Heriot-Watt Mathematics Report Series
HWM09-14, 10 Dec 2009
Algebraic deformations of toric varieties I. General constructions
L Cirio, G Landi and R J Szabo
Abstract
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 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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