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Torus Fibrations, Gerbes, and Duality
Ron Donagi and Tony Pantev, University of Pennsylvania, Philadelphia, PA
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Memoirs of the American Mathematical Society
2008; 90 pp; softcover
Volume: 193
ISBN-10: 0-8218-4092-4
ISBN-13: 978-0-8218-4092-4
List Price: US$64
Individual Members: US$38.40
Institutional Members: US$51.20
Order Code: MEMO/193/901
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Let \(X\) be a smooth elliptic fibration over a smooth base \(B\). Under mild assumptions, the authors establish a Fourier-Mukai equivalence between the derived categories of two objects, each of which is an \(\mathcal{O}^{\times}\) gerbe over a genus one fibration which is a twisted form of \(X\). The roles of the gerbe and the twist are interchanged by the authors' duality. The authors state a general conjecture extending this to allow singular fibers, and they prove the conjecture when \(X\) is a surface. The duality extends to an action of the full modular group. This duality is related to the Strominger-Yau-Zaslow version of mirror symmetry, to twisted sheaves, and to non-commutative geometry.

Table of Contents

  • Introduction
  • The Brauer group and the Tate-Shafarevich group
  • Smooth genus one fibrations
  • Surfaces
  • Modified \(T\)-duality and the SYZ conjecture
  • Appendix A. Duality for representations of \(1\)-motives, by Dmitry Arinkin
  • Bibliography
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