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Non-commutative tori and Fourier–Mukai duality

Published online by Cambridge University Press:  26 March 2007

O. Ben-Bassat ,
J. Block  and
T. Pantev
O. Ben-Bassat
Affiliation:
Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104-6395, USA orenb@math.upenn.edu
J. Block
Affiliation:
Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104-6395, USA blockj@math.upenn.edu
T. Pantev
Affiliation:
Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104-6395, USA tpantev@math.upenn.edu
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Abstract

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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 that these two deformations are Fourier–Mukai equivalent.

Keywords

deformation quantizationgerbeFourier–Mukai transformanalytic sheaves14A2253D5514A2058B34
Type
Research Article
Information
Compositio Mathematica , Volume 143 , Issue 2 , March 2007 , pp. 423 - 475
Copyright
Foundation Compositio Mathematica 2007