ℤₚᵐ-Action of Type (d;p,n) in Complex Geometry

• The topic defines ℤₚᵐ-actions of type (d;p,n) as structures where a finite abelian group acts faithfully on a compact complex manifold, producing a Galois orbifold cover of ℙᵈ with a branched hyperplane arrangement.
• It employs generalized Fermat varieties and specific numerical conditions (2 ≤ d ≤ m ≤ n) to construct these covers, ensuring the branch locus is composed of n+1 hyperplanes with branching order p.
• Key results include the normality of the acting group in Aut(S), the uniqueness of the cover construction, and the demonstration of non-hyperbolicity when n ≤ 2d–1, supported by explicit examples.

A ${\mathbb Z}_{p}^{m}$-action of type $(d;p,n)$, where $2 \leq d \leq m \leq n$ are integers and $p \geq 2$, is a structure in complex geometry consisting of a pair $(S,N)$. Here, $S$ is a compact $d$-dimensional complex manifold, and $N \cong {\mathbb Z}_p^m$ is a group of holomorphic automorphisms of $S$ such that $N$ acts faithfully and properly discontinuously. The quotient orbifold $S/N$ is isomorphic to the complex projective space ${\mathbb P}^d$, with a branch locus given by $n+1$ hyperplanes in general position, each with branch order $p$. This construction generalizes the classical theory of abelian Galois covers and brings together perspectives from group actions, branched coverings, and the theory of orbifolds.

1. Precise Definition and Orbifold Structure

Let $p \ge 2$ and $2 \le d \le m \le n$. A ${\mathbb Z}_{p}^{m}$-action of type $(d;p,n)$ is defined by the following data:

• $S$, a compact complex manifold of complex dimension $d$,
• $N \cong {\mathbb Z}_p^m$, a subgroup of the holomorphic automorphism group $\operatorname{Aut}(S)$, acting faithfully and properly discontinuously.

The action is specified so that the quotient orbifold $\pi_N: S \to S/N = {\mathbb P}^d$ is a Galois cover with deck group $N$. The branch locus is the union $$\mathcal{B} = \Sigma_1 \cup \cdots \cup \Sigma_{n+1}$$, where each $\Sigma_j$ is a hyperplane in general position, and the local branching order along each $\Sigma_j$ is $p$.

Equivalently, the orbifold pair can be written as $({\mathbb P}^d, \Delta)$, where

$\Delta = \frac{p-1}{p} \sum_{j=1}^{n+1} \Sigma_j$

so the orbifold fundamental group has the presentation

$$\pi_{1}^{\mathrm{orb}}({\mathbb P}^d, \Delta) = \langle \gamma_1,\ldots,\gamma_{n+1} ~|~ \gamma_j^p = 1,~ \gamma_1\cdots\gamma_{n+1}=1 \rangle$$

The manifold $S$ is the ${\mathbb Z}_p^m$-cover corresponding to this group via deck transformations.

2. Numerical Constraints and Geometric Interpretation

For such a structure to exist, several numerical inequalities must be satisfied:

• $2 \le d \le m \le n$, $n \ge d$.
• Branching along $n+1$ hyperplanes in ${\mathbb P}^d$ imposes $n \ge d$ due to the general-position requirement.

Each hyperplane contributes a $\mathbb{Z}_p$-generator in the maximal abelian cover, so $m \le n$.

Borderline and illustrative cases include:

• If $n = d$, then $S \cong {\mathbb P}^d$ and $m = d$: the cover is trivial,
• If $n = d+1$, then $S$ is the Fermat hypersurface of degree $p$ in ${\mathbb P}^{d+1}$,

$X = \{x_1^p + \cdots + x_{d+2}^p = 0\}$

with deck group $H \cong {\mathbb Z}_p^{d+1}$.

Generally, for $n \ge d + 1$, nontrivial actions and covers arise. This formalism unifies a variety of classical and new constructions in the topology of complex manifolds.

3. Fundamental Theorems: Normality, Uniqueness, and Hyperbolicity

Assuming $d \geq 2$, $d+1 \leq n$, and excluding the special surface cases $(d,p,n) = (2,2,5)$ or $(2,4,3)$, the following hold:

A. Normality of $N$ in $\operatorname{Aut}(S)$: $\operatorname{Aut}(S)$ is finite, and $N$ is normal ($N \triangleleft \operatorname{Aut}(S)$). Every holomorphic automorphism of $S$ normalizes $N$.

B. Uniqueness: If $(S, M)$ is any other ${\mathbb Z}_{\hat{p}}^{\hat{m}}$-action of type $(d;\hat{p},\hat{n})$ on the same $S$, so that $S/M \cong {\mathbb P}^d$ has $n+1$ hyperplanes of order $\hat p$, then necessarily $\hat p = p$, $\hat m = m$, $\hat n = n$, and $M = N$.

C. (Non-)Hyperbolicity: If $d+1 \leq n \leq 2d-1$, then $S$ is not algebraically hyperbolic: $S$ admits a rational curve, specifically, a genus $0$ cover of a line in ${\mathbb P}^d$.

This places severe rigidity constraints on possible group actions and the geometric type of the resulting manifolds.

4. Construction via Generalized Fermat Varieties

Realization of these actions proceeds through quotients of generalized Fermat varieties. Let $X := X^p_n(\Lambda) \subset {\mathbb P}^n$ be the complete intersection defined by

$$\begin{cases} x_1^p + \cdots + x_{d+1}^p + x_{d+2}^p = 0 \ \lambda_{1,1} x_1^p + \cdots + \lambda_{1,d} x_d^p + x_{d+1}^p + x_{d+3}^p = 0 \ \vdots \end{cases}$$

with deck group $H \cong {\mathbb Z}_p^n$ acting linearly. There exists a subgroup $K \cong {\mathbb Z}_p^{n-m}$ acting freely, such that $S \cong X/K$ and $N = H/K \cong \mathbb{Z}_p^m$.

Since $X$ is simply connected for $d \geq 2$, every automorphism of $S$ lifts to an automorphism of $X$ normalizing $K$. The uniqueness theorem for generalized Fermat groups implies $H$ is the only ${\mathbb Z}_p^n$-subgroup in $\operatorname{Aut}(X)$, so $N$ is normal in the finite group $\operatorname{Aut}(S)$.

For the uniqueness property: any other appropriate ${\mathbb Z}_p^r$ subgroup in $\operatorname{Aut}(S)$ must correspond to a subgroup of $H$, and by analysis of the fixed loci (each fixed by codimension-$1$ subgroups), one obtains $M = H/K = N$.

The non-hyperbolicity result is demonstrated by explicit construction: for $n \leq 2d-1$, one can find lines in the branch locus whose preimages in $S$ yield rational (or low genus) curves.

5. Explicit Examples

Example 1: Fermat Hypersurface Quotient

Let $$X = \{x_1^p + \cdots + x_{d+2}^p = 0\} \subset {\mathbb P}^{d+1}$$ and $$H = \langle \varphi_j: x_j \mapsto \omega_p x_j \rangle \cong {\mathbb Z}_p^{d+1}$$. If $p \geq d+2$, the subgroup $$K = \langle \varphi_1 \varphi_2^{-1}, \ldots, \varphi_{d+1} \varphi_{d+2}^{-1} \rangle \cong \mathbb{Z}_p$$ acts freely, and the quotient $S = X/K$ admits $N = H/K \cong \mathbb{Z}_p^d$, yielding a ${\mathbb Z}_p^d$-action of type $(d;p,d+1)$.

Example 2: Case $(d;p,n) = (2;2,6),~m=3$

For the generalized Fermat surface $X = X_6^2(\Lambda) \subset \mathbb{P}^6$, the subgroup $$K = \langle \varphi_1\varphi_2\varphi_4,\, \varphi_1\varphi_3\varphi_5,\, \varphi_2\varphi_3\varphi_6\rangle \cong \mathbb{Z}_2^3$$ acts freely, and $S = X/K$, $N = H/K \cong \mathbb{Z}_2^3$ gives a ${\mathbb Z}_2^3$-action of type $(2;2,6)$. This construction leads to a six-parameter family of surfaces with the specified symmetry and covering properties.

6. Context and Connections to Broader Theory

This class of actions generalizes a variety of classical constructions: for dimension $d=1$, the situation corresponds to abelian covers of the Riemann sphere branched at several points, aligning with the rich theory of $(\mathbb{Z}_p)^m$ actions on Riemann surfaces as in quadrangular actions (Hidalgo, 2021). The case $d=2$ includes generalized Fermat surfaces, which have been the focus of deep paper due to their automorphism groups and moduli.

The rigidity, normality, and uniqueness results yield a strong classification in terms of orbifold data. The theory interconnects with the paper of the field of moduli and the structure of the Jacobian, although, for $d > 1$, the Jacobian decompositions analogous to the $d=1$ case do not extend directly.

A further implication is the geometric non-hyperbolicity for $n\leq 2d-1$, which situates these manifolds outside the landscape of algebraically hyperbolic varieties. This result constrains potential applications in the context of higher-dimensional hyperbolicity, emphasizing the abundance of rational curves in these covers for lower $n$.

7. Future Directions and Open Problems

The data suggest several plausible research directions: classifications for exceptional parameter values (such as $(2;2,5)$ and $(2;4,3)$), further exploration of higher symmetry loci (where the automorphism group is larger than $N$), and extending detailed analysis to cases with $n \gg d$. Questions regarding moduli, arithmetic properties, and connections to broader phenomena such as rigidity in algebraic and differential geometry remain open.

The explicit geometry of these covers, especially for small $p$, $d$, and $n$, continues to yield new families of complex manifolds with prescribed group actions and quotient orbifolds. The interplay of group theory, algebraic geometry, and orbifold fundamental groups characteristic of ${\mathbb Z}_p^m$-actions of type $(d;p,n)$ presents a framework for understanding higher-dimensional, highly symmetric manifolds with controlled branching loci.