论文标题
奇异符号歧管上的集成系统：从本地到全局
Integrable systems on singular symplectic manifolds: From local to global
论文作者
论文摘要
在本文中，我们考虑了具有命令一的奇异符号结构的流形上的可集成系统。这些结构是从超出表面的符合性的，其中符号体积以横向方式（一个命令的奇异性）变为无穷大或零，这是$ b $ sym-Symplectic形式或折叠的符号形式。形式退化的高度表面称为关键集。我们通过证明用于折叠的符号符合性集成系统的动作 - 角度定理来对这些结构的作用角度坐标的存在进行新的冲动。与期望相反，折叠式折射率的动作角度坐标定理不能以[km]中的symplectic和$ b $ symplectic形式作为旋转型提升。提供了可集成系统的全球结构，并在两种情况下都研究了全球动作角度坐标存在的障碍。新的拓扑障碍物发现，奇异符号歧管的关键集合$ z $的拓扑结构。这些障碍的存在反过来意味着在$ z $上存在综合系统的奇异性。
In this article we consider integrable systems on manifolds endowed with singular symplectic structures of order one. These structures are symplectic away from an hypersurface where the symplectic volume goes either to infinity or to zero in a transversal way (singularity of order one) resulting either in a $b$-symplectic form or a folded symplectic form. The hypersurface where the form degenerates is called critical set. We give a new impulse to the investigation of the existence of action-angle coordinates for these structures initiated in [KM] and [KMS] by proving an action-angle theorem for folded symplectic integrable systems. Contrary to expectations, the action-angle coordinate theorem for folded symplectic manifolds cannot be presented as a cotangent lift as done for symplectic and $b$-symplectic forms in [KM]. Global constructions of integrable systems are provided and obstructions for global existence of action-angle coordinates are investigated in both scenarios. The new topological obstructions found emanate from the topology of the critical set $Z$ of the singular symplectic manifold. The existence of these obstructions in turn implies the existence of singularities for the integrable system on $Z$.
