AI 中文总结
本文针对李型广义格拉斯曼流形，结合根与权系几何等方法，证明其非零次数自映射的上同调诱导作用仅为亚当斯运算或 Dynkin 对称性两类，完成了此类自映射的分类。
AI 中文摘要
广义格拉斯曼流形是与李群相关的一类基本旗流形。本文旨在根据非零次数的广义格拉斯曼流形自映射在上同调上的诱导作用对其进行分类。我们证明了一个刚性定理，该定理表明，尽管这类流形的上同调环具有丰富而复杂的结构，但诱导的上同调自同态仅分为两种自然类型：由次数确定的亚当斯运算，以及由 Dynkin 图自同构产生的 Dynkin 对称性。结合根与权系的几何、外尔与 Dynkin 对称性在舒伯特类上的作用，以及舒伯特簇的 Bott–Samelson 解消，我们的方法可统一应用于所有李型的广义格拉斯曼流形。
英文摘要
Generalized Grassmannians form a fundamental class of flag manifolds associated with Lie groups. The purpose of this paper is to classify self-maps of generalized Grassmannians of nonzero degree in terms of their induced actions on cohomology. We prove a rigidity theorem showing that, despite the rich and intricate structure of their cohomology rings, the induced cohomology endomorphisms fall into only two natural types: Adams operations determined by the degree and Dynkin symmetries arising from automorphisms of the Dynkin diagram. Combining the geometry of root and weight systems, the actions of Weyl and Dynkin symmetries on Schubert classes, and Bott--Samelson desingularizations of Schubert varieties, our approach applies uniformly to generalized Grassmannians of all Lie types.
Comments27 pages
