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Torus fibrations, gerbes, and duality

Ron Donagi·Tony Pantev
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Abstract

Let X be a smooth elliptic fibration over a smooth base B. Under mild assumptions, we establish a Fourier-Mukai equivalence between the derived categories of two objects, each of which is an O^* gerbe over a genus one fibration which is a twisted form of X. The roles of the gerbe and the twist are interchanged by our duality. We state a general conjecture extending this to allow singular fibers, and we 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.

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Ron Donagi, Tony Pantev. 2004-11-30. Torus fibrations, gerbes, and duality. https://arxiv.org/abs/math/0306213

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