Since Strominger, Yau, and Zaslow conjectured an interpretation of mirror symmetry as a duality of special Lagrangian torus fibrations [SYZ96], there has been considerable progress toward proving the topological consequences of this conjecture. Recently, for a Calabi-Yau family in a neighborhood of the large complex structure point, Kontsevich and Soibelman [KS01] and Gross and Wilson [GW00] conjectured the existence of an affine Kähler structure on the (complement of a codimension two locus of the) limiting metric space of the Gromov-Hausdorff collapse. This metric space should be identified with the base of the SYZ fibration (with McLean’s metric). In particular, the collapse picture asserts the existence of an integral affine structure on the base, which is enough to reconstruct the non-degenerate part of the topological torus fibration of the CY family.

An integral affine structure on the base of the conjectural SYZ torus fibration has been described by Ruan [Rua99] and Gross [Gro01] in the quintic case, followed by work of Ruan [Rua00] for general toric hypersurfaces. The present paper provides a short and explicit combinatorial description of an affine structure with mirror duality built in, based on an idea of David Morrison [Mor00].

Given a dual pair of $d$-dimensional reflexive polytopes with coherent triangulations of their boundaries, we construct in §2.1 a $(d-1)$-dimensional polytopal complex $\Sigma$, topologically a sphere, with a codimension 2 subcomplex $D$, which we call a discriminant locus. The manifold $\Sigma\backslash D$ possesses an integral affine structure (§2.3). That is, the tangent space at any point in $\Sigma\backslash D$ contains a natural integral lattice, and one can form a $(d-1)$-torus fibration $W\to\Sigma\backslash D$ by taking fiber wise quotients. The nerve of the covering of $\Sigma\backslash D$ by affine charts is a two-colored graph, whose nodes are labeled by the vertices of the triangulations and an edge connects any two which live in dual faces of the polytopes.

The more technical §2.2 is concerned with the sphericity of $\Sigma$. We develop a generalization of barycentric subdivisions which might be of independent interest to a combinatorially inclined reader.

In Section 3 we link the model to the topology of toric hypersurfaces $H_{s}$ constructed from our input data (§3.1). The main result Theorem 3.7 asserts that for any neighborhood $N$ of $D$, and a hypersurface with large enough complex structure, there is a torus fibration of $H_{s}^{\mathrm{sm}}$, a portion of the hypersurface, over $\Sigma\backslash N$, which is diffeomorphic to the restriction of our model fibration.

In the second part of the paper we will develop a connection of our model to the geometry of the hypersurfaces, conjectured in [GW00] and [KS01]. Namely, we will show that the above diffeomorphism provides, in fact, an “almost” holomorphic embedding (there is a preferred choice of complex structures on $W$). Moreover, we will construct a family of Kähler forms on $H_{s}$ in the expected class, so that the pairs $(H_{s},H_{s}\backslash H_{s}^{\mathrm{sm}})$ with the induced metrics converge in the Gromov-Hausdorff sense to the pair $(\Sigma,D)$.