**Memoirs of the American Mathematical Society**

2008;
90 pp;
Softcover

MSC: Primary 14;

Print ISBN: 978-0-8218-4092-4

Product Code: MEMO/193/901

List Price: $68.00

Individual Member Price: $40.80

**Electronic ISBN: 978-1-4704-0507-6
Product Code: MEMO/193/901.E**

List Price: $68.00

Individual Member Price: $40.80

# Torus Fibrations, Gerbes, and Duality

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*Ron Donagi; Tony Pantev*

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

# Table of Contents

## Torus Fibrations, Gerbes, and Duality

- Contents v6 free
- Chapter 1. Introduction 18 free
- Chapter 2. The Brauer group and the Tate-Shafarevich group 1320 free
- Chapter 3. Smooth genus one fibrations 4350
- Chapter 4. Surfaces 6370
- Chapter 5. Modified T-duality and the SYZ conjecture 7986
- Appendix A. Duality for representations of 1-motives, by Dmitry Arinkin 8390
- Bibliography 8794