标题： 关于与曲线乘积同构的刚性代数簇 显示英文标题

标题： On Rigid Varieties Isogenous to a Product of Curves

摘要： 在本文中，我们研究作为曲线乘积的商空间的刚性复流形，该商空间由有限群的自由作用得到。它们作为Beauville曲面的高维类似物。利用统一化理论，我们概述了通过与群相关的特定组合数据来表征这些流形的理论，假设作用是斜对角的且流形为一般型。这导致了$n$-重Beauville结构的概念。我们定义了一个作用于给定有限群的所有$n$-重Beauville结构集合上的作用，使我们能够区分底层刚性流形的共形类。作为应用，我们给出了三维情况下由群$\mathbb Z_5^2$产生的这些流形的分类，并证明这是允许在三个曲线的乘积上进行刚性、自由和斜对角作用的最小可能群。此外，我们提供了由群在每个因子上忠实作用所给出的刚性3-重流形$X$的分类，对于全纯欧拉数$\chi(\mathcal O_X) \geq -5$的任何值都适用。 显示英文摘要

摘要： 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 $\chi(\mathcal O_X) \geq -5$.