分层梯度哈密顿向量场与集体可积系统
Stratified Gradient Hamiltonian Vector Fields and Collective Integrable Systems
AI总结：
本文在紧李群对偶上构造推广 Gelfand-Zeitlin 性质的完全可积系统，通过推广环面退化方法至奇异簇，证明分层梯度向量场流的极限存在。
摘要：
我们在任意紧李群 $K$ 的李代数对偶空间上，针对标准李-泊松结构，构造了完全可积系统。这些系统推广了 Gelfand-Zeitlin 系统的关键性质：A) 拉回到任何哈密顿 $K$-流形上，会在一个开稠密子集上定义哈密顿环面作用；B) 若该 $K$-流形是无重数的，则所得环面作用是\textit{完全}可积的；C) 集体矩映射具有凸性和纤维连通性。这些系统通过实极化的几何量子化，推广了 Gelfand-Zeitlin 系统与 Gelfand-Zeitlin 典范基之间的关系。为构造这些系统，我们将 Harada 和 Kaveh 关于光滑射影簇上通过环面退化构造可积系统的构造，推广到奇异拟射影簇。在某些条件下，我们证明此类退化的分层梯度哈密顿向量场（其定义为分段形式）的流具有极限，且该极限定义了连续退化映射。
英文摘要：
We construct completely integrable systems on the dual of the Lie algebra of any compact Lie group $K$ with respect to the standard Lie-Poisson structure. These systems generalize key properties of Gelfand-Zeitlin systems: A) the pullback to any Hamiltonian $K$-manifold defines a Hamiltonian torus action on an open dense subset, B) if the $K$-manifold is multiplicity-free, then the resulting torus action is \textit{completely} integrable, and C) the collective moment map has convexity and fiber connectedness properties. These systems generalize the relationship between Gelfand-Zeitlin systems and Gelfand-Zeitlin canonical bases via geometric quantization by a real polarization. To construct these systems, we generalize Harada and Kaveh's construction of integrable systems by toric degeneration on smooth projective varieties to singular quasi-projective varieties. Under certain conditions, we show that the stratified-gradient Hamiltonian vector field of such a degeneration, which is defined piece-wise, has a flow whose limit exists and defines continuous degeneration map.
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