Hamiltonian classification of toric fibres and symmetric probes

Brendel, Joé¹

¹Department of Mathematics, ETH Zürich, Zürich, Switzerland

Algebraic and Geometric Topology, Tome 25 (2025) no. 3, pp. 1839-1876

Résumé

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, ℂP2, ℂ × S2, ℂ2 × T∗S1, T∗S1 × S2 and monotone S2 × 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.

Détail

DOI : 10.2140/agt.2025.25.1839

Keywords: symplectic geometry, symplectic topology, toric geometry, Hamiltonian group actions, Hamiltonian torus actions, probes, symmetric probes, toric fibres, classification of toric fibres, Hamiltonian monodromy

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