Reduced Delzant spaces and a convexity theorem

Bong H. Lian Brandeis University Bailin Song University of California Los Angeles

Differential Geometry mathscidoc:1608.10076

Topology, 46, (6), 554–576, 2007
The convexity theorem of Atiyah and Guillemin-Sternberg says that any connected compact manifold with Hamiltonian torus action has a moment map whose image is the convex hull of the image of the fixed point set. Sjamaar-Lerman proved that the Marsden-Weinstein reduction of a connected Hamitonian G -manifold is a stratified symplectic space. Suppose $1\ra A\ra G\ra T\ra 1$ is an exact sequence of compact Lie groups and T is a torus. Then the reduction of a Hamiltonian G -manifold with respect to A yields a Hamiltonian T -space. We show that if the A -moment map is proper, then the convexity theorem holds for such a Hamiltonian T -space, even when it is singular. We also prove that if, furthermore, the T -space has dimension 2dimT and T acts effectively, then the moment polytope is sufficient to essentially distinguish their homeomorphism type, though not their diffeomorphism types. This generalizes a theorem of Delzant in the smooth case.
Reduced Delzant Spaces, Hamiltonian torus action, Marsden-Weinstein reduction, connected Hamitonian G -manifold
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  title={Reduced Delzant spaces and a convexity theorem},
  author={Bong H. Lian, and Bailin Song},
Bong H. Lian, and Bailin Song. Reduced Delzant spaces and a convexity theorem. 2007. Vol. 46. In Topology. pp.554–576.
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