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24 June 2024 

   

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Hamiltonian Classification of toric fibres and symmetric probes  Joé Brendel
;  Date: 
1 Feb 2023  Abstract:  In 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  Services:  Forum  Review  PDF  Favorites 


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