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 http://arXiv.org/abs/math:SG/9904179, and that the
induced symplectic structure is compatible with the complex one, thus defining
on M the structure of a Kaehler 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.Comment: LaTeX, 19 pages, some changes, final version to appear in Intern.
Math. Res. Notice