Let $(X,L)$ be a polarized family of $n$-dimensional Calabi–Yau varieties over the punctured disk $\mathbb{D}^{*}\subset\mathbb{C}$; each fiber $X_{t}$ additionally carries a unique Ricci-flat Kähler-metric $\omega_{t}\in c_{1}(L_{t})$, according to the celebrated Yau theorem. We will be primarily interested in such families that are maximally degenerate, in the following sense: the monodromy acting on the degree $n$ cohomology of the general fiber has a Jordan block of maximal (that is, $n+1$) size.

In this setting, the Strominger-Yau-Zaslow conjecture predicts that the general fiber $X_{t}$ admits a fibration $\rho_{t}:X_{t}\longrightarrow S$, called an SYZ fibration, whose base $S$ is a real $n$-dimensional topological manifold (even a sphere if the $X_{t}$ are strict Calabi–Yau), and whose fibers are special Lagrangian tori away from a discriminant locus of codimension $2$ in $S$.
An SYZ fibration endows $S$ with a singular integral affine structure, induced by action-angle coordinates. This means that, in the complement of the discriminant locus of the fibration, the transition functions between charts of $S$ are affine transformations in $\textrm{GL}_{n}(\mathbb{Z})\ltimes\mathbb{R}^{n}$.
Moreover, the limit for $t\rightarrow 0$ of the metric spaces $(X_{t},\omega_{t})$ should correspond to the metric collapse of the torus fibers of $\rho_{t}$. Then, the (suitably rescaled) Gromov-Hausdorff limit of $(X_{t},\omega_{t})$ should coincide with the space $S$, endowed with a metric which in affine coordinates satisfies a real Monge–Ampère equation away from the discriminant locus.

While some examples of special Lagrangian torus fibrations can be produced, dealing with the general case seems very difficult. The insight of Kontsevich and Soibelman is to replace the above conjecture by an analogous one in the non-archimedean world, and to interpret the latter as an asymptotic limit of the complex phenomenon when $t\rightarrow 0$. We now elaborate on this idea.

Consider the field $K=\mathbb{C}((t))$ of Laurent power series, which comes equipped with the non-archimedean valuation $\ord_{t}$, order of vanishing at $t=0$; the family $X$ can be viewed as a variety over $K$. Within this framework, we associate with $X/K$ a topological space, called the Berkovich space $X^{\an}$ of $X$; this is a space of real (semi)valuations on $X$ (see Section 1.3).
A way to construct and visualize points of $X^{\an}$ is to consider models of $X$ over $R=\mathbb{C}[[t]]$. Indeed, any suitably regular (dlt) model $\mathscr{X}$ of $X$ has an associated simplicial subset $\Sk(\mathscr{X})\subset X^{\an}$, called the skeleton of $\mathscr{X}$ and homeomorphic to the dual (intersection) complex of the degenerate fiber $\mathscr{X}_{0}$ of $\mathscr{X}$, and a continuous retraction $\rho_{\mathscr{X}}:X^{\an}\rightarrow\Sk(\mathscr{X})$ (see Sections 1.4 and 1.5 for more details). It follows that $X^{\an}$ encodes geometric information coming from degenerations of $X$ and about combinatorics of models of $X$.
Among various models and associated skeletons, minimal (in the sense of MMP) models $\mathscr{X}$ of $X$ determine a canonical skeleton $\Sk(X)=\Sk(\mathscr{X})$, called the essential skeleton of $X$ and independent of the choice of the minimal model. The essential skeleton and the retractions $\rho_{\mathscr{X}}:X^{\an}\rightarrow\Sk(\mathscr{X})=\Sk(X)$, which do depend on $\mathscr{X}$, are of particular relevance in the non-archimedean reformulation of the SYZ conjecture as the following conjectures point out.

The key idea is that Berkovich theory should allow to construct (non-unique) non-archimedean avatars of SYZ fibrations. More precisely, in [KS06] Kontsevich and Soibelman conjecture that the essential skeleton $\Sk(X)$ can be endowed with an integral affine structure outside of a codimension 2 piecewise-affine subset $\Gamma\subset\Sk(X)$, such that the following holds. The space $\Sk(X)$ can be recovered from the Kähler geometry of $X$, as a (suitably rescaled) Gromov-Hausdorff limit of the metric spaces $(X_{t},\omega_{t})$. Moreover, the limiting metric on $\Sk(X)$ should satisfy the following: outside of $\Gamma$, the metric is given locally in affine coordinates by the Hessian of a convex function, satisfying a real Monge-Ampère equation. It is furthermore expected that this limiting affine structure can be recovered by a map $\rho:X^{\an}\rightarrow\Sk(X)$, which is a non-archimedean analog of the SYZ fibration.
The construction of the above fibration is made more rigorous in [NXY19]. The authors prove that the retraction $\rho_{\mathscr{X}}$ associated with a minimal model $\mathscr{X}$ is an affinoid torus fibration away from a codimension 2 locus of the base - the non-archimedean analog of a smooth torus fibration - and induces an integral affine structure there, as the SYZ heuristic and the conjecture by Kontsevich and Soibelman predict. Here in particular the transition functions of the integral affine structure are in $\textrm{GL}_{n}(\mathbb{Z})\ltimes\mathbb{Z}^{n}$.

The local model for affinoid torus fibrations is the tropicalization map $\textrm{val}:\mathbb{T}^{\an}\rightarrow N_{\mathbb{R}}$, where $\mathbb{T}=\mathbb{G}_{m,K}^{n}$ and $N$ is the cocharacter lattice of the torus $\mathbb{T}$. Global examples of such retractions are given as follows: given a (non-proper) toric variety $\mathscr{Y}$ over $R=\mathbb{C}[[t]]$ which is a model of $\mathbb{T}$, the retraction $\rho_{\mathscr{Y}}$ is a restriction of $\mathrm{val}$. This reduces the proof of the result in [NXY19] to showing that minimal models $\mathscr{X}$ are in fact toric along one-dimensional strata of the special fiber when the latter is reduced.

At this point the base of the SYZ fibration appears to be well identified - as the essential skeleton or equivalently the dual complex of any minimal model - while the affine structure and the metric are not. In fact, the construction in [NXY19] yields integral affine structures that depend on the additional choice of a model, while the Kontsevich–Soibelman conjecture predicts uniqueness, at least of the metric space. Moreover, the location and the nature of the singularities obtained in [NXY19] differ from previous constructions in mirror symmetry.
Such discrepancy already appears in the case of quintic three-folds in $\mathbb{P}^{4}$. On one side, the constructions in [Rua01, Gro01] - using symplectic and toric geometry - yield an affine structure on a triangulated $3$-sphere whose singularities are located away from the vertices. On the other side, the discriminant locus of the non-archimedean SYZ fibration constructed in [NXY19] passes through the vertices of the triangulation.
Moreover, the recent work in [Li19] provides evidence that for a degeneration of Fermat hypersurfaces, the affine structure on the Gromov-Hausdorff limit of the Kähler Ricci-flat metric on the nearby fibers has its singularities located inside the cells of codimension one and away from the vertices.

In this paper we deal with the apparent incompatibility raised by the expected affine structures on the essential skeleton and the ones induced by non-archimedean SYZ fibration. To this purpose, we further develop the non-archimedean approach, and produce examples of a new type of non-archimedean retractions. This allows us to construct singular integral affine structures which are both compatible with SYZ mirror symmetry, and built by means of non-archimedean tools. In this respect, our results provide new evidence for the dictionary between the SYZ heuristic and the non-archimedean interpretation of mirror symmetry.
Inspired by the example [KS06, §4.2.5] of an integral affine structure on the sphere $\mathbb{S}^{2}$ with $24$ singular points, associated with a degeneration of $K3$ surfaces, we move to the $3$-dimensional case and consider the quintic $3$-fold as testing ground of our results. More precisely, let $X\rightarrow\mathbb{D}^{*}$ be a generic family of quintics:

$$X=\{tF_{5}(z_{1},z_{2},z_{3},z_{4},z_{5})+z_{1}z_{2}z_{3}z_{4}z_{5}=0\}\subset\mathbb{P}^{4}\times\mathbb{D}^{*}.$$

We endow $\Sk(X)\simeq\mathbb{S}^{3}$ with the simplicial structure induced by the identification with the dual complex of $\mathscr{X}_{0}$, with $\mathscr{X}$ being the closure of $X$ in $\mathbb{P}^{4}\times\mathbb{D}$, and $\mathscr{X}_{0}$ the fiber over $0$.

The theorem holds in particular for the Fermat family of quintics; in this case, the Gromov-Hausdorff limit of the family is known [Li19], and naturally induces a singular affine structure on $\mathbb{S}^{3}$. We are able to show (see Proposition 4.8.1) that the non-archimedean retraction in Theorem A, the metric limit as determined by Li, and the works of Gross and Ruan, all induce the same integral affine structure on $\mathbb{S}^{3}$; this provides a new piece of evidence for the dictionary between SYZ mirror symmetry and Berkovich geometry.

The main idea behind our construction is to consider the Berkovich retractions $\rho_{\mathscr{X}_{i}}$ associated with several minimal models $\mathscr{X}_{1},\ldots,\mathscr{X}_{N}$ of $X$, adapted to different regions of $\Sk(X)$ and glued together. In order to prove Theorem A and describe the singular locus of the retraction $\pi$, we establish the following result, which generalizes [NXY19, Proposition 5.4]:

Note that the dlt assumption, combined with the fact that $Z$ is toric, imply that $Z$ is smooth (see Remark 2.1.1). By assumption, $Z$ is (a connected component of) the intersection of the divisors $D_{j}\subset\mathscr{X}_{k}$ containing $Z$ and the $D_{j}$’s are Cartier, so that $\nu^{*}_{Z/\mathscr{X}}=\bigoplus_{j\in J}\mathcal{O}_{Z}(-D_{j})$ and the nef assumption simply means that each of the $\mathcal{O}_{Z}(-D_{j})$ is a nef divisor.
Using the positivity of the conormal bundle, we then prove that in a formal neighbourhood of $Z$, $\mathscr{X}$ is isomorphic to the normal bundle of $Z$, which is a toric variety. This is similar in spirit to the classical work of [Gra62, Satz 7, p. 363] on holomorphic tubular neighbourhoods, as well as Grothendieck’s algebraization theorem [Gro61, Theorem 5.1.4]; the key technical point being the vanishing of the higher cohomology groups of the powers of $\nu^{*}_{Z/\mathscr{X}}$ which allows us to extend combinatorial data from $Z$ to a formal neighbourhood.

The subset $\Star(\tau_{Z})$ is the open star of the face determined by $Z$ (see Definition 1.4.2). Theorem B and Corollary C show that the discriminant locus of the retraction $\rho_{\mathscr{X}}$ measures the defect of a stratum to being toric. Therefore, to prove Theorem A, we combine retractions $\rho_{\mathscr{X}_{i}}$ coming from different models $\mathscr{X}_{1},\ldots,\mathscr{X}_{N}$ with the following property: for each region of $\Sk(X)\setminus\Gamma$, there exists a model $\mathscr{X}_{i}$ such that $\pi=\rho_{\mathscr{X}_{i}}$ and $\mathscr{X}_{i}$ satisfies the hypothesis of Theorem B, hence defines an affinoid torus fibration over the corresponding region.
If $\dim Z=1$ and $Z$ is a rational curve, then the positivity assumption can always be achieved via a finite number of blow-ups and Corollary C holds; this was established in [NXY19, Proposition 5.4].

The connection between toric geometry and mirror symmetry has been explored in several ways; in particular, the Gross–Siebert program considers toric degenerations of Calabi–Yau varieties. Such degenerations satisfy assumptions similar to the ones in Theorem B, as the irreducible components of the special fiber are all assumed to be toric varieties; however, note that they are not assumed to be $\mathbb{Q}$-Cartier, so that there may not be an associated Berkovich retraction. In [GS06] the authors then glue together the fans of the various components of the special fiber to combinatorially construct an affine structure on the skeleton of the degeneration; see Section 3.1.1 for an example in dimension 2. Using tools from non-archimedean geometry, Theorem B in particular enables us to generalize this construction to degenerations that are not necessarily toric.

The retraction in Theorem A should be closely related to the tropical contractions constructed in [Yam21], from a tropical Calabi–Yau variety $V$ to an associated integral affine manifold $B$. In particular, the setting of Yamamoto applies to toric degenerations of Calabi–Yau varieties constructed by Gross [Gro05]; in such case, the tropical contraction maps onto the dual complex of the degeneration and induces on it the singular integral affine structure defined in [Gro05]. Relations between (co)homology groups of $V$ and $B$ are also studied in [Yam21].

Finally, the non-archimedean retraction $\pi$ of Theorem A yields an integral affine structure whose discriminant locus is of codimension 2. We recall that the $2$-codimensionality of the discriminant is expected from the SYZ heuristic at the topological level; in Theorem A, this is achieved by construction of $\pi^{\prime}$, building on the results in [NXY19]. In the setting of the Gross–Siebert program, given a singular integral affine manifold $B$, one can produce, using classical moment maps, a topological torus fibration over $B$, which however has a discriminant locus of codimension 1, see [RS20, §2.1]. In the series of recent or upcoming papers [RZ21a, RZ21b, RZ], Ruddat and Zharkov develop a strategy which solves this problem - at least at the topological level - and works in arbitrary dimension.
More precisely, the authors are able to construct a torus fibration with discriminant locus of codimension 2 in $B$, isotopic to the previous fibration, and with a symplectic structure on the complement of a codimension 2 subset in the total space. Note that the total space of this resulting fibration will not be a manifold in general, as it could have orbifold singularities.

Let us briefly describe the organization of the paper.
In Section 1, we introduce some notation and collect some basic facts about toric varieties. We also define Berkovich spaces and recall the definition of skeletons and retractions we will be using. Section 2 is devoted to the proof of Theorem B. In Section 3 we give a detailed description of the monodromy of the integral affine structures induced by Berkovich retractions, or combinations of them. Finally, in Section 4 we study in detail the example of the degeneration of quintic 3-folds and prove Theorem A by applying the results of the previous sections. We also compare our results to various constructions existing in the literature.

Acknowledgements. We would like to thank Sébastien Boucksom and Mirko Mauri for their comments on the first version of this paper. We are also grateful to Omid Amini, Johannes Nicaise, Helge Ruddat, Yuto Yamamoto for helpful conversations. Enrica Mazzon was partially supported by Max Planck Institute for Mathematics in Bonn during the preparation of this paper.