Moment maps on symplectic cones

Abstract

We analyze some convexity properties of the image maps on symplectic cones, similar to the ones obtained by GuilleminSternberg and Atiyah for compact symplectic manifolds in the early 80’s. We prove the image of the moment map associated to the symplectic action of an n-torus on a symplectic cone is a polytopic convex cone in R n : Then, we generalize these results to symplectic manifolds obtained by special perturbations of the symplectic structure of a cone: we obtain sucient (and essentially necessary) conditions for the image of the moment map associated to the perturbed form to remain unchanged. Hamiltonian actions of tori and the images of their moment maps have been intensely studied in the eighties. According to the fundamental result, obtained independently by Atiyah [2] and Guillemin and Sternberg [4], the moment map of a Hamiltonian action of a torus on a compact symplectic manifold has for its image a convex polytope, spanned by the images of the xed points of the action. More recently, Prato [10] proved a convexity result concerning the image of moment maps of torus actions on non-compact symplectic manifolds. Theorem [10]. Let the torus T r act in a Hamiltonian fashion on the symplectic manifold (X;!) and denote by : X !(LieT r ) = R r the corresponding moment map. Suppose that there exists a circle S 1 =fe t 0g T r for some 02 LieT r such that 0 =h; 0iis a proper function having a minimum as its unique critical value. Then (X) is the convex hull of a nite number of rays in (LieT r ). In this paper, we prove a dierent kind of result, closer in spirit to perturbation theory: We start with a special non-compact symplectic manifold, described below, for which a similar convexity theorem holds, and consider the changes of the underlying symplectic structure which keep the image of the resulting moment maps unchanged. Let (X;!) be a symplectic cone with homothety groupft ;t 2 R + gso that t! = t!, for positive t, and compact base X=R + . Suppose that the torus T r acts symplectically on (X;!) and that this action commutes with