Generalized Toric Varieties for Simple Nonrational Convex Polytopes

We call complex quasifold of dimension k a space that is locally isomorphic to the quotient of an open subset of the space C^k by the holomorphic action of a discrete group; the analogue of a complex torus in this setting is called a complex quasitorus. We associate to each simple polytope, rational or not, a family of complex quasifolds having same dimension as the polytope, each containing a dense open orbit for the action of a suitable complex quasitorus. We show that each of these spaces M is diffeomorphic to one of the symplectic quasifolds defined in [P], and that the induced symplectic structure is compatible with the complex one, thus defining on M the structure of a Kähler quasifold. These spaces may be viewed as a generalization of the toric varieties that are usually associated to those simple convex polytopes that are rational. [P] E. Prato, Simple Non-Rational Convex Polytopes via Symplectic Geometry, Topology 40 (2001), 961-975