Generalized Toric Varieties for Simple Non-Rational 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. Introduction Consider a vector space d of dimension n. To each simple convex polytope ∆ ⊂ d∗ that is rational with respect to a lattice in d there corresponds a toric variety with at worst quotient singularities. What happens in the case that the simple convex polytope is no longer rational? To answer this question we consider a special class of spaces, called quasifolds, which were first introduced by one of the authors in [P]. A quasifold is not necessarily a Hausdorff space: it is locally modeled by orbit spaces of the action of discrete, possibly infinite, groups on open subsets of R. A quasitorus, on the other hand, is the natural replacement of a torus in this geometry. In this article we define the notions of complex quasifold and complex quasitorus and we associate to each simple convex polytope ∆ ⊂ d∗ a family of compact complex quasifolds of dimension n, each endowed with the holomorphic action of a complex quasitorus DC having a dense open orbit. Our construction is explicit: each space M is the topological quotient of a suitable open subset of C by the action of a suitable subgroup NC ⊂ T d C , and DC is isomorphic to T d C/NC, d being the number of facets of the polytope. We show that M is a complex quasifold by covering it with mutually compatible local models of the type C modulo the action of a discrete group. If the polytope is rational our procedure matches the standard one for constructing toric varieties as quotients (see Chapter VI in [A] or Appendix 1 in [G]).