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

Ruben A. HIdalgo, Maximiliano Leyton-Alvarez

TL;DR

This work analyzes ${Z_p^m}$-actions of type $(d;p,n)$ on compact complex manifolds whose quotient by the action is $P^d$ with $n+1$ hyperplanes of branch order $p$. It develops a framework via generalized Fermat varieties $X_n^p(\Lambda)$ and freely acting subgroups $K\le H$, showing that any such action arises as $S\cong X_n^p(\Lambda)/K$ with $N\cong H/K$, and that the automorphism structure lifts through exact sequences, yielding normality of $N$ in ${Aut}(S)$ and rigidity $M=N$ for isomorphic actions when $(d;p,n)\notin \{(2;2,5),(2;4,3)\}$. The paper also constructs the parameter space $\Omega_{n,d}$, provides explicit equations for the generalized Fermat varieties, analyzes fixed-point loci of canonical generators, and proves non-hyperbolicity in broad parameter ranges, notably when $d+1\le m\le n\le 2d-1$. Together these results give a precise description of when ${Aut}(S)$ is finite, when the ${Z_p^m}$-action is unique up to conjugacy, and how hyperbolicity properties fail in many cases, contributing to the understanding of abelian group actions on higher-dimensional projective varieties. The findings have potential implications for rigidity, automorphism group structure, and hyperbolicity questions in complex geometry and algebraic geometry.

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

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

TL;DR

This work analyzes -actions of type on compact complex manifolds whose quotient by the action is with hyperplanes of branch order . It develops a framework via generalized Fermat varieties and freely acting subgroups , showing that any such action arises as with , and that the automorphism structure lifts through exact sequences, yielding normality of in and rigidity for isomorphic actions when . The paper also constructs the parameter space , provides explicit equations for the generalized Fermat varieties, analyzes fixed-point loci of canonical generators, and proves non-hyperbolicity in broad parameter ranges, notably when . Together these results give a precise description of when is finite, when the -action is unique up to conjugacy, and how hyperbolicity properties fail in many cases, contributing to the understanding of abelian group actions on higher-dimensional projective varieties. The findings have potential implications for rigidity, automorphism group structure, and hyperbolicity questions in complex geometry and algebraic geometry.

Abstract

A -action of type , where are integers, is a pair where is a -dimensional compact complex manifold, is a group of holomorphic automorphisms of such that the quotient orbifold is the -dimensional projective space whose branch locus consists of hyperplanes in general position, each one of branch order . If and , then we prove that: (i) is a normal subgroup of and (ii) if is a -action of type , then . If, moreover, , then we observe that is not algebraically hyperbolic

Paper Structure

This paper contains 25 sections, 18 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Generalized Fermat varieties
  3. The group $H$
  4. Generalized Fermat pairs
  5. Case $n=d$
  6. Case $n=d+1$
  7. Case $n \geq d+2$
  8. The parameter space $\Omega_{n,d}$
  9. A family of algebraic varieties parametrized by $\Omega_{n,d}$
  10. Algebraic equations of all generalized Fermat varieties
  11. A simple remark on the cohomological information of generalized Fermat varieties
  12. Uniqueness of generalized Fermat groups
  13. Automorphisms of generalized Fermat varieties
  14. Fixed points of elements of $H$
  15. ${\mathbb Z}_{p}^{m}$-actions of type $(d;p,n)$, $d \geq 2$
  16. ...and 10 more sections

Key Result

Theorem 1

Let $(S,N)$ is a ${\mathbb Z}_{p}^{m}$-action of type $(d;p,n) \notin \{(2;2,5),(2;4,3)\}$ and $3 \leq d+1 \leq n$. Then (i) ${\rm Aut}(S)$ is finite, (ii) $N$ is a normal subgroup of ${\rm Aut}(S)$, and (iii) if $(S,M)$ is a ${\mathbb Z}_{q}^{r}$-action, then $M=N$.

Theorems & Definitions (27)

  • Theorem 1
  • Theorem 2
  • Theorem 3: HHL23
  • Theorem 4: GHL09, HHL23
  • Proposition 1
  • proof
  • Theorem 5: HKLP17
  • Theorem 6: HHL23
  • proof
  • Corollary 1
  • ...and 17 more