INTRODUCTION xiii several points of view which appear to be yielding new insights into mirror sym- metry: notably, the rigid analytic program initiated by Kontsevich and Soibelman in [69, 70] and the program developed by Siebert and myself using log geometry, [47, 48, 51, 49]. These ideas which grew out of the SYZ conjecture focus on the base of the SYZ fibration even though we do not know an SYZ fibration exists, we have a good guess as to what these bases look like. In particular, they should be aﬃne manifolds, i.e., real manifolds with an atlas whose transition maps are aﬃne linear transformations. In general, these manifolds have a singular locus, a subset not carrying such an aﬃne structure. It is not diﬃcult to write down examples of such manifolds which we expect to correspond, say, to hypersurfaces in toric varieties. More precisely, Definition 0.1. An aﬃne manifold B is a real manifold with an atlas of coordinate charts {ψi : Ui → Rn} with ψi ◦ ψ−1 j ∈ Aff(Rn), the aﬃne linear group of Rn. We say B is tropical (respectively integral) if ψi◦ψj −1 ∈ Rn GLn(Z) ⊆ Aff(Rn) (respectively ψi◦ψj −1 ∈ Aff(Zn), the aﬃne linear group of Zn). In the tropical case, the linear part of each coordinate transformation is integral, and in the integral case, both the translational and linear parts are integral. Given a tropical manifold B, we have a family of lattices Λ ⊆ TB generated locally by ∂/∂y1,...,∂/∂yn, where y1,...,yn are aﬃne coordinates. The condition on transition maps guarantees that this is well-defined. Dually, we have a family of lattices ˇ ⊆ T ∗ B generated by dy1,...,dyn, and then we get two torus bundles f : X(B) → B ˇ : ˇ(B) → B with X(B) = TB/Λ, ˇ(B) = TB/Λ.∗ Now X(B) carries a natural complex structure. Sections of Λ are flat sections of a connection on TB, and the horizontal and vertical tangent spaces of this connection are canonically isomorphic. Thus we can write down an almost complex structure J which interchanges these two spaces, with an appropriate sign-change so that J2 = − id. It is easy to see that this almost complex structure on TB is integrable and descends to X(B). On the other hand, T ∗ B carries a canonical symplectic form which descends to ˇ(B), so ˇ(B) is canonically a symplectic manifold. We can think of X(B) and ˇ(B) as forming a mirror pair this is a simple version of the SYZ conjecture. In this simple situation, however, there are few interesting compact examples, in the K¨ ahler case being limited to the possibility that B = Rn/Γ for a lattice Γ (shown in [15]). Nevertheless, we can take this simple case as motivation, and ask some basic questions: (1) What geometric structures on B correspond to geometric structures of interest on X(B) and ˇ(B)? (2) If we want more interesting examples, we need to allow B to have singu- larities, i.e., have a tropical aﬃne structure on an open set B0 ⊆ B with

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