3.3.2 Integral affine structure induced combining more models [04PT]

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3.3.2 Integral affine structure induced combining more models

We recall a construction from [KS06, §4.2.5]. Consider the resolution $h:\mathscr{Z}\rightarrow\mathscr{X}$ obtained by blowing-up the $24$ singular points of $\mathscr{X}$ , which in particular dominates any model $\mathscr{X}_{ijk}$ . Then the special fiber is $\mathscr{Z}_{k}=\sum_{i=1}^{4}D_{i}+\sum_{q=1}^{24}E_{q}$ and the associated dual complex is the boundary of a tetrahedron with four additional $2$ -cells glued along each edge of the tetrahedron; following Kontsevich–Soibelman we call such $2$ -cells wings.

We parametrize each edge $e$ of $\mathcal{D}(\mathscr{X}_{k})$ by the interval $[-1,1]$ , and each wing $W_{q}$ glued to $e$ by the $2$ -simplex in $\mathbb{R}^{2}_{(x,y)}$ bounded by $e$ and $0\leqslant y\leqslant 1-|x|$ .

Lemma 3.3.2.

Let $W_{q}$ be a wing over the edge $e_{lh}$ , for $l\in\{i,j,k\}$ and assume $i,j,k,h$ all distinct. Then the retraction $\rho_{\mathscr{X}_{ijk}}:W_{q}\rightarrow e_{lh}$ is the contraction of $W_{q}$ to the edge $e_{lh}$ parallel to the edge $e_{lq}$ :

	$\displaystyle\rho_{\mathscr{X}_{ijk}}:W_{q}$	$\displaystyle\rightarrow e_{lh}$
	$\displaystyle(x,y)$	$\displaystyle\mapsto(x-y,0)$

Proof.

The morphism $\mathscr{Z}\rightarrow\mathscr{X}_{ijk}$ is the blow-up of the 24 exceptional curves of $g_{ijk}$ . In particular, the exceptional divisor $E_{q}$ is the preimage in $\mathscr{Z}$ of a curve contained in $\tilde{D}_{l}$ ; it follows that $v_{E_{q}}(\tilde{z}_{l})=1$ and $v_{E_{q}}(\tilde{z}_{h})=0$ , where $\tilde{z}_{l},\tilde{z}_{h}$ are local equations for $\tilde{D}_{l},\tilde{D}_{h}$ on $\mathscr{X}_{ijk}$ . The Berkovich retraction $\rho_{\mathscr{X}_{ijk}}$ is linear on $W_{q}$ and hence depends only on the image of $v_{q}$ , which is determined by $v_{E_{q}}(\tilde{z}_{l})$ and $v_{E_{q}}(\tilde{z}_{h})$ . Thus we conclude that $\rho_{\mathscr{X}_{ijk}}(v_{q})=v_{l}$ and we have the result. ∎

Kontsevich and Soibelman define a retraction

\rho:X^{\textrm{an}}\xrightarrow{\rho_{\mathscr{Z}}}\Sk(\mathscr{Z})\xrightarrow{\rho^{\prime}}\mathbb{S}^{2}\simeq\Sk(X)=\Sk(\mathscr{X})

where $\rho_{\mathscr{Z}}$ is the Berkovich retraction onto the skeleton $\Sk(\mathscr{Z})$ , and $\rho^{\prime}$ is a retraction of the 24 wings of $\Sk(\mathscr{Z})$ to the sphere given as follows. For each edge $e$ of $\mathcal{D}(\mathscr{X}_{k})$ we choose a point $a_{e}=(a_{e},0)$ in the interior of $e$ , and define the retraction of $W_{q}$ onto $e$ by

	$(x+y,0)$	if $x+y\leqslant a_{e}$
$\rho^{\prime}:\,(x,y)\mapsto$	$(x-y,0)$	if $x-y\geqslant a_{e}$
	$(a_{e},0)$	otherwise.

We note that

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over the interior of any $2$ -dimensional face $\tau\subset\Sk(\mathscr{X})$ , $\rho$ is equal to $\rho_{\mathscr{Z}}$ , thus it is an affinoid torus fibration (see Example 1.6.2).
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Around any vertex $v_{D}$ , $\rho$ is equal to $\rho_{\mathscr{X}_{ijk}}$ for any triple such that $D\neq D_{i},D_{j},D_{k}$ , as follows from the previous lemma. Thus, from Section 3.3.1, $\rho$ is an affinoid torus fibration around $v_{D}$ , and the affine structure induced there is the fan structure induced by $D$ , by Corollary 2.6.1.

For any edge $e$ corresponding to $C_{e}=D_{i_{m}}\cap D_{i_{m^{\prime}}}$ , adopting the notation of Section 3.2,

\rho=\begin{cases}\rho_{\mathscr{X}_{ijk}}&\textrm{ for $i_{m}\neq i,j,k$, over }\textrm{Int}(\tau_{p_{0}})\cup\textrm{Int}(\tau_{p_{\infty}})\cup[v_{i_{m}},a_{e})\\ \rho_{\mathscr{X}_{i^{\prime}j^{\prime}k^{\prime}}}&\textrm{ for $i_{m^{\prime}}\neq i^{\prime},j^{\prime},k^{\prime}$, over }\textrm{Int}(\tau_{p_{0}})\cup\textrm{Int}(\tau_{p_{\infty}})\cup[v_{i_{m^{\prime}}},a_{e}),\end{cases}

and thus is an affinoid torus fibration over the union of these two open sets.

We conclude that $\rho$ induces an integral affine structure on $\Sk(X)$ away from the points $a_{e}$ . By Corollary 3.2.4, we can compute the monodromy around the singularities. As all these computations are analogous, we exhibit the case $C_{e}=D_{1}\cap D_{2}$ :

	$\displaystyle T_{\rho}(\gamma_{a_{e}})$	$\displaystyle=\left(\begin{matrix}1&0\\ b_{1,\mathscr{X}_{234}}-b_{1,\mathscr{X}_{134}}&1\end{matrix}\right)$
		$\displaystyle=\left(\begin{matrix}1&0\\ 3-(-1)&1\end{matrix}\right)$
		$\displaystyle=\left(\begin{matrix}1&0\\ 4&1\end{matrix}\right)$

with respect to the basis $(v_{3},v_{1})$ and origin $v_{2}$ . This formula was already stated in [KS06, §4.2.5].