The complex and Ricci-flat structure conjecture for the dual torus family

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Let YY be a Calabi--Yau manifold with a special Lagrangian TnT^n-fibration, and let MD(Tn,Y){\cal M}_D(T^n,Y) be the family of dual tori over MsL(Tn,Y){\cal M}_{sL}(T^n,Y). The dual torus family conjecture. The family MD(Tn,Y){\cal M}_D(T^n,Y) admits complex structures and Ricci-flat Kähler metrics. In particular, it has a nowhere vanishing holomorphic nn-form. This is presented as the most accessible part of the preceding SYZ compactification conjectures, while the source does not establish it.

References

Primary source

David R. Morrison, “The Geometry Underlying Mirror Symmetry”, arXiv:alg-geom/9608006 (1997).

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