February 7, 2007

Bailin Song (Brandies)

Title: A Convexity Theorem and Reduced Delzant Spaces

Abstract: 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 image of the T-moment map for such a Hamiltonian T-space is convex, even when it is singular. This is a generalization of the convexity theorem of Atiyah and Guillemin-Sternberg. We also proved that if, furthermore, the T-space has dimension 2 dim 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.