Algebraic deformations of toric varieties I. General constructions

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 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

N=2 gauge theories, instanton moduli spaces and geometric representation theory

“…We shall apply this equivalence to the noncommutative toric deformation of the quotient stack [P 2 /Z k ]. For this, we recall the construction of the homogeneous coordinate algebra A = A(P 2 θ ) of the noncommutative projective plane P 2 θ from [33,34,35]. It is the graded polynomial algebra in three generators w i , i = 0, 1, 2 of degree one with the quadratic relations w 0 w i = w i w 0 for i = 1, 2 and w 1 w 2 = q 2 w 2 w 1 .…”

A 4-Sphere With Noncentral Radius and its Instanton Sheaf

“…The study of instantons in terms of bundles and connections with (anti)selfdual curvatures goes back to the late 70's. The prime example is that of SU(2)-instantons on the four-sphere S 4 and their underlying Hopf bundle S 7 → S 4 [1].…”

Hochschild cohomology and deformation quantization of affine toric varieties

“…Consider the toric surface A n given by g(x, y, z) = xy − z n+1 . We would like to express π g in the form (9). We see that it holds π g (x, y) = −(n + 1)z n , π g (z, x) = x and π g (y, z) = y.…”

“…In this case Λ is generated by S 1 := (0, 1), S 2 := (1, 1) and S 3 := (n + 1, n), with the relation S 1 + S 3 = (n + 1)S 2 . We would like to find p of the form (9), such that p = π g . With a simple computation, we see that p is of degree −S 2 :…”