Turkish Journal of Mathematics




We outline work in progress suggesting an algebro-geometric version of the Strominger-Yau-Zaslow conjecture. We define the notion of a toric degeneration, a special case of a maximally unipotent degeneration of Calabi-Yau manifolds. We then show how in this case the dual intersection complex has a natural structure of an affine manifold with singularities. If the degeneration is polarized, we also obtain an intersection complex, also an affine manifold with singularities, related by a discrete Legendre transform to the dual intersection complex. Finally, we introduce log structures as a way of reversing this construction: given an affine manifold with singularities with a suitable polyhedral decomposition, we can produce a degenerate Calabi-Yau variety along with a log structure. Hopefully, in interesting cases, this object will have a well-behaved deformation theory, allowing us to use the discrete Legendre transform to construct mirror pairs of Calabi-Yau manifolds. We also connect this approach to the topological form of the Strominger-Yau-Zaslow conjecture.

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