Convexity of Multi-valued Momentum Maps

Abstract

A famous theorem of Atiyah, Guillemin and Sternberg states that, given a Hamiltonian torus action, the image of the momentum map is a convex polytope. We prove that this result can be extended to the case in which the action is non-Hamiltonian. Our generalization of the theorem states that, given a symplectic torus action, the momentum map can be defined on an appropriate covering of the manifold and its image is the product of a convex polytope along a rational subspace times the orthogonal vector space. We also prove that this decomposition in direct product is stable under small equivariant perturbations of the symplectic structure; this, in particular, means that the property of being Hamiltonian is locally stable. The technique developed allows us to extend the result to any compact group action and also to deduce that any symplectic n-torus action, with fixed points, on a compact 2n-dimensional manifold, is Hamiltonian. (2000). 37Jxx, 37Kxx. Mathematics Subject Classifications

Hamiltonian and symplectic symmetries: An introduction

“…Other convexity theorems were proven later by Birtea-Ortega-Ratiu [20], Kirwan [93] (in the case of compact, non-abelian group actions), Benoist [15], and Giacobbe [58], to name a few. Convexity in the case of Poisson actions has been studied by Alekseev, Flaschka-Ratiu, Ortega-Ratiu and Weinstein [4,47,118,158] among others.…”

“…Theorem 3.22 (Giacobbe [58]). A symplectic action of a n-torus on a compact connected symplectic 2n-manifold with fixed points must be Hamiltonian.…”

Remarks on Symplectic Geometry

“…Theorem 5.5. (Giacobbe [26]) An effective symplectic action of an n-dimensional torus on a compact connected symplectic 2n-dimensional manifold with some fixed point must be Hamiltonian.…”

Generalization of Hamiltonian Monodromy: Quantum Manifestations

“…The resulting fractional monodromy was first introduced in [36,37] for a problem of nonlinearly coupled resonant oscillators and was illustrated immediately on quantum example by the evolution of a multiple (double) cell along a closed path crossing once the singular stratum. Much more detailed analysis of the fractional monodromy is given in several recent publications [37][38][39][40].…”