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

TL;DR

This paper classifies certain complex manifolds with specific group actions, showing their automorphism groups and hyperbolicity properties, with implications for the structure of these manifolds and their symmetries.

Contribution

It establishes conditions under which the group actions are normal in the automorphism group and characterizes the uniqueness of these actions, also analyzing hyperbolicity.

Findings

N is a normal subgroup of Aut(S) under specified conditions

Any other group action of the same type coincides with N

S is not algebraically hyperbolic when n is between d+1 and 2d-1

Abstract

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