AI 中文总结
本文研究复化实构形补空间有限循环覆盖的上同调群，利用Yoshinaga的室上链复形得到贝蒂数上界，引入类CDO组合条件证明此类覆盖整上同调无挠，部分推广了作者与Liu的相关结果。
AI 中文摘要
本文研究复化实构形补空间的有限循环覆盖的上同调群。一个未解决的问题是，超平面构形补空间的有限覆盖空间（包括经典Milnor纤维）的（上）同调中的挠是否由组合确定。利用吉永（Yoshinaga）构造的室上链复形，我们得到这些覆盖在任意域上的贝蒂数的显式上界。此外，我们引入一个类似Cohen-Dimca-Orlik（CDO）条件的组合条件，在该条件下，我们证明有限循环覆盖的整上同调群是无挠的，这部分推广了作者与刘（Liu）在复直线构形上获得的近期结果[\u200bLX26，定理1.3]。
英文摘要
In this paper, we study the cohomology groups of finite cyclic covers of complexified real arrangement complements. An open problem is whether the torsion in the (co)homology of finite covering spaces of hyperplane arrangement complements, including the classical Milnor fiber, is combinatorially determined. Using the chamber cochain complex constructed by Yoshinaga, we obtain explicit upper bounds for the Betti numbers of these covers over arbitrary fields. Furthermore, we introduce a combinatorial condition analogous to the Cohen-Dimca-Orlik (CDO) condition. Under this condition, we prove that the integral cohomology groups of the finite cyclic covers are torsion-free. This partially generalizes the recent results on complex line arrangements obtained by the author and Liu \cite[Theorem 1.3]{LX26}.
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