Hamiltonian classification of toric fibres and symmetric probes
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2025
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Journal Article
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Abstract
In a toric symplectic manifold, regular fibres of the moment map are Lagrangian tori which are called toric fibres. We discuss the question of 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. 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 Cn, C P 2, C x S2, C2 x T*S1, T*S1 x S2 and monotone S2 x S2. This problem is intimately related to determining the Hamiltonian monodromy group of toric fibres, ie 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.
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25 (3)
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1839 - 1876
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Mathematical Sciences Publishers
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03839 - Biran, Paul I. / Biran, Paul I.