Electronic ISBN:  9781470405076 
Product Code:  MEMO/193/901.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 193; 2008; 90 ppMSC: Primary 14;
Let \(X\) be a smooth elliptic fibration over a smooth base \(B\). Under mild assumptions, the authors establish a FourierMukai 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 StromingerYauZaslow version of mirror symmetry, to twisted sheaves, and to noncommutative geometry.

Table of Contents

Chapters

1. Introduction

2. The Brauer group and the TateShafarevich group

3. Smooth genus one fibrations

4. Surfaces

5. Modified $T$duality and the SYZ conjecture


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Let \(X\) be a smooth elliptic fibration over a smooth base \(B\). Under mild assumptions, the authors establish a FourierMukai 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 StromingerYauZaslow version of mirror symmetry, to twisted sheaves, and to noncommutative geometry.

Chapters

1. Introduction

2. The Brauer group and the TateShafarevich group

3. Smooth genus one fibrations

4. Surfaces

5. Modified $T$duality and the SYZ conjecture