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Convexity of multi-valued momentum map

Andrea Giacobbe
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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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Andrea Giacobbe. 2000-04-10. Convexity of multi-valued momentum map. https://arxiv.org/abs/math/0004061

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