Hamiltonian Classification of toric fibres and symmetric probes
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916818518540288 |
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| author | Brendel, Joé |
| author_facet | Brendel, Joé |
| contents | 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 $\mathbb{C}^n,\mathbb{C}P^2, \mathbb{C} \times S^2, \mathbb{C}^2 \times T^*S^1, T^*S^1 \times S^2$ and monotone $S^2 \times S^2$.
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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_00334 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hamiltonian Classification of toric fibres and symmetric probes Brendel, Joé Symplectic Geometry Primary 53D12, Secondary 53D20 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 $\mathbb{C}^n,\mathbb{C}P^2, \mathbb{C} \times S^2, \mathbb{C}^2 \times T^*S^1, T^*S^1 \times S^2$ and monotone $S^2 \times S^2$. 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. |
| title | Hamiltonian Classification of toric fibres and symmetric probes |
| topic | Symplectic Geometry Primary 53D12, Secondary 53D20 |
| url | https://arxiv.org/abs/2302.00334 |