Hamiltonian Classification of toric fibres and symmetric probes

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 ${\mathbb{C}}^n,\mathbb{C}{P}^2,\mathbb{C}\times S^2,{\mathbb{C}}^2\times T^{\ast }{S}^1,T^{\ast }{S}^1\times S^2$ 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.

39 pages, 7 figures

References in corpus (1)

• Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle