By [NXY19] the non-archimedean SYZ fibration $\rho_{\mathscr{X}_{ijk}}:S^{\an}\rightarrow\Sk(\mathscr{X}_{ijk})=\Sk(X)\simeq\mathbb{S}^{2}$ is an affinoid torus fibration (at least) away from the vertices of the triangulation of $\Sk(X)$ induced by the special fiber of $\mathscr{X}_{ijk}$, i.e. away from the $v_{D_{m}}$’s.
By Theorem B, $\rho_{\mathscr{X}_{ijk}}$ is an affinoid torus fibration over $\Star(\tau_{D_{h}})$ for $h\neq i,j,k$, as $D_{h}\simeq\mathbb{P}^{2}$ and $g_{ijk}$ is an isomorphism on the strict transform of $D_{h}$. Moreover, by Remark 3.1.7 and Eq. 3.3.1 the integral affine structure induced by $\rho_{\mathscr{X}_{ijk}}$ does not extend to $v_{D_{i}}$, $v_{D_{j}}$ and $v_{D_{k}}$.

We conclude that the singular points of the affine structure on $\Sk(X)$ induced by $\rho_{\mathscr{X}_{ijk}}$ are precisely $v_{D_{i}}$, $v_{D_{j}}$ and $v_{D_{k}}$. Corollary 3.1.6 establishes that the monodromies around these vertices are

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 $0\leqslant y\leqslant 1-|x|$.

Kontsevich and Soibelman define a retraction

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

We note that

• -

  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).

• -

  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 $C_{e}=D_{1}\cap D_{2}$:

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

We construct a third singular integral affine structure on $\Sk(X)$ pushing forward the techniques developed so far. This can be viewed as a dispersion of singularieties with respect to the integral affine structure studied in Section 3.3.2: on each edge we pass from one singular point around which the monodromy is $\begin{pmatrix}1&0\\ 4&1\end{pmatrix}$, to $4$ singular points around each of which the monodromy is $\begin{pmatrix}1&0\\ 1&1\end{pmatrix}$. in the literature Such singularities are called focus-focus and are the most standard examples of singularities for $\mathbb{Z}$-affine structures in dimension 2. Those arise for instance when considering the hyperkähler rotation of a generic elliptic K3 surface $f:S\longrightarrow\mathbb{C}\mathbb{P}^{1}$, with $f$ an elliptic fibration: the hyperkähler rotation $S^{\HK}$ is a complex surface with same underlying topological space as $S$, and hence comes with a map $f^{\HK}:S^{\HK}\longrightarrow\mathbb{S}^{2}$, induced by $f$ at the level of topological spaces. The map $f^{\HK}$ is no longer holomorphic, but is a symplectic torus fibration inducing a $\mathbb{Z}$-affine structure with 24 focus-focus singularities on $\mathbb{S}^{2}$ and acting as an SYZ fibration for $S^{\HK}$. We refer the reader to [GW00] for more details.

Let $e$ be an edge of $\Sk(\mathscr{X})$, let $C_{e}=D_{e_{1}}\cap D_{e_{2}}$ be the corresponding stratum curve in $\mathscr{X}_{k}$. We recall that as the degree four polynomial $F_{4}$ is generic, $C_{e}$ contains four singular points $p_{1},\ldots,p_{4}$ of $\mathscr{X}$, which are ordinary double points. Around each $p_{i}$, $\mathscr{X}$ is étale locally of the form $\{xy=wt\}\subset\mathbb{A}^{3}_{R}$, with $x$ and $y$ being local equations for $D_{e_{1}}$ and $D_{e_{2}}$ away from $p_{i}$. Blowing-up the singular point $p_{i}$ yields an exceptional divisor $E\simeq\mathbb{P}^{1}\times\mathbb{P}^{1}$. Contracting one or the other ruling of $E$, we obtain two distinct small resolutions of $\widehat{\mathscr{X}_{/C_{e}}}$ around $p_{i}$, respectively with an exceptional curve inside $D_{e_{1}}$ or $D_{e_{2}}$.

For $j\in\{0,\ldots,4\}$, we denote by $\mathscr{X}_{e,j}$ the following small resolution of $\widehat{\mathscr{X}_{/C_{e}}}$: around $p_{i}$ for $i\leqslant j$ we consider the small resolution such that the exceptional curve over $p_{i}$ lies in $D_{e_{1}}$, while for $i>j$ the small resolution such that the exceptional curves lie in $D_{e_{2}}$. The gluing of these local small resolutions is done in the étale topology, so that in general the obtained models are no longer schemes but only algebraic spaces. Nevertheless, $\mathscr{X}_{e,j}$ is dominated by $\mathscr{Z}$ (defined in Section 3.3.2) and still induces a Berkovich retraction $\rho_{\mathscr{X}_{e,j}}$, as described in Section 5. In particular, by Proposition 5.0.4, $\rho_{\mathscr{X}_{e,j}}$ is an affinoid torus fibration over $\Star(\tau_{C_{e}})$.

We construct the following continuous retraction

where $\rho_{\mathscr{Z}}$ is the Berkovich retraction onto the skeleton $\Sk(\mathscr{Z})$, and $\overline{\rho}^{\prime}$ is a retraction of the 24 wings of $\Sk(\mathscr{Z})$ to the sphere given as follows. We fix four distinct, ordered, interior points $a_{e,1},\ldots,a_{e,4}$ of each edge $e$. Then the map $\overline{\rho}^{\prime}$ on the wing $W_{i}$ attached to $e$ is defined as the map $\rho^{\prime}$ of Section 3.3.2, setting $a_{e}=a_{e,i}$, for each $i\in\{1,\ldots,4\}.$

Note that for a generic family of quartic surfaces $X/K$, the metric aspects of the Kontsevich-Soibelman conjecture suggest that there should exist a distinguished singular affine structure on $\Sk(X)$ (coming from the Gromov-Hausdorff limit of the family), and hence a canonical choice of interior points $a_{e,i}$ for each edge. To the authors’ knowledge there does not exist a way to produce such a canonical set of $a_{e,i}$’s using non-archimedean techniques.

In Section 3.3.2, the retraction $\rho^{\prime}$ depends on the choice of the points $a_{e}$; the same holds for the induced integral affine structure, whose singular locus consists indeed of the points $a_{e}$. Moving a point $a_{e}$ in the interior of the edge $e$ affects the location of the singular points, but it does not change the monodromy around the point (see Section 3.3.3). We observe now, in two examples, what happens if we let a point $a_{e}$ move to a vertex of $\Sk(\mathscr{X})$. We write $\rho=\rho_{a_{e}}$ to emphasize the dependency on the choice of singular points.

The computations above suggest that the singularities and the monodromy representation induced by the non-archimedean SYZ fibration $\rho_{\mathscr{X}_{ijk}}$ can be viewed respectively as a collision of singular points and a product of monodromies induced by the $\rho_{a_{e}}$ when the $a_{e}$’s collide. The affine structure induced by $\rho$ turns out to be more symmetric and simpler, as all the singular points have the same monodromy and are of focus-focus type.