In the previous sections, we constructed a $\mathbb{Z}$-affine structure for generic quintic hypersurfaces in $\mathbb{P}^{4}_{K}$ via minimal models. This applies in particular to the hypersurfaces in the Fermat family, i.e. to

$$\mathscr{X}=\{z_{0}\ldots z_{n+1}+t(z_{0}^{n+2}+\ldots+z_{n+1}^{n+2})=0\}\subset\mathbb{P}^{n+1}_{\mathbb{C}}\times\mathbb{D}$$

with $n=3$. For the hypersurfaces $X_{t}\subset\mathscr{X}$ and for arbitrary $n$, in [Li19] Li constructs special Lagrangian torus fibrations on generic regions of $X_{t}$. More precisely, endow $X_{t}$ with the unique Calabi–Yau metric $\omega_{t}$ in the class induced by $\mathcal{O}_{\mathbb{P}}(1)$. Then the family of rescaled metrics $(\log\lvert t\rvert^{-1})^{-1}\omega_{t}$ on $X_{t}$ has bounded diameter and converges in the Gromov-Hausdorff sense to a smooth metric on $\mathbb{S}^{n}\setminus\Gamma$ as $t\rightarrow 0$; here $\mathbb{S}^{n}$ is triangulated as the boundary of a standard simplex of dimension $n+1$, and $\Gamma$ is the complement of the open stars of the vertices of $\mathbb{S}^{n}$ in the first barycentric subdivision (see Definition 3.2.1). The metric limit obtained this way is a real Monge–Ampère metric with respect to a certain affine structure on $\mathbb{S}^{n}\setminus\Gamma$, which is described in [Li19, §3.2, §3.5].

In this final section we prove that the integral affine structure constructed by Li coincides with the one from Theorem A when $n=3$, which further motivates the main results of this paper. Indeed, it shows that for Fermat quintics, the essential skeleton equipped with the new type of retraction we built recovers the Gromov-Hausdorff limit with the affine structure induced by an SYZ fibration, as expected from the conjecture by Kontsevich and Soibelman.

To recall the details of Li’s construction we start by fixing some notation. The toric variety $Z=\mathbb{P}_{K}^{n+1}$ has homogeneous coordinates $[z_{0}:\ldots:z_{n+1}]$, and we write $\mathbb{T}\subset Z$ the open dense torus. We denote by $N$ the abelian group of 1-parameter subgroups of $\mathbb{T}$, and its dual $M=\Hom(N,\mathbb{Z})$. We identify $M_{\mathbb{R}}$ with $\{\sum_{i=0}^{n+1}m_{i}=0\}\subset\mathbb{R}^{n+2}$, so that $(m_{0},\ldots,m_{n+1})$ defines the character $z^{m}=\prod_{i=0}^{n+1}z_{i}^{m_{i}}$. It follows that $N_{\mathbb{R}}\simeq\mathbb{R}^{n+2}/(1,\ldots,1)$.
The variety $X=\mathscr{X}_{K}$ is embedded in $Z$ and the toric structure of $Z$ allows us to realize $\Sk(X)$ as a simplicial subset of $N_{\mathbb{R}}$. Indeed, recall that the analytification of the torus $\mathbb{T}^{\an}$ comes with a tropicalization map

$$\val:\mathbb{T}^{\an}\longrightarrow N_{\mathbb{R}},$$

defined in Section 1.5. The generic point of $X$ lies in $\mathbb{T}$, thus the set of birational points of $X^{\an}$ and in particular $\Sk(X)$ is contained inside $\mathbb{T}^{\an}$. This yields a well-defined continuous map

$$\val:\Sk(X)\longrightarrow N_{\mathbb{R}},$$

which we claim to be an embedding. Let $i\in\{0,\ldots,n+1\}$ and $\tau_{i}\subset\Sk(\mathscr{X})=\Sk(X)$ be a top-dimensional face, corresponding to a zero-dimensional stratum $p_{i}=\cap_{j\neq i}D_{j}$ of $\mathscr{X}_{k}=\sum_{i=0}^{n+1}D_{i}$. The points of $\tau_{i}$ are quasi-monomial valuations $v_{w}$ with weights $w=(w_{j})_{j\neq i}$ such that $\sum_{j\neq i}w_{j}=1$, where $w_{j}=v_{w}(z_{j}/z_{i})$. By definition of the tropicalization map, we have

$$\langle\val(v_{w}),m\rangle=v_{w}(z^{m})=v_{w}\Big(\prod_{j=0}^{n+1}z_{j}^{m_{j}}\Big)=v_{w}\Big(\prod_{j\neq i}(z_{j}/z_{i})^{m_{j}}\Big)=\sum_{j\neq i}m_{j}w_{j}$$

for any $m\in M_{\mathbb{R}}$. Hence the tropicalization map sends the face $\tau_{i}$ to the $n$-simplex

$$\theta_{i}=\{x_{i}=0\}\cap\{\sum_{j\neq i}x_{j}=1\}\subset N_{\mathbb{R}}\simeq\mathbb{R}^{n+2}/(1,\ldots,1),$$

and the image of $\Sk(X)$ by $\val$ is the boundary $\partial\Delta^{\vee}$ of the standard $(n+1)$-simplex $\Delta^{\vee}$ generated by the vertices $e_{i}=(0,\ldots,1,\ldots,0)$, for $i=0,\ldots,n+1$, inside $N_{\mathbb{R}}$ (note that this is still an $(n+1)$-simplex when passing to the quotient). Moreover, $\Delta^{\vee}$ is the dual polytope of the convex hull $\Delta$ of the characters $(-1,\ldots,(n+1),\ldots,-1)\in M_{\mathbb{R}}$, i.e.

$$\Delta^{\vee}=\{x\in N_{\mathbb{R}}\,|\,\langle x,m\rangle\leq 1,\,\forall m\in\Delta\}.$$

It now follows from an elementary computation that the simplex $\Delta^{\vee}_{\lambda}$ defined in [Li19] by the formula

$$\Delta^{\vee}_{\lambda}=\{x\in N_{\mathbb{R}}\,|\,\max_{i=0,\ldots,n+1}(n+1)x_{i}-\sum_{j\neq i}x_{j}=1\}$$

is such that $-\partial\Delta^{\vee}_{\lambda}=\partial\Delta^{\vee}(=\Sk(X)\simeq\mathbb{S}^{n}$); observe that in $\mathbb{R}^{n+2}$, the preimage of $-\Delta^{\vee}_{\lambda}$ by the quotient map is the Minkowski sum of the standard simplex and $\mathbb{R}(1,\ldots,1)$. The discrepancy in sign conventions is due to the fact that in Li’s work, the tropicalisation map is taken to be $\log\lvert\cdot\rvert$, instead of $\val=-\log\lvert\cdot\rvert$, which is the standard non-archimedean convention.

We can now describe the integral affine structure constructed by Li on $\partial\Delta^{\vee}_{\lambda}$. Fix a vertex $v_{i}\in\Sk(X)$, $v_{i}$ is identified via the tropicalization map with the vertex $e_{i}\in\partial\Delta^{\vee}\subset N_{\mathbb{R}}$, and corresponds to a codimension 1 face of $\Delta$. Then the $\mathbb{Z}$-linear functions on $\Star(v_{i})$ are generated by

	$\displaystyle\{m$	$\displaystyle\in M\,|\,\langle m,e_{i}\rangle=0\}=\{m\in\mathbb{Z}^{n+2}\,|\,\sum_{i=0}^{n+1}m_{i}=1\,,\,m_{i}=0\}$
		$\displaystyle=\{(1,0,\ldots,\underset{i}{0},\ldots,-1),(0,1,\ldots,\underset{i}{0},\ldots,-1),\ldots,(0,0,\ldots,\underset{i}{0},\ldots,1,-1)\}$

This yields an atlas of $(n+1)$ charts $U_{i}=\Star(v_{i})$ on $\Sk(X)$

	$\displaystyle f_{i}:U_{i}$	$\displaystyle\simeq\Star(e_{i})\subset\partial\Delta^{\vee}\subset N_{\mathbb{R}}\rightarrow\mathbb{R}^{n}$
	$\displaystyle x$	$\displaystyle\mapsto(x_{0}-x_{n+1},x_{1}-x_{n+1},\ldots,x_{i-1}-x_{n+1},x_{i+1}-x_{n+1},\ldots,x_{n}-x_{n+1})$

whose overlaps are the $U_{ij}=\Star(e_{ij})$, with $e_{ij}$ being the edge joining $v_{i}$ to $v_{j}$. One can easily check that the transition functions between those charts are piecewise-linear on $U_{ij}$, but not linear as they induce a corner precisely along the codimension 1 faces of $U_{ij}$.
To overcome this problem, Li uses the additional $\mathfrak{S}_{n+2}$ symmetry of the Fermat hypersurface to extend the affine structure in codimension 1, as follows. Consider the first barycentric subdivision of $\Sk(X)$, and denote by $V_{i}$ the open star of a vertex $v_{i}$ for this new simplicial structure. We now endow $\Sk(X)$ with the atlas of charts consisting of $(V_{i},f_{i}|_{V_{i}})$ and of the top-dimensional open faces of $\Sk(X)$. This atlas covers precisely $\Sk(X)\setminus\Gamma$, and since the overlaps between the charts are always contained in a top-dimensional face, this yields a $\mathbb{Z}$-affine structure on $\Sk(X)\setminus\Gamma$.