Reduction theory for singular symplectic manifolds and singular forms on moduli spaces

Matveeva, Anastasia; Miranda, Eva

doi:10.1016/j.aim.2023.109161

Advances in Mathematics

2023

DOI: 10.1016/j.aim.2023.109161

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Reduction theory for singular symplectic manifolds and singular forms on moduli spaces

Anastasia Matveeva

Eva Miranda

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Cited by 2 publications

References 47 publications

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The Arnold conjecture for singular symplectic manifolds

Brugués,

Miranda,

Oms

2024

J. Fixed Point Theory Appl.

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In this article, we study the Hamiltonian dynamics on singular symplectic manifolds and prove the Arnold conjecture for a large class of $$b^m$$ 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 of 1-periodic Hamiltonian orbits for $$b^{2m}$$ b 2 m -symplectic manifolds depending only on the topology of the manifold. Moreover, for $$b^m$$ 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.

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The Arnold conjecture for singular symplectic manifolds

Brugués,

Miranda,

Oms

2024

J. Fixed Point Theory Appl.

Self Cite

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Constructions of b-semitoric systems

Joaquim

Hohloch

Mir³

et al. 2023

Journal of Mathematical Physics

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In this article, we introduce b-semitoric systems as a generalization of semitoric systems, specifically tailored for b-symplectic manifolds. The objective of this article is to furnish a collection of examples and investigate the distinctive characteristics of these systems. A b-semitoric system is a four-dimensional b-integrable system that satisfies certain conditions: one of its momentum map components is proper and generates an effective global S1-action and all singular points are non-degenerate and devoid of hyperbolic components. To illustrate this concept, we provide five examples of b-semitoric systems by modifying the coupled spin oscillator and the coupled angular momenta, and we also classify their singular points. Additionally, we describe the dynamics of these systems through the image of their respective momentum maps.

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Reduction theory for singular symplectic manifolds and singular forms on moduli spaces

Cited by 2 publications

References 47 publications

The Arnold conjecture for singular symplectic manifolds

The Arnold conjecture for singular symplectic manifolds

Constructions of b-semitoric systems

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