Convexity of multi-valued momentum map

2000/04/10 by Andrea Giacobbe, Giacobbe, Andrea
Mathematics · #37J #53D #FOS: Mathematics #Symplectic Geometry (math.SG) #math.SG #msc:37J #msc:53D

paper · pdf · doi:10.48550/arxiv.math/0004061

21 pages, 2 pictures

arxiv created 2000/04/10 · arxiv updated 2009/11/30

Abstract

We extend the famous convexity theorem of Atiyah, Guillemin and Sternberg to the case of non-Hamiltonian actions. We show that the image of a generalized momentum map is a bounded polytope times a vector space. We prove that this picture is stable for small perturbations of the symplectic form. We also observe that an n-dimensional torus symplectic action on a 2n-dimensional symplectic manifold, with fixed point, is Hamiltonian. We finally prove that an extension of Kirwan's convexity theorem (for compact group actions) is also true.

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