For a smooth finite cyclic covering over a projective space of dimension greater than one, we show that its group of automorphisms faithfully acts on its cohomology except for a few cases. In characteristic zero, we study the equivariant deformation theory and groups of automorphisms for complex cyclic coverings. The proof uses the decomposition of the sheaf of differential forms due to Esnault and Viehweg. In positive characteristic, a lifting criterion of automorphisms reduce the faithfulness problem to characteristic zero. To apply this criterion, we prove the degeneration of the Hodge–de Rham spectral sequences for a family of smooth finite cyclic coverings, and the infinitesimal Torelli theorem for finite cyclic coverings defined over an arbitrary field.

中文翻译：

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关于射影空间有限循环覆盖的自同构和上同调的注解

对于维度大于 1 的射影空间上的平滑有限循环覆盖，我们表明，除了少数情况外，它的自同构群忠实地作用于其上同调。在特征零中，我们研究了复杂循环覆盖的等变变形理论和自同构群。证明使用了 Esnault 和 Viehweg 的微分形式束的分解。在正特征中，自同构的提升标准将忠实性问题减少到特征零。为了应用这个标准，我们证明了平滑有限循环覆盖族的 Hodge-de Rham 谱序列的退化，以及在任意域上定义的有限循环覆盖的无穷小 Torelli 定理。