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24 June 2024
 
  » arxiv » 2302.00334

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Hamiltonian Classification of toric fibres and symmetric probes
Joé Brendel ;
Date 1 Feb 2023
AbstractIn a toric symplectic manifold, regular fibres of the moment map are Lagrangian tori which are called toric fibres. We discuss the question which two toric fibres are equivalent up to a Hamiltonian diffeomorphism of the ambient space. On the construction side of this question, we introduce a new method of constructing equivalences of toric fibres by using a symmetric version of McDuff’s probes (see arXiv:0904.1686 and arXiv:1203.1074). On the other hand, we derive some obstructions to such equivalence by using Chekanov’s classification of product tori together with a lifting trick from toric geometry. Furthermore, we conjecture that (iterated) symmetric probes yield all possible equivalences and prove this conjecture for $mathbb{C}^n,mathbb{C}P^2, mathbb{C} imes S^2, mathbb{C}^2 imes T^*S^1, T^*S^1 imes S^2$ and monotone $S^2 imes S^2$.
This problem is intimately related to determining the Hamiltonian monodromy group of toric fibres, i.e. determining which automorphisms of the homology of the toric fibre can be realized by a Hamiltonian diffeomorphism mapping the toric fibre in question to itself. For the above list of examples, we determine the Hamiltonian monodromy group for all toric fibres.
Source arXiv, 2302.00334
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