Hamiltonian classification of toric fibres and symmetric probes

Brendel, Joé

doi:10.2140/agt.2025.25.1839

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Hamiltonian classification of toric fibres and symmetric probes

Joé Brendel

Algebraic & Geometric Topology 25 (2025) 1839–1876

DOI: 10.2140/agt.2025.25.1839

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 $ℂ^{n}$ , $ℂ P^{2}$ , $ℂ \times S^{2}$ , $ℂ^{2} \times T^{*} S^{1}$ , $T^{*} S^{1} \times S^{2}$ and monotone $S^{2} \times S^{2}$ .

This problem is intimately related to determining the Hamiltonian monodromy group of toric fibres, 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.

Keywords

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

Mathematical Subject Classification

Primary: 53D12

Secondary: 53D20

References

Publication

Received: 10 July 2023

Accepted: 19 February 2024

Published: 20 June 2025

Authors

Joé Brendel
Department of Mathematics ETH Zürich Zürich Switzerland

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Volume 25, issue 3 (2025)