关于双曲3-流形和乐簇的异常子簇
On anomalous subvarieties of holonomy varieties of hyperbolic 3-manifolds
AI总结：
研究n尖点双曲3-流形的和乐簇的异常子簇结构，证明最大异常子簇对应保持一个尖点完整，并分类2尖点情形。
摘要：
本文旨在探索两个看似不同领域之间的相互作用。更精确地说，设$\mathcal{M}$是一个具有有理独立尖点形状的$n$尖点双曲$3$-流形，$\mathcal{X}$是其和乐簇。我们研究$\mathcal{X}$的异常子簇结构，这一概念源于算术几何，并将其与$\mathcal{M}$的各种几何性质联系起来。首先，我们证明$\mathcal{X}$的每个包含单位元的极大异常子簇都是其余维数为$1$的子簇，该子簇通过保持$\mathcal{M}$的一个尖点完整而产生。其次，我们证明，如果$\mathcal{X}$被其异常子簇退化（即$\mathcal{X}^{oa}=\emptyset$），那么$\mathcal{M}$的某些尖点在保持其他尖点完整时，与其余部分强几何隔离。最后，我们完全分类并刻画了任意$2$尖点双曲$3$-流形的和乐簇$\mathcal{X}$满足$\mathcal{X}^{oa}=\emptyset$的情形。
英文摘要：
The goal of this paper is to explore the interplay between two seemingly distinct fields. More precisely, let $\mathcal{M}$ be an $n$-cusped hyperbolic $3$-manifold with rationally independent cusp shapes, and $\mathcal{X}$ be its holonomy variety. We study the structure of anomalous subvarieties of $\mathcal{X}$, a concept originating in arithmetic geometry, and relate it to various geometric properties of $\mathcal{M}$. First, we show that every maximal anomalous subvariety of $\mathcal{X}$ containing the identity is its subvariety of codimension $1$ which arises by keeping one cusp of $\mathcal{M}$ complete. Second, we show that, if $\mathcal{X}$ is degenerated by its anomalous subvarieties (i.e., $\mathcal{X}^{oa}=\emptyset$), then $\mathcal{M}$ has cusps which are, while keeping some other cusps of it complete, strongly geometrically isolated from the rest. Finally, we completely classify and characterize the case $\mathcal{X}^{oa}=\emptyset$ for the holonomy variety $\mathcal{X}$ of any $2$-cusped hyperbolic $3$-manifold.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
