The cohomology jump loci of a space $X$ are of two basic types: the characteristic varieties, defined in terms of homology with coefficients in rank one local systems, and the resonance varieties, constructed from information encoded in either the cohomology ring, or an algebraic model for $X$. We explore here the geometry of these varieties and the delicate interplay between them in the context of closed, orientable 3-dimensional manifolds and link complements. The classical multivariable Alexander polynomial plays an important role in this analysis. As an application, we derive some consequences regarding the formality and the existence of finite-dimensional models for such 3-manifolds.

中文翻译：

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三流形的上同调跳跃轨迹

空间 $X$ 的上同调跳跃轨迹有两种基本类型：特征变体，根据与一级局部系统中系数的同源性定义，以及共振变体，由编码在上同调环或$X$ 的代数模型。我们在这里探索了这些变体的几何形状以及它们在封闭的、可定向的 3 维流形和链接互补的背景下的微妙相互作用。经典的多变量亚历山大多项式在此分析中起着重要作用。作为一个应用，我们推导出了关于这种 3 流形的有限维模型的形式和存在的一些后果。