1. Introduction [03EK]

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1. Introduction

In this paper we endow the topological model $\Sigma$ constructed in [HZ02] with the structure of a metric space which is a Kähler affine manifold away from the (codimension 2) discriminant locus and relate it to the geometry of the Calabi-Yau family of toric hypersurfaces $Z_{s}$ near large complex structure limit point.

The main result is the following. Given an integral Kähler affine structure on ${\Sigma\backslash D}$ which is in the right class and satisfies certain (bi-polyhedral) compatibility conditions, we construct a family of Kähler metrics on $Z_{s}$ such that as $s\to\infty$ (the large complex structure limit):

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The embedding $\psi:Z_{s}^{\mathrm{sm}}\hookrightarrow W_{s}$ of “smooth” portions of the hypersurfaces into the model torus bundle (with the right twist) identifies the scalar products and the complex structures on the tangent spaces $T_{x}Z_{s}$ and $T_{\phi(x)}W_{s}$ up to terms of order $o(1)$ uniformly in $x\in Z_{s}^{\mathrm{sm}}$ .
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the Gromov-Hausdorff distance between the pairs $(Z_{s},Z_{s}\backslash Z_{s}^{\mathrm{sm}})$ and $(\Sigma,D)$ is of order $o(1)$ .

Interchanging the input data $(\Delta,S,\lambda)$ with $(\Delta^{\vee},T,\nu)$ and repeating the construction of the Kähler metrics for the dual Calabi-Yau family gives rise to the same limiting metric space $\Sigma$ and the dual Kähler affine structure. This result constitutes a significant part of the the limiting mirror symmetry conjecture (cf. [KS01]).

The major missing part toward proving the metric collapse is that our metrics are not Ricci-flat. To establish the Ricci-flatness away from the discriminant one would need the affine Calabi conjecture (see Section 2.3). But even assuming a Monge-Ampère solution on ${\Sigma\backslash D}$ the behavior of true Calabi-Yau metrics is not expected to be approximated by a semi-flat construction near the discriminant locus. A further development in this direction requires some local estimates on CY metrics near singularities. We hope to address these issues by using generalized Gibbons-Hawking ansatz elsewhere (cf. [Zha02]).

Notations.

We continue to use notations from [HZ02].

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$(\Delta,S),(\Delta^{\vee},T)$ is a dual pair of $d$ -dimensional reflexive polytopes with coherent triangulations of their boundaries.
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$\Delta_{\mathbb{Z}},\Delta^{\vee}_{\mathbb{Z}}$ are the sets of integral points of $\Delta,\Delta^{\vee}$ .
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$\operatorname{SC}(S),\operatorname{SC}(T)$ are the secondary cones in $\mathbb{R}^{\Delta_{\mathbb{Z}}},\mathbb{R}^{\Delta^{\vee}_{\mathbb{Z}}}$ corresponding to the triangulations $S\ast\{0\},T\ast\{0\}$ of $\Delta,\Delta^{\vee}$ .
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$\lambda$ and $\nu$ are integral vectors in the interiors of the respective secondary cones.
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$X_{T}$ is the toric variety associated to the simplicial fan in $(\mathbb{R}^{d})^{*}$ given by the triangulation $T$ .

Acknowledgments.

We would like to thank Lev Borisov, Robert Bryant, Mark Gross, Maxim Kontsevich, Grisha Mikhalkin, Dave Morrison, Svetlana Roudenko, Mark Stern and Stephanos Venakides for valuable conversations, and the entire Duke CGTP group for a stimulating environment.