Z}_{p}^{m}$ is a group of holomorphic automorphisms of $S$ such that the quotient orbifold $S/N$ is the $d$-dimensional projective space ${\mathbb P}^{d}$ whose branch locus consists of $n+1$ hyperplanes in general position, each one of branch order $p$. If $(d;p,n) \notin \{(2;2,5),(2;4,3)\}$ and $d+1 \leq n$, then we prove that: (i) $N$ is a normal subgroup of ${\rm Aut}(S)$ and (ii) if $(S,M)$ is a ${\mathbb Z}_{\hat{p}}^{\hat{m}}$-action of type $(d;\hat{p},\hat{n})$, then $M=N$. If, moreover, $d+1 \leq n \leq 2d-1$, then we observe that $S$ is not algebraically hyperbolic |
| title | ${\mathbb Z}_{p}^{m}$-actions of type $(d;p,n)$ |
| topic | Algebraic Geometry 14J50, 32Q40, 53C15 |
| url | https://arxiv.org/abs/2511.04462 |