Torus fibrations, gerbes, and duality

Details Torus fibrations, gerbes, and duality Torus fibrations, gerbes, and duality

2003-06-13

arxiv.org/abs/math/0306213v2

Mathematics

Algebraic Geometry

74 pages, LaTeX 2e, with an appendix by D.Arinkin, minor corrections

Scientific paper

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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