Affine structures, torus fibrations and Mirror Symmetry

Starting with a real manifold B whose transition maps are integral affine linear transformations, one can easily construct either a symplectic manifold fibred by tori over B or a complex manifold fibred by tori over B. Furthermore, these fibrations are naturally dual to each other. This exhibits a simple version of Mirror Symmetry as predicted by Strominger, Yau and Zaslow.

The above situation is too simple, though. One does not get interesting examples because the fibrations obtained using this method do not include singular fibres. To get interesting examples --such as Mirror Symmetry for complete intersections in toric varieties-- one should study affine manifolds with singularities. In this case it is much more difficult to construct symplectic or complex manifolds.

In this talk I shall focus on the `symplectic side' of Mirror Symmetry. I will explain a method to construct interesting examples of Lagrangian 3-torus fibrations of compact, simply connected, symplectic 6-manifolds, starting from suitable affine manifolds with singularities.