复射影空间上的稳定性与对数叶状结构

中文摘要 AI 辅助

我们研究了复射影空间上余维数为1的全纯叶状结构的不变代数除子上的Lyapunov型稳定性。受\cite{LeonScardua2018}中针对平面奇点引入的$L$-稳定性概念的启发，我们定义了一个射影稳定性条件，该条件与对数设置相容，并控制不变除子正则部分沿叶片的和乐（holonomy），同时放弃对其奇点轨迹的指定邻域内的任何稳定性要求。我们的主要结果是\cite{LeonScardua2018}中局部分类的全局射影对应：对于极大不变代数除子，射影$L$-稳定性加上在一般平面截面上非临界非节点广义曲线奇点的条件，迫使环境叶状结构是全局对数的。作为应用，我们在温和一般条件下获得了$\mathbb P^2$上对数叶状结构的拓扑刚性结果。主要定理的证明基于通过显式Dulac输运论证将稳定性假设的后果传播到约化树，并结合一维复微分同胚芽的$L$-稳定群的分类。

英文摘要

We study Lyapunov-type stability for invariant algebraic divisors of codimension-one holomorphic foliations on complex projective spaces. Motivated by the notion of $L$-stability introduced in \cite{LeonScardua2018} for plane singularities, we define a projective stability condition which is compatible with the logarithmic setting and controls leafwise holonomy along the regular part of the invariant divisor, while discarding any stability requirement inside prescribed neighborhoods of its singular locus. Our main result then is a global projective counterpart of the local classification in \cite{LeonScardua2018}: for a maximal invariant algebraic divisor, projective $L$-stability together with non-dicritical non-nodal generalized-curve singularities on a generic plane section forces the ambient foliation to be globally logarithmic. As an application we obtain a topological rigidity consequence on $\mathbb P^2$ for logarithmic foliations under mild generic conditions. The proof of the main theorem is based on propagating the consequences of the stability hypothesis through the reduction tree by means of an explicit Dulac transport argument, together with the classification of $L$-stable groups of germs of one-dimensional complex diffeomorphisms.

发表机构

• Universidade Federal da Integração Latino-Americana（拉丁美洲联邦大学）
• Instituto de Matemática, Universidade Federal do Rio de Janeiro（里约热内卢联邦大学数学研究所）

机构由 AI 辅助整理，请以论文原文为准。