On Rigid Varieties Isogenous to a Product of Curves

TL;DR

The work extends the theory of Beauville surfaces to higher dimensions by studying rigid varieties isogenous to a product $(C_1\times\cdots\times C_n)/G$ with free, diagonal group actions. It develops a purely group-theoretic framework using $n$-fold Beauville structures and analyzes biholomorphism classes via the action of $\mathrm{Aut}(G)\times (\mathcal{B}_3\wr \mathfrak{S}_n)$ on the set of unmixed Beauville structures $\mathcal{UB}_n(G)$. The paper establishes existence and uniqueness of minimal realizations, introduces the notion of Beauville dimension, and provides explicit classifications for $3$-folds with $G=\mathbb{Z}_5^2$ (eight classes with $\chi(\mathcal{O}_X)=-1$ and 76 with $\chi(\mathcal{O}_X)=-5$) along with a finite table of cases for other Euler characteristics, all computed with MAGMA using the CGP23 database. These results yield concrete, computable invariants and a scalable approach for classifying rigid higher-dimensional varieties isogenous to products, broadening the scope of Beauville-type constructions and their interaction with finite group theory. The methods provide tools for deciding when two such manifolds are biholomorphic and offer insights into the minimal dimension required for a given group to realize a Beauville manifold.

Abstract

In this note, we study rigid complex manifolds that are realized as quotients of a product of curves by a free action of a finite group. They serve as higher-dimensional analogues of Beauville surfaces. Using uniformization, we outline the theory to characterize these manifolds through specific combinatorial data associated with the group under the assumption that the action is diagonal and the manifold is of general type. This leads to the notion of a $n$-fold Beauville structure. We define an action on the set of all $n$-fold Beauville structures of a given finite group that allows us to distinguish the biholomorphism classes of the underlying rigid manifolds. As an application, we give a classification of these manifolds with group $\mathbb Z_5^2$ in the three dimensional case and prove that this is the smallest possible group that allows a rigid, free and diagonal action on a product of three curves. In addition, we provide the classification of rigid 3-folds $X$ given by a group acting faithfully on each factor for any value of the holomorphic Euler number $χ(\mathcal O_X) \geq -5$.

Figures (1)

• Figure 1: The three generators $\gamma$, $\delta$, $\epsilon$ of the fundamental group of $\mathbb{P}^1 \setminus \mathcal{B}$.

Theorems & Definitions (46)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• Definition 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Definition 2.5