Automorphisms of Generalized Fermat manifolds

TL;DR

This paper offers a shorter proof for the uniqueness of generalized Fermat groups acting on certain complex algebraic varieties and explores the fixed point loci of their subgroups.

Contribution

It presents a new, concise proof of the uniqueness of generalized Fermat groups and investigates the fixed point sets of their subgroups.

Findings

Uniqueness of generalized Fermat groups for most parameters

Shorter proof method compared to previous work

Analysis of fixed point loci of subgroup actions

Abstract

Let $d \geq 1$, $k \geq 2$ and $n\geq d+1$ be integers. A $d$-dimensional smooth complex algebraic variety $M$ is called a generalized Fermat variety of type $(d;k,n)$ if there is a Galois holomorphic branched covering $\pi:M \to {\mathbb P}^{d}$, with deck group $H\cong {\mathbb Z}_{k}^{n}$, whose branch divisor consists of $n+1$ hyperplanes in general position, each one of branch order $k$. In this case, $H$ is called a generalized Fermat group of type $(d;k,n)$. In previous work, we proved that the generalized Fermat group $H$ is unique in the following cases: (i) $d=1$ and $(k-1)(n-1)>2$, or (ii) $d \geq 2$ and $(d;k,n) \notin \{(2;2,5), (2;4,3)\}$. To obtain this uniqueness fact, we used a differential method due to Kontogeorgis. This paper provides a different and shorter proof of the uniqueness of $H$. We also study the locus of fixed points of subgroups of $H$.