标题： 标准局部齐性空间的形变 显示英文标题

标题： Deformations of Standard Locally Homogeneous Spaces

摘要： 设$X=G/H$为一个齐性空间，其中$G \supset H$为半单李群。 我们提出问题：在$\Gamma \backslash G/H$为标准商的情况下，离散子群$\Gamma$在保持$\Gamma$作用在$X$上的离散性条件下，可以被变形到何种程度？ 我们提供了几个分类结果，包括：在紧致标准商$\Gamma\backslash X$的情况下局部刚性成立的条件；标准商可以变形为非标准商的条件；在小变形下不连续群的极大 Zariski 闭包的刻画；以及 Zariski 稠密变形发生的条件。 本文所述结果的证明在 arXiv:2507.03476 中详细提供。 显示英文摘要

摘要： Let $X=G/H$ be a homogeneous space, where $G \supset H$ are reductive Lie groups. We ask: in the setting where $\Gamma \backslash G/H$ is a standard quotient, to what extent can the discrete subgroup $\Gamma$ be deformed while preserving the proper discontinuity of the $\Gamma$-action on $X$? We provide several classification results, including: conditions under which local rigidity holds for compact standard quotients $\Gamma\backslash X$; criteria for when a standard quotient can be deformed into a nonstandard one; a characterization of the maximal Zariski-closure of discontinuous groups under small deformations; and conditions under which Zariski-dense deformations occur. Proofs of the results stated in this paper are provided in detail in arXiv:2507.03476.