Mirror symmetry for the Tate curve via tropical and log corals

Log Mirror Symmetry and Local Mirror Symmetry

We study Mirror Symmetry of log Calabi-Yau surfaces. On one hand, we consider the number of “affine lines” of each degree in P\B, where B is a smooth cubic. On the other hand, we consider coefficients of a certain expansion of a function obtained from the integrals of dx/x ∧ dy/y over 2chains whose boundaries lie on Bφ, where {Bφ} is a family of smooth cubics. Then, for small degrees, they coin...

Mirror Symmetry : The Elliptic Curve

We describe an isomorphism of categories conjectured by Kontsevich. If M and M are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on M and a suitable version of Fukaya's category of Lagrangian submanifolds on M. We prove this equivalence when M is an elliptic curve and M is its dual curve, exhibiting the dictionary in detail. Mirror symmetry – ...

Mirror Symmetry for P and Tropical Geometry

In [13, 14, 15, 16], Bernd Siebert and myself have been working on a program designed to understand mirror symmetry via an algebro-geometric analogue to the Strominger-YauZaslow program [29]. The basic idea is that the controlling objects in mirror symmetry are integral affine manifolds with singularities. One can view an integral affine manifold as producing a mirror pair of manifolds, one a s...

Affine Manifolds, Log Structures, and Mirror Symmetry

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 a...

Manifolds , Log Structures , and Mirror Symmetry