Convex Polytopes and Quasilattices from the Symplectic Viewpoint

Abstract. We construct, for each convex polytope, possibly nonrational and nonsimple, a family of compact spaces that are stratified by quasifolds, i.e. each of these spaces is a collection of quasifolds glued together in an suitable way. A quasifold is a space locally modelled on R^k modulo the action of a discrete, possibly infinite, group. The way strata are glued to each other also involves the action of an (infinite) discrete group. Each stratified space is endowed with a symplectic structure and a moment mapping having the property that its image gives the original polytope back. These spaces may be viewed as a natural generalization of symplectic toric varieties to the nonrational setting. If we restrict to the rational case we obtain, from the point of view of singularities, what is expected from the classical theory of toric varieties, and in addition we gain further insight on the symplectic geometry of them; it should also be noticed that these spaces provide a wide range of explicit examples of symplectic stratified spaces. Communicated by A. Connes. Partially supported by GNSAGA (CNR). Received: 5 May 2005/Accepted: 5 May 2006. Published online: 4 November 2006.