It would be interesting to know whether the subdivision of $|\Sigma|$ given by the $F\times F^{\vee}$ can be realized as the boundary of a $d$-dimensional polytope (as the picture suggests). The dual of the face lattice would be given by the lattice of intervals in the face lattice of $\Delta$ (or of $\Delta^{\vee}$).

In fact, if there was a realization of the combinatorial type of $\Delta$ with an identification $\mathbb{R}^{d}\cong(\mathbb{R}^{d})^{*}$ such that the normal cones of $F$ and $F^{\vee}$ intersect in their relative interiors, then the Minkowski sum $\Delta+\Delta^{\vee}$ would do the trick. For $d=3$, Koebe’s Theorem [Thu80] guarantees such a realization.

There is a natural family of complex structures $J_{s}$ on the model torus fibration $h\colon W\to\Sigma\backslash D$. Namely, for a given $s$ we can take $dy_{i}+\frac{\sqrt{-1}}{\log|s|}d\theta_{i}$ to be the holomorphic 1-forms on $W$, where $\{y_{i}\}$ are the affine coordinates on $\Sigma\backslash D$ and $\{\theta_{i}\}$ are the corresponding coordinates on the torus fibers. Also, given a Riemannian metric $\sum g_{ij}dy_{i}\otimes dy_{j}$ on $\Sigma\backslash D$, one can define the Kähler metric on $W$ by $\omega_{ij}=\sum g_{ij}(dy_{i}+\frac{\sqrt{-1}}{\log|s|}d\theta_{i})\otimes(dy_{j}-\frac{\sqrt{-1}}{\log|s|}d\theta_{j})$. If, in addition, $g_{ij}$ satisfy the real Monge-Ampère equation: $\det g_{ij}\equiv 1$, then the induced metric on $W$ is Ricci-flat.

In the second part of the paper we will show that $H_{s}^{\mathrm{sm}}$ embeds into $(W,J_{s})$ “almost” holomorphically. Moreover, we will construct a family of $\mathbb{T}$-invariant Kähler forms $\omega_{s}$ on $X_{\Delta_{\nu}}$ in the class $\frac{[\nu]}{\log|s|}$, such that the pairs $(H_{s},H_{s}^{\mathrm{sing}})$ with the induced metrics converge in the Gromov-Hausdorff sense to the pair $(\Sigma,D)$. Here $\Sigma$ will carry a compact metric space structure which will restrict to a Riemannian metric on $\Sigma\backslash D$.

The family $H^{\operatorname{af{}f}}_{s}$ can be thought of as an affine algebraic hypersurface in $(K^{*})^{d}$ defined over a complete algebraically closed field $K$ that contains $\mathbb{C}((s))$. Then, as the name suggests, the image of $H_{s}^{\operatorname{af{}f}}$ under the valuation map $val:(K^{*})^{d}\to R^{d}$ will be the non-Archimedean amoeba $\mathcal{A}^{\lambda}_{\infty}$ (cf. [Kap00]).

Interestingly, the potential functions for our Kähler forms $\omega_{s}$ will come from smoothing a convex piecewise linear function on $\mathbb{R}^{d}$. This gives another evidence that there may be a reformulation of mirror symmetry purely in non-Archimedean terms. There is a partial understanding of this approach [Kon00], but the entire program still remains wide open.