从解析观点看非交换上同调跳跃轨迹
Nonabelian cohomology jump loci from an analytic viewpoint
摘要：
对于一个拓扑空间，我们研究了它的上同调支撑轨迹，它位于基本群的各种（非交换）表示之内。为了做到这一点，对于一个CDG（交换微分分次）代数，我们定义它的上同调跳轨迹，坐在各种（代数）平坦连接。证明了表示簇原点1处的解析芽由空间的Sullivan 1-极小模型决定。在弱有限性假设下，证明了当空间和代数具有相同的$q$ -极小模型时，在一定程度上$q$，两种类型的跳跃轨迹在原点具有相同的解析芽.我们将这种一般方法应用于形式空间（为此我们建立了Farber-Novikov谱序列的退化）、拟投射流形和非线性生成的幂零群幂群。当CDG代数具有正权时，我们阐明了（秩一复形）拓扑和代数跳跃轨迹的一些结构：直到次数$q$，它们所有通过原点的不可约分量都是连通的仿射子环面，分别是有理线性子空间.此外，全球指数映射发送所有代数上同调跳转轨迹，直到程度$q$，到他们的拓扑对应。
英文摘要：
For a topological space, we investigate its cohomology support loci, sitting inside varieties of (nonabelian) representations of the fundamental group. To do this, for a CDG (commutative differential graded) algebra, we define its cohomology jump loci, sitting inside varieties of (algebraic) flat connections. We prove that the analytic germs at the origin 1 of representation varieties are determined by the Sullivan 1-minimal model of the space. Under mild finiteness assumptions, we show that, up to a degree $q$, the two types of jump loci have the same analytic germs at the origins, when the space and the algebra have the same $q$-minimal model. We apply this general approach to formal spaces (for which we establish the degeneration of the Farber-Novikov spectral sequence), quasi-projective manifolds, and finitely generated nilpotent groups. When the CDG algebra has positive weights, we elucidate some of the structure of (rank one complex) topological and algebraic jump loci: up to degree $q$, all their irreducible components passing through the origin are connected affine subtori, respectively rational linear subspaces. Furthermore, the global exponential map sends all algebraic cohomology jump loci, up to degree $q$, into their topological counterpart.
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