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.

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Correspondence to Alvaro Pelayo.

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A. Pelayo was partially supported by an NSF Postdoctoral Fellowship.

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Pelayo, A., Vũ Ngọc, S. Semitoric integrable systems on symplectic 4-manifolds. Invent. math. 177, 571–597 (2009). https://doi.org/10.1007/s00222-009-0190-x

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Keywords

  • Local Diffeomorphism
  • Singular Foliation
  • Symplectic Invariant
  • Elliptic Singularity
  • Equivariant Normal Form