Generalized toric varieties for simple non-rational convex polytopes

Details Generalized toric varieties for simple non-rational convex polytopes Generalized toric varieties for simple non-rational convex polytopes

2000-04-11

arxiv.org/abs/math/0004066v2

Mathematics

Complex Variables

LaTeX, 19 pages, some changes, final version to appear in Intern. Math. Res. Notices

Scientific paper

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.

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