闭叶全纯叶状结构的稳定性：通过胚群扭转行为
Stability of closed leaves holomorphic foliations via torsion behavior of groups of germs
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中文总结 AI 辅助
本文通过引入胚群扭转轨迹，证明了闭叶全纯叶状结构中存在高维紧致叶不变解析集，且叶体积一致有界，并给出射影情形的全局化。
中文摘要 AI 辅助
考虑一个全纯叶状结构 $\mathcal{F}$ 的紧致叶 $\mathcal{L}$，使得 $\mathcal{F}$ 在 $\mathcal{L}$ 的某个邻域 $U$ 上的限制的所有叶在 $U$ 中都是闭的。我们证明，在 $\mathcal{L}$ 的邻域中存在一个解析集 $V$ 的不变胚，其维数高于 $\dim(\mathcal{F})$，由紧致叶组成，使得 $V$ 中叶的体积一致有上界。如果环境流形是射影的，我们还提供了该结果的全局化。为了研究闭叶叶状结构，并通过和乐表示，我们引入了一个独立有趣的新概念，用于全纯微分同胚胚群 $G$ 的子群，即所谓的扭转轨迹。我们证明，如果 $G$ 具有有限轨道，则它是非平凡的。
英文摘要
Consider a compact leaf $\mathcal{L}$ of a holomorphic foliation $\mathcal{F}$ such that all the leaves of the restriction of $\mathcal{F}$ to some neighborhood $U$ of $\mathcal{L}$ are closed in $U$. We show that there exists an invariant germ of analytic set $V$ in a neighborhood of $\mathcal{L}$, of dimension higher than $\dim (\mathcal{F})$, consisting of compact leaves, such that the volume of the leaves in $V$ is uniformly bounded by above. We also provide a globalization of this result if the ambient manifold is projective. In order to study closed leaves foliations, and via the holonomy representation, we introduce a new concept, of independent interest, for subgroups $G$ of germs of holomorphic diffeomorphisms, the so called {\it torsion locus}. We show that it is non-trivial if $G$ has finite orbits.
