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Torus Fibrations, Gerbes, and Duality
 
Ron Donagi University of Pennsylvania, Philadelphia, PA
Tony Pantev University of Pennsylvania, Philadelphia, PA
Front Cover for Torus Fibrations, Gerbes, and Duality
Available Formats:
Electronic ISBN: 978-1-4704-0507-6
Product Code: MEMO/193/901.E
List Price: $68.00
MAA Member Price: $61.20
AMS Member Price: $40.80
Front Cover for Torus Fibrations, Gerbes, and Duality
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  • Front Cover for Torus Fibrations, Gerbes, and Duality
  • Back Cover for Torus Fibrations, Gerbes, and Duality
Torus Fibrations, Gerbes, and Duality
Ron Donagi University of Pennsylvania, Philadelphia, PA
Tony Pantev University of Pennsylvania, Philadelphia, PA
Available Formats:
Electronic ISBN:  978-1-4704-0507-6
Product Code:  MEMO/193/901.E
List Price: $68.00
MAA Member Price: $61.20
AMS Member Price: $40.80
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1932008; 90 pp
    MSC: Primary 14;

    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
     
     
    • Chapters
    • 1. Introduction
    • 2. The Brauer group and the Tate-Shafarevich group
    • 3. Smooth genus one fibrations
    • 4. Surfaces
    • 5. Modified $T$-duality and the SYZ conjecture
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Volume: 1932008; 90 pp
MSC: Primary 14;

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.

  • Chapters
  • 1. Introduction
  • 2. The Brauer group and the Tate-Shafarevich group
  • 3. Smooth genus one fibrations
  • 4. Surfaces
  • 5. Modified $T$-duality and the SYZ conjecture
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