We use the non-proper Morse theory of Palais-Smale to investigate the topology of smooth closed subvarieties of complex semi-abelian varieties, and that of their infinite cyclic covers. As main applications, we obtain the finite generation (except in the middle degree) of the corresponding integral Alexander modules, as well as the signed Euler characteristic property and generic vanishing for rank-one local systems on such subvarieties. Furthermore, we give a more conceptual (topological) interpretation of the signed Euler characteristic property in terms of vanishing of Novikov homology. As a byproduct, we prove a generic vanishing result for the $L^2$-Betti numbers of very affine manifolds. Our methods also recast Jun Huh's extension of Varchenko's conjecture to very affine manifolds, and provide a generalization of this result in the context of smooth closed subvarieties of semi-abelian varieties.

中文翻译：

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复杂半阿贝尔变种的子变种拓扑

我们使用 Palais-Smale 的非真 Morse 理论来研究复杂半阿贝尔簇的光滑封闭子变体的拓扑结构，以及它们的无限循环覆盖的拓扑结构。作为主要应用，我们获得了相应积分 Alexander 模的有限代（除中间阶），以及此类子变体上的秩一局部系统的带符号欧拉特征和泛型消失。此外，我们根据 Novikov 同源性的消失对带符号的欧拉特性进行了更概念化（拓扑）的解释。作为副产品，我们证明了非常仿射流形的 $L^2$-Betti 数的通用消失结果。我们的方法还将 Jun Huh 对 Varchenko 猜想的扩展重新定义为非常仿射流形，