论文标题
奇异符号歧管的阿诺德猜想
The Arnold conjecture for singular symplectic manifolds
论文作者
论文摘要
在本文中，我们研究了有关单数符号歧管的哈密顿动力学，并证明了大型$ b^m $ $ symplectic歧管的Arnold猜想。在某些轻度条件下，引入了新型技术，将平滑的符号形式与原始的奇异象征结构相关联。这些技术在多种情况下产生了Arnold猜想对单数符号歧管的有效性。更确切地说，我们证明了$ b^{2m} $ - 仅取决于歧管的拓扑的1个周期的哈密顿轨道的下限。此外，对于$ b^m $ -symplectic表面，我们根据对$（m，z）$的拓扑改善下限。然后，我们冒险研究这个奇异领域的浮子同源性研究，并以一系列开放的问题结束。
In this article, we study the Hamiltonian dynamics on singular symplectic manifolds and prove the Arnold conjecture for a large class of $b^m$-symplectic manifolds. Novel techniques are introduced to associate smooth symplectic forms to the original singular symplectic structure, under some mild conditions. These techniques yield the validity of the Arnold conjecture for singular symplectic manifolds across multiple scenarios. More precisely, we prove a lower bound on the number 1-periodic Hamiltonian orbits for $b^{2m}$-symplectic manifolds depending only on the topology of the manifold. Moreover, for $b^m$-symplectic surfaces, we improve the lower bound depending on the topology of the pair $(M,Z)$. We then venture into the study of Floer homology to this singular realm and we conclude with a list of open questions.
