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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 Click above image for expanded view 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.

• 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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