Profinite 完备化与上同调跳跃轨迹
Profinite completions and cohomology jump loci
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- Northeastern University（东北大学）
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中文总结 AI 辅助
本文证明 profinite 完备化在特定假设下决定 CW 复形的上同调跳跃轨迹及言语子群分次商，并应用于超平面排列，揭示其不区分算术 Zariski 对且不决定排列群。
中文摘要 AI 辅助
设 $X$ 为具有基本群 $G$ 的连通有限型 CW 复形。我们证明，在两种分别处理的假设下，profinite 完备化 $\widehat{G}$ 决定了上同调跳跃轨迹 $\mathcal{V}^q_s(X,\mathbb{C})$：其一，这些轨迹是有限个扭转平移子环面的并集，这适用于光滑拟射影簇；其二，$\widehat{G}$ 决定了 $X$ 的有限循环覆盖在度数 $\le q$ 上的 Betti 数，这在 $q=1$ 时无条件成立，且当 $X$ 为可缩空间且 $G$ 在 Serre 意义下为良群时对所有度数成立。我们还证明，$\widehat{G}$ 决定了每个言语子群 $W(G)$ 的分次阿贝尔群 $\mathrm{gr}_r(G/W(G))$（包括挠元）；$W(G)=1$ 和 $W(G)=G''$ 的情形分别给出下中心级数商和 Chen 群。对于超平面排列 $\mathcal{A}$，由此可知，任何算术 Zariski 对都不能通过这些不变量中的任何一个来区分，而两个已知的格同构对表明，排列群 $G(\mathcal{A})$ 的 profinite 完备化并非组合决定的，且不决定 $G(\mathcal{A})$。
英文摘要
Let $X$ be a connected finite-type CW-complex with fundamental group $G$. We show that the profinite completion $\widehat{G}$ determines the cohomology jump loci $\mathcal{V}^q_s(X,\mathbb{C})$ under two hypotheses: that the loci are finite unions of torsion-translated subtori, as for smooth quasi-projective varieties, and that $\widehat{G}$ determines the Betti numbers of the finite cyclic covers of $X$ in degrees $\le q$, which holds unconditionally for $q=1$, and in all degrees when $X$ is aspherical and $G$ is good in the sense of Serre. When the isomorphism of completions is compatible with an identification of the abelianizations, the loci correspond exactly; in general, they correspond up to an isogeny. Applied to finite covers, this shows that the tropical bounds for the Bieri--Neumann--Strebel--Renz invariants are profinite invariants, and recovers the profinite invariance of the BNS invariant of Kähler groups due to Hughes, Llosa Isenrich, Py, Stover, and Vidussi. We also show that $\widehat{G}$ determines the graded abelian groups $\mathrm{gr}_r(G/W(G))$, torsion included, for every verbal subgroup $W(G)$; the cases $W(G)=1$ and $W(G)=G''$ give the lower central series quotients and the Chen groups. For hyperplane arrangements, it follows that no arithmetic Zariski pair is distinguished by any of these invariants, while two known lattice-isomorphic pairs show, respectively, that the profinite completion of an arrangement group is not combinatorially determined, and that it does not determine the group.
