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Article Dans Une Revue International Mathematics Research Notices Année : 2001

Generalized toric varieties for simple non-rational convex polytopes

Résumé

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.

Dates et versions

hal-00019448 , version 1 (21-02-2006)

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Fiammetta Battaglia, Elisa Prato. Generalized toric varieties for simple non-rational convex polytopes. International Mathematics Research Notices, 2001, 24, pp.1315-1373. ⟨hal-00019448⟩
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