${\mathbb{Z}}_p^m$-actions of type $(d;p,n)$

Abstract

A ${\mathbb{Z}}_p^m$-action of type $(d;p,n)$, where $2\le d\le m\le n$ 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 $d+1\le n$, then we prove that: (i) $N$ is a normal subgroup of $\mathrm{Aut}(S)$ 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, $d+1\le n\le 2d-1$, then we observe that $S$ is not algebraically hyperbolic