Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

From compact semi-toric systems to Hamiltonian $S^1$-spaces

Pages: 247 - 281, Volume 35, Issue 1, January 2015      doi:10.3934/dcds.2015.35.247

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Sonja Hohloch - Section de Mathématiques, EPFL, SB MATHGEOM CAG, Station 8, 1015 Lausanne, Switzerland (email)
Silvia Sabatini - CAMGSD, Instituto Superior Técnico, Av. Rovisco Pais, Lisboa, 1049-001, Portugal (email)
Daniele Sepe - Instituto de Matemática, Universidade Federal Fluminense, Rua Mário Santos Braga, 24020-240 Niteroi, RJ, Brazil (email)

Abstract: We show how any labeled convex polygon associated to a compact semi-toric system, as defined by Vũ ngọc, determines Karshon's labeled directed graph which classifies the underlying Hamiltonian $S^1$-space up to isomorphism. Then we characterize adaptable compact semi-toric systems, i.e. those whose underlying Hamiltonian $S^1$-action can be extended to an effective Hamiltonian $\mathbb{T}^2$-action, as those which have at least one associated convex polygon which satisfies the Delzant condition.

Keywords:  Hamiltonian torus actions, semi-toric systems, symplectic toric manifolds, integrable Hamiltonian systems, symplectic invariants.
Mathematics Subject Classification:  Primary: 37J05, 37J35, 53D20.

Received: December 2013;      Revised: March 2014;      Available Online: August 2014.