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Here is a result whose proof uses Fourier-Mukai duality:

Consider a family of abelian varieties $A \rightarrow X$, its dual $\check{A} \rightarrow X$, and a torsor $\mathcal{T}$ (for $A \rightarrow X$) and a gerbe $\mathcal{G}$ on $\mathcal{T}$.

Then there exists a "dual torsor" $\check{\mathcal{T}} \rightarrow X$ (for $\check{A} \rightarrow X$), and a "dual" gerbe $\check{\mathcal{G}}$ on $\check{\mathcal{T}}$, such that the derived categories of coherent sheaves on $\mathcal{G}$ and $\check{\mathcal{G}}$ are equivalent.

My question is how to define the dual "gerbe" and its dual "torsor". My rough understanding is that the dual "torsor" $\check{\mathcal{T}}$ is defined as follows: a point in $\check{\mathcal{T}}$ corresponds to a splitting of the restriction of the gerbe $\mathcal{G}$ to a fiber of $\mathcal{T} \rightarrow X$. I'm not sure about how to construct the dual "gerbe".

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This is the topic of the paper arxiv.org/abs/math/0306213 of Donagi and Pantev (with appendix by Arinkin). There's also a chart I love in notes of a talk of Arinkin on Fourier-Mukai transform here math.uchicago.edu/~mitya/langlands.html --- it summarizes compactly the various deformations of the Fourier-Mukai transform on an abelian variety, very handy to have around. –  David Ben-Zvi Feb 20 '13 at 1:10
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There are also several papers of Oren Ben-Bassat on the subject. –  Sasha Feb 20 '13 at 5:45
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See also work (in the smooth setting) on T-duality, e.g. work by Bouwknegt, Mathai and collaborators, and Bunke and Schick. –  David Roberts Feb 20 '13 at 7:00
    
Thanks David! I'll have another look at that paper of Donagi & Pantev. –  Vinoth Feb 22 '13 at 15:46
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