极大短复杂度1空间的骨架与环面扩张
Skeletons and Toric Extensions of Maximally Short Complexity One Spaces
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中文总结 AI 辅助
本文研究极大短复杂度1空间的骨架，证明其矩像连通并给出相关事实的初等证明，还利用该结果估计了对应辛环面流形的数量。
中文摘要 AI 辅助
复杂度1的T-空间是满足$\frac{1}{2}\text{dim}\thinspace M - \text{dim}\thinspace T=1$的哈密顿T-空间$(M,\boldsymbol{\theta},\boldsymbol{\theta})$。复杂度1的T-空间的骨架是分类中的重要不变量，编码非通用轨道的信息。本文证明：紧致连通极大短复杂度1的T-空间（其实际为GKM空间）的骨架的矩像连通，该证明依赖“恰当矩映射的正则值的每个连通分支是凸局部多面体集”这一熟知事实，过程中还给出了该事实的初等证明。随后利用连通性结果，估计了辛环面$(T \times S^1)$-流形的数量，这类流形的底层复杂度1的T-空间与给定的极大短复杂度1的T-空间相同。
英文摘要
Complexity one $T$-spaces are Hamiltonian $T$-spaces $(M,ω,Φ)$ such that $\frac{1}{2}\dim M -\dim T=1$. The skeleton of a complexity one $T$-space is an important invariant in the classification and encodes the information about non-generic orbits. In this paper, we prove that the moment image of the skeleton of a compact, connected maximally short complexity one $T$-space, which is in fact a GKM space, is connected. The proof relies on the well-known fact that each connected component of regular values of a proper moment map is a convex locally polyhedral set. We also gave an elementary proof of that fact along the way. Then we use the connectedness result to estimate the number of symplectic toric $(T \times S^1)$-manifolds whose underlying complexity one $T$-space is the same as the given maximally short complexity one $T$-space.
