We give two applications of the exponential Ax-Lindemann Theorem to local systems. One application is to show that for a connected topological space, the existence of a finite model of the real homotopy type implies linearity of the cohomology jump loci around the trivial local system. Another application is the linearity of the cohomology jump loci of rank one local systems on quasi-compact K\"ahler manifolds.

中文翻译：

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准紧致 Kähler 流形的上同调跳跃轨迹

我们给出了指数 Ax-Lindemann 定理在局部系统中的两个应用。一个应用是证明对于连通的拓扑空间，实同伦类型的有限模型的存在意味着围绕平凡局部系统的上同调跳跃轨迹的线性。另一个应用是准紧致 K\"ahler 流形上一级局部系统的上同调跳跃轨迹的线性。