A Convexity Theorem and Reduced Delzant Spaces

Details A Convexity Theorem and Reduced Delzant Spaces A Convexity Theorem and Reduced Delzant Spaces

2005-09-19

arxiv.org/abs/math/0509429v2

Mathematics

Differential Geometry

43 pages; a few corrections made

Scientific paper

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 $2dim T$ 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.

Affiliated with

Also associated with

No associations

Rating

A Convexity Theorem and Reduced Delzant Spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with A Convexity Theorem and Reduced Delzant Spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A Convexity Theorem and Reduced Delzant Spaces will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-298365