Semitoric integrable systems on symplectic 4-manifolds

Abstract

Let (M, ω) be a symplectic 4-manifold. A semitoric integrable system on (M, ω) is a pair of smooth functions J, H ∈ C ∞ (M, R) for which J generates a Hamiltonian S 1 -action and the Poisson brackets {J, H } vanish. We shall introduce new global symplectic invariants for these systems; some of these invariants encode topological or geometric aspects, while others encode analytical information about the singularities and how they stand with respect to the system. Our goal is to prove that a semitoric system is completely determined by the invariants we introduce.

Semiclassical quantization and spectral limits of ħ-pseudodifferential and Berezin-Toeplitz operators

“…In the -pseudodifferential setting (see, for instance, [17]), the analysis is more complicated due to the possible unboundedness of the operators. Here, we work with a standard class of symbols in R 2n (see [17]) and with the Kohn-Nirenberg-Hörmander class S m 1,0 (T * X) of symbols a(x, ξ, ) on T * X with closed X (see equation (27) for the explicit definition of S m 1,0 (T * X))…”

“…Proposition 17. The -pseudodifferential quantization on X, where X is either R n or a closed manifold equipped with a density, is a semiclassical quantization of T * X in the sense of Definition 4, where A 0 is either the usual Hörmander class of symbols for R 2n , or the Kohn-Nirenberg-Hörmander class S m 1,0 (T * X) in equation (27) when X is a closed manifold, I = (0, 1], the Hilbert space H is L 2 (X) (it is independent of ). If X = R n , then Op (f ) is given by the Weyl quantization.…”

Topology of symplectic torus actions with symplectic orbits

“…Orlik-Raymond's [28,29] and Pao's [31,32] studied smooth actions of 2-tori on compact connected smooth 4-manifolds; Karshon and Tolman classified centered complexity one Hamiltonian torus actions in [18] and also studied Hamiltonian torus actions with 2-dimensional symplectic quotients in [17]; Kogan [20] worked on completely integrable systems with local torus actions; most recently, Pelayo and Vũ Ngo . c [34,35] have studied integrable systems on symplectic 4-manifolds in which one component of the integrable system comes from a Hamiltonian circle action. There are many other papers which relate integrable systems and Hamiltonian torus actions, for instance Duistermaat's paper on global action-angle coordinates [9] and Zung's work on the topology of integrable Hamiltonian systems [42,43].…”

Removing a Ray from a Noncompact Symplectic Manifold

“…In many cases, a diffeomorphism between the bases of two singular Lagrangian fibrations preserving certain structures can be uniquely lifted to a fiberwise symplectomorphism, up to the Hamiltonian flow of momentum maps (the fiberwise translation). The principle, reducing symplectomorphisms between 2n-dimensional manifolds to diffeomorphisms between n-dimensional spaces, has been used in [3,4,7,8,11,12,14,16].…”