A Convexity Theorem and Reduced Delzant Spaces
نویسندگان
چکیده
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 → A → G → T → 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 2dim T 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. 2 Bong H. Lian & Bailin Song
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