The Arnold conjecture for singular symplectic manifolds

Abstract

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.

arXiv commentoverall improvement of the article, new results included, 50 pages, 7 figures, typos corrected, final version to appear at the Journal of Fixed Point Theory and Applications

References in corpus (14)

• Symplectic and Poisson geometry on b-manifolds
• Toric actions on b-symplectic manifolds
• Examples of integrable and non-integrable systems on singular symplectic manifolds
• Cotangent models for integrable systems
• Deformations of log-symplectic structures
• Action-angle variables and a KAM theorem for b-Poisson manifolds
• On geometric quantization of $b$-symplectic manifolds
• Desingularizing $b^m$-symplectic structures
• Examples and counter-examples of log-symplectic manifolds
• Convexity of the moment map image for torus actions on $b^m$-symplectic manifolds
• The geometry of E-manifolds
• Reduction theory for singular symplectic manifolds and singular forms on moduli spaces
• Bounds for tentacular Hamiltonians
• The Sigma Form for the PII Hierarchy