Algebraic deformations of toric varieties I. General constructions

Details Algebraic deformations of toric varieties I. General constructions Algebraic deformations of toric varieties I. General constructions

2010-01-08

arxiv.org/abs/1001.1242v2

Mathematics

Quantum Algebra

58 pages; v2: Presentation of Grassmann and flag varieties improved, minor corrections

Scientific paper

We construct and study noncommutative deformations of toric varieties by combining techniques from toric geometry, isospectral deformations, and noncommutative geometry in braided monoidal categories. Our approach utilizes the same fan structure of the variety but deforms the underlying embedded algebraic torus. We develop a sheaf theory using techniques from noncommutative algebraic geometry. The cases of projective varieties are studied in detail, and several explicit examples are worked out, including new noncommutative deformations of Grassmann and flag varieties. Our constructions set up the basic ingredients for thorough study of instantons on noncommutative toric varieties, which will be the subject of the sequel to this paper.

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