Non-commutative tori and Fourier-Mukai duality

Abstract

The classical Fourier-Mukai duality establishes an equivalence of categories between the derived categories of sheaves on dual complex tori. In this article we show that this equivalence extends to an equivalence between two dual objects. Both of these are generalized deformations of the complex tori. In one case, a complex torus is deformed formally in a non-commutative direction specified by a holomorphic Poisson structure. In the other, the dual complex torus is deformed in a B-field direction to a formal gerbe. We show these two deformations are Fourier-Mukai equivalent.

80 pages, LaTeX2e

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Cited by in corpus (20)

• Deformation quantization modules
• Duality and equivalence of module categories in noncommutative geometry I
• T-duality with H-flux: non-commutativity, T-folds and G x G structure
• Dagger Geometry As Banach Algebraic Geometry
• Infinitesimal Derived Torelli Theorem for K3 surfaces
• Deformation quantization modules I:Finiteness and duality
• Duality and equivalence of module categories in noncommutative geometry II: Mukai duality for holomorphic noncommutative tori
• Automorphisms and autoequivalences of generic analytic K3 surfaces
• Moduli Stacks of Bundles on Local Surfaces
• Hodge modules on complex tori and generic vanishing for compact Kähler manifolds
• Milnor descent for cohesive dg-categories
• Gerbes and the Holomorphic Brauer Group of Complex Tori
• Mirror duality and noncommutative tori
• Deformations of vector bundles on coisotropic subvarieties via the Atiyah class
• Fourier-Mukai transforms, mirror symmetry, and generalized K3 surfaces
• Equivariant Gerbes on Complex Tori
• Cohomology Groups of Deformations of Line Bundles on Complex Tori
• DG-resolutions of NC-smooth thickenings and NC-Fourier-Mukai transforms
• Integral representation theorems for DQ-modules
• Deformations of categories of coherent sheaves via quivers with relations