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 | Donagi R. Torus fibrations, gerbes, and duality / Donagi R., Pantev T.; with an appendix by Arinkin D. - Providence, R.I.: American Mathematical Society, 2008. - 90 p. - (Memoirs of the American Mathematical Society; Vol. 193, N 901). - ISBN 0821840924
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Chapter 1. Introduction ......................................... 1
Chapter 2. The Brauer group and the Tate-Shafarevich group ..... 13
1. Brauer groups and  -gerbes ................................ 13
2. Tate-Shafarevich groups and genus one fibrations ............ 28
3. Complementary fibrations .................................... 33
Chapter 3. Smooth genus one fibrations ......................... 43
1. -gerbes .................................................. 43
2. The class of the lifting gerbe .............................. 47
3. The class of the extension gerbe ............................ 49
4. Duality between the lifting and extension presentations ..... 53
Chapter 4. Surfaces ............................................ 63
1. The lifting presentation .................................... 63
2. The extension presentation .................................. 65
3. Duality for gerby genus one fibered surfaces ................ 67
Chapter 5. Modified T-duality and the SYZ conjecture ........... 79
Appendix A. Duality for representations of 1-motives,
by Dmitry Arinkin .............................................. 83
Bibliography ................................................... 87
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