(1.1) The theory of mirror symmetry emanated from string theory and has had a fundamental impact on algebraic geometry ever since the groundbreaking work of Candelas, de la Ossa, Green and Parkes [COGP91]. The mirror symmetry heuristic predicts that every complex Calabi-Yau manifold $X$ has a mirror partner $\check{X}$ of the same dimension whose complex geometry is equivalent, in a suitable sense, to the symplectic geometry of $X$, and vice versa. A celebrated application of these ideas was the prediction of the numbers of rational curves of fixed degree (more precisely, Gromov-Witten invariants) of the quintic threefold in [COGP91] by means of period integral calculations on the mirror partner. An important challenge in the theory of mirror symmetry is to give an exact definition of what it means to be a mirror pair of Calabi-Yau manifolds, and to devise techniques to construct such pairs.

(1.2) In recent years, much progress has been made, in particular by Kontsevich–Soibelman [KS00, KS06] and Gross–Siebert [GS11a]. Both of these programs are based on a conjectural geometric explanation of mirror symmetry due to Strominger, Yau and Zaslow, known as the SYZ conjecture [SYZ96]. Since its appearance, the conjecture has been amended in certain ways; a common way to formulate it today is the following. Let $\mathcal{X}^{\ast}$ be a projective family of $n$-dimensional complex Calabi-Yau varieties over a punctured disk $\Delta^{\ast}$, and assume that this family is maximally degenerate. The latter condition means that the monodromy transformation on the degree $n$ cohomology of the general fiber $\mathcal{X}_{t}$ of $\mathcal{X}^{\ast}$ has a Jordan block of rank $n+1$. Then, up to rescaling the metrics, the family $\mathcal{X}_{t}$ is conjectured to converge in the Gromov-Hausdorff limit to an $n$-dimensional topological manifold $S$. Moreover, a general fiber $\mathcal{X}_{t}$ should admit a fibration $\rho\colon\mathcal{X}_{t}\to S$, called an SYZ fibration, whose fibers are special Lagrangian tori in $\mathcal{X}_{t}$, except over a discriminant locus of codimension at least $2$ in the base $S$. The mirror partner of $\mathcal{X}_{t}$ can then be constructed by dualizing the torus fibration $\rho$ over the smooth locus and compactifying the result in an appropriate way (this involves deforming the dual fibration by so-called quantum corrections). We refer to the excellent survey paper [Gr13] for a more precise statement and additional background on the SYZ conjecture, as well as the Gross–Siebert program.

(1.3) The SYZ conjecture remains largely open, and is quite difficult even in basic cases; see for instance [GW00]. A fundamental insight of Kontsevich and Soibelman in [KS06] is that one should be able to construct a close analog of the SYZ fibration in the world of non-archimedean geometry, more precisely in the context of Berkovich spaces. Here, the base $S$ of the fibration arises as a so-called skeleton in the Berkovich analytification of the degeneration. Let us emphasize that the non-archimedean SYZ fibration is not merely an analog of the conjectural structure in a different context; it can effectively be used to realize the original goal of constructing mirror partners over the complex numbers, since one can go back from the non-archimedean world to the complex world by means of non-archimedean GAGA and algebraization techniques. In the non-archimedean approach, the quantum corrections are provided by non-archimedean enumerative geometry and wall-crossing structures [KS06, Yu16a, Yu16b, KY18]. The non-archimedean SYZ fibration induces an affine structure with singularities on the base $S$, and Kontsevich and Soibelman made the striking conjecture that this affine manifold should be related to the Gromov-Hausdorff limit of $\mathcal{X}$ (Conjecture 3 in [KS06]) – see [BJ17] for interesting results towards that conjecture.

(1.4) The aim of the present paper is to construct the non-archimedean SYZ fibration in full generality, and to prove some of its conjectural properties. This paves the way for a better understanding of the Gromov-Hausdorff limits and the SYZ conjecture. Our construction of the SYZ fibration builds upon the original work of Kontsevich and Soibelman and the relations with the Minimal Model Program discovered by the first two authors in [NX16a]. This discovery has led to a surprising dictionary where the SYZ heuristic can be translated into precise predictions about the structure of minimal models, which can then be proven with techniques from the Minimal Model Program – see for instance [KX16] and [NX16b]. Our main new result here is that the non-archimedean SYZ fibration is a smooth affinoid torus fibration away from a codimension two subset of the base (Theorem 6.1), as implied by Conjectures 1 and 3 in [KS06]. This amounts to proving that minimal dlt models with reduced special fiber of Calabi-Yau varieties are snc along the one-dimensional strata of the special fiber (Theorem 4.5), and have a toric structure along these strata (Proposition 5.4).

(1.5) We fix an algebraically closed field $k$ of characteristic $0$ and we set $R=k[\negthinspace[t]\negthinspace]$ and $K=k(\negthinspace(t)\negthinspace)$. We also fix an algebraic closure $K^{a}$ of $K$. We denote by $\mathrm{ord}_{t}$ the $t$-adic valuation on $K$ and we define an absolute value $|\cdot|$ on $K$ by setting $|a|=\exp(-\mathrm{ord}_{t}a)$ for every $a\in K^{\times}$. This turns $K$ into a complete non-archimedean field. We denote by $(\cdot)^{\mathrm{an}}$ the analytification functor from the category of $K$-schemes of finite type to Berkovich’s category of $K$-analytic spaces. For every $R$-scheme $\mathscr{X}$, we will denote by $\mathscr{X}_{k}=\mathscr{X}\times_{R}k$ and $\mathscr{X}_{K}=\mathscr{X}\times_{R}K$ its special and generic fiber.

(1.6) If $\mathscr{X}$ is a Noetherian $R$-scheme and $C$ is a subscheme of $\mathscr{X}_{k}$, then we will denote by $\widehat{\mathscr{X}_{/C}}$ the formal completion of $\mathscr{X}$ along $C$. If $\mathscr{X}$ is of finite type over $R$, then $\widehat{\mathscr{X}_{/C}}$ is formally of finite type over $R$ (or special, in the terminology of [Be96]). That is, it has a finite cover by open formal subschemes of the form $\mathrm{Spf}\,(A)$ where $A$ is a quotient of a topological $R$-algebra of the form $R\{x_{1},\ldots,x_{m}\}[\negthinspace[y_{1},\ldots,y_{n}]\negthinspace]$. Every Noetherian formal scheme $\mathfrak{X}$ has a unique maximal ideal of definition $\mathscr{I}$, consisting of all the topologically nilpotent elements in $\mathcal{O}_{\mathfrak{X}}$. The closed subscheme of $\mathfrak{X}$ defined by $\mathscr{I}$ will be denoted by $\mathfrak{X}_{\mathrm{red}}$. This construction induces a functor from the category of Noetherian formal schemes to the category of reduced Noetherian schemes. If $\mathfrak{X}$ is a scheme, then $\mathfrak{X}_{\mathrm{red}}$ is the maximal reduced closed subscheme of $\mathfrak{X}$.

(1.7) A separated flat $R$-scheme of finite type $\mathscr{Y}$ is called toric if there exists a toric morphism of toric varieties

such that $\mathscr{Y}$ is isomorphic to $Y\times_{k[t]}R$. Such a toric scheme can be defined by giving a finite fan $\Sigma$ of strongly convex rational polyhedral cones in $\mathbb{R}^{n}\times\mathbb{R}_{\geq 0}$ for some $n\geq 0$, together with a positive integer $\iota$; then one can take $Y$ to be the toric $k$-variety associated with $\Sigma$ and $Y\to\mathbb{A}^{1}_{k}$ to be the toric morphism induced by the morphism

(1.8) A Calabi-Yau variety over $K$ is a smooth, proper, geometrically connected $K$-scheme $X$ such that the canonical line bundle $\omega_{X}$ is trivial. In particular, our definition also includes abelian varieties. A volume form on a Calabi-Yau variety $X$ is a nowhere vanishing differential form of maximal degree, that is, a global generator for the canonical line bundle $\omega_{X}$.

(1.9) Let $\mathscr{X}$ be a Noetherian scheme, and let $D$ be an effective divisor on $\mathscr{X}$, with prime components $D_{i},\,i\in I$. A stratum of $D$ is a connected component of the schematic intersection $D_{J}=\cap_{j\in J}D_{j}$, for some non-empty subset $J$ of $I$. An open stratum is a stratum $S$ minus the union of the prime components of $D$ that do not contain $S$.

(1.10) Let $X$ be a smooth and proper $K$-scheme. A model of $X$ is a proper flat $R$-scheme $\mathscr{X}$ endowed with an isomorphism $\mathscr{X}_{K}\to X$. An snc-model of $X$ is a regular model $\mathscr{X}$ such that $\mathscr{X}_{k}$ is a divisor with strict normal crossings. An snc-model is called semistable if $\mathscr{X}_{k}$ is reduced. By the semistable reduction theorem [KKMS73, Ch4§3], there exists a finite extension $K^{\prime}$ of $K$ such that $X\times_{K}K^{\prime}$ has a semistable snc-model over the integral closure of $R$ in $K^{\prime}$.

A dlt-model of $X$ is a normal model $\mathscr{X}$ such that the pair $(\mathscr{X},\mathscr{X}_{k,\mathrm{red}})$ is divisorially log terminal (dlt). We say that a dlt-model is good if every prime component of $\mathscr{X}_{k,\mathrm{red}}$ is $\mathbb{Q}$-Cartier; this is slightly weaker than the usual condition that $\mathscr{X}$ is $\mathbb{Q}$-factorial, but it is sufficient for our purposes. In particular, every snc-model is also a good dlt-model. A dlt-model $\mathscr{X}$ is called minimal if the logarithmic relative canonical divisor $K_{\mathscr{X}/R}+\mathscr{X}_{k,\mathrm{red}}$ is semi-ample. When $X$ is Calabi-Yau, this is equivalent to saying that $K_{\mathscr{X}/R}+\mathscr{X}_{k,\mathrm{red}}$ is torsion; when, moreover, $\mathscr{X}_{k}$ is reduced, then it is equivalent to saying that $K_{\mathscr{X}/R}\sim 0$.

(1.12) An integral affine function on an open subset of $\mathbb{R}^{n}$ is a continuous real-valued function that can locally be written as a degree one polynomial with coefficients in $\mathbb{Z}$. Beware that some authors, including [KS06], allow a constant term in $\mathbb{R}$ in the degree one polynomial; our more restrictive definition is better suited for the purposes of this paper.

(2.1) Let $X$ be a Calabi-Yau variety over $K$. The essential skeleton $\mathrm{Sk}(X)$ of $X$ was first defined by Kontsevich and Soibelman in [KS06]. The construction was then refined and generalized in [MN15]. Let $\omega$ be a volume form on $X$. Then one can attach to the pair $(X,\omega)$ a weight function

that measures the degeneration of $(X,\omega)$ at $t=0$ along points of $X^{\mathrm{an}}$; see [MN15, §4.5]. The essential skeleton $\mathrm{Sk}(X)$ is the locus of points in $X^{\mathrm{an}}$ where $\mathrm{wt}_{\omega}$ reaches its minimal value. This definition only depends on $X$, and not on $\omega$, because multiplying $\omega$ with a scalar $\lambda\in K^{\ast}$ shifts the weight function by the constant $\mathrm{ord}_{t}\lambda$. The essential skeleton is a non-empty compact subspace of $X^{\mathrm{an}}$, which can be explicitly computed in the following way. Let $\mathscr{X}$ be an snc-model of $X$, with special fiber $\mathscr{X}_{k}=\sum_{i\in I}N_{i}E_{i}$. If we view $\omega$ as a rational section of the line bundle $\omega_{\mathscr{X}/R}(\mathscr{X}_{k,\mathrm{red}})$, then it defines a Cartier divisor on $\mathscr{X}$ that we denote by $\mathrm{div}_{\mathscr{X}}(\omega)$. It is supported on $\mathscr{X}_{k}$ because $\omega$ is nowhere vanishing on $X$; thus we can write $\mathrm{div}_{\mathscr{X}}(\omega)=\sum_{i\in I}\nu_{i}E_{i}$. If we denote by $\Delta(\mathscr{X})$ the dual intersection complex of $\mathscr{X}_{k}$, then $\mathrm{Sk}(X)$ is canonically homeomorphic to the sub-$\Delta$-complex of $\Delta(\mathscr{X})$ spanned by the vertices corresponding to the components $E_{i}$ for which $\nu_{i}/N_{i}$ is minimal (see Theorem 3 in [KS06, §6.6] and Theorem 4.7.5 in [MN15]). In particular, $\mathrm{Sk}(X)$ is homeomorphic to a finite $\Delta$-complex of dimension $\leq\dim(X)$.

(2.2) Kontsevich and Soibelman postulated that $\mathrm{Sk}(X)$ should be the base of the non-archimedean SYZ fibration, but the definition of $\mathrm{Sk}(X)$ does not provide us with a map $X^{\mathrm{an}}\to\mathrm{Sk}(X)$. To construct such a map, we will use an alternative description of the essential skeleton that appeared in [NX16a]. Let $\mathscr{X}$ be a minimal dlt-model of $X$, and denote by $\mathscr{X}^{\mathrm{snc}}$ the open subscheme of $\mathscr{X}$ consisting of the points where $\mathscr{X}$ is regular and $\mathscr{X}_{k}$ has strict normal crossings. Then the dual intersection complex $\Delta(\mathscr{X}^{\mathrm{snc}})$ of $\mathscr{X}^{\mathrm{snc}}_{k}$ can be canonically embedded into $X^{\mathrm{an}}$ (see [MN15, §3]). It follows from [NX16a, 3.3.3] that the image of this embedding is exactly the essential skeleton $\mathrm{Sk}(X)$. To be precise, it is assumed in the statement of [NX16a, 3.3.3] that $\mathscr{X}$ is $\mathbb{Q}$-factorial and defined over an algebraic curve, but these assumptions are not used in the proof. If the minimal dlt-model $\mathscr{X}$ is good, we will now construct a continuous retraction $\rho_{\mathscr{X}}\colon X^{\mathrm{an}}\to\mathrm{Sk}(X)$ by generalizing the construction for snc-models in [MN15, 3.1.5].

(2.3) Let $\mathscr{X}$ be a good minimal dlt-model of $X$. We need to make the following technical assumption: the strata of $\mathscr{X}_{k}$ are precisely the log canonical centers of the pair $(\mathscr{X},\mathscr{X}_{k,\mathrm{red}})$ that are contained in $\mathscr{X}_{k}$. By the definition of a dlt-model, every log canonical center of $(\mathscr{X},\mathscr{X}_{k,\mathrm{red}})$ is a stratum. The converse implication is known when $\mathscr{X}$ is defined over an algebraic curve [Ko13, 4.16]. We will prove in Corollary 4.4 that it also holds when $\mathscr{X}_{k}$ is reduced, which is the most important case for our purposes. We expect that the assumption is always satisfied, but the relevant parts of the Minimal Model Program have not been written down for $R$-schemes. In any case, if our technical assumption holds, we can proceed in the following way.

(2.4) Let $x$ be a point in $X^{\mathrm{an}}$ and let $\mathrm{red}_{\mathscr{X}}(x)$ be its reduction on $\mathscr{X}_{k}$ (see [MN15, 2.2.2]). Let $Z$ be the unique minimal stratum of $\mathscr{X}_{k}$ that contains $\mathrm{red}_{\mathscr{X}}(x)$. By our assumption (2), $Z$ is a log canonical center of $(\mathscr{X},\mathscr{X}_{k,\mathrm{red}})$. Then $Z\cap\mathscr{X}^{\mathrm{snc}}$ is a non-empty stratum of $\mathscr{X}^{\mathrm{snc}}_{k}$ by the definition of a dlt pair. Thus, it determines a unique face $\tau$ of the dual intersection complex $\Delta(\mathscr{X}^{\mathrm{snc}})$. Let $E_{1},\ldots,E_{r}$ be the prime components of $\mathscr{X}_{k}$ that contain $Z$, and let $N_{1},\ldots,N_{r}$ be their multiplicities in $\mathscr{X}_{k}$. Then $E_{1},\ldots,E_{r}$ correspond precisely to the vertices $v_{1},\ldots,v_{r}$ of $\tau$. We choose a positive integer $m$ such that $mE_{i}$ is Cartier at the point $\mathrm{red}_{\mathscr{X}}(x)$ for every $i$, and we choose a local equation $f_{i}=0$ for $mE_{i}$ at $\mathrm{red}_{\mathscr{X}}(x)$. Then $\rho_{\mathscr{X}}(x)$ is the point of the simplex $\tau$ with barycentric coordinates

with respect to the vertices $(v_{1},\ldots,v_{r})$. Under the embedding of $\Delta(\mathscr{X}^{\mathrm{snc}})$ into $X^{\mathrm{an}}$, the point $\rho_{\mathscr{X}}(x)$ corresponds to the monomial point represented by $(\mathscr{X},(E_{1},\ldots,E_{r}),\xi)$ and the tuple

in the terminology of [MN15, 2.4.5]. It is obvious that this definition does not depend on the choices of $m$ and the local equations $f_{i}$. It is also straightforward to check that $\rho_{\mathscr{X}}$ is continuous, and that it is a retraction onto $\Delta(\mathscr{X}^{\mathrm{snc}})=\mathrm{Sk}(X)$.

(2.6) Beware that, even though the subspace $\mathrm{Sk}(X)$ of $X^{\mathrm{an}}$ only depends on $X$, the $\Delta$-structure on $\Delta(\mathscr{X}^{\mathrm{snc}})=\mathrm{Sk}(X)$ and the retraction $\rho_{\mathscr{X}}\colon X^{\mathrm{an}}\to\mathrm{Sk}(X)$ depend on the choice of the (good) minimal dlt-model $\mathscr{X}$; we will illustrate this in Example 2.7 below. However, the essential skeleton $\mathrm{Sk}(X)$ does carry a canonical piecewise integral affine structure, which is induced by the embedding into the $K$-analytic space $X^{\mathrm{an}}$: see [MN15, §3.2]. If $\mathscr{X}$ is a minimal dlt-model for $X$, then this piecewise integral affine structure coincides with the one induced by the $\Delta$-complex structure on $\Delta(\mathscr{X}^{\mathrm{snc}})=\mathrm{Sk}(X)$, provided that the barycentric coordinates on the faces of $\Delta(\mathscr{X}^{\mathrm{snc}})$ are weighted by the multiplicities of the prime components in $\mathscr{X}_{k}$ as in [MN15, 3.2.1].

(2.9) Let $X$ be a Calabi-Yau variety over $K$ of dimension $n$. We say that $X$ is maximally degenerate if $X$ has a semistable snc-model over $R$ and the essential skeleton $\mathrm{Sk}(X)$ has dimension $n$. This is the class of Calabi-Yau varieties where we expect the SYZ mirror symmetry picture to appear. If $X$ is projective, then the condition $\dim(\mathrm{Sk}(X))=n$ is equivalent to the property that, for any topological generator $\sigma$ of $\mathrm{Gal}(K^{a}/K)\cong\widehat{\mu}(k)$ and any prime number $\ell$, the action of $\sigma$ on the étale cohomology space

has a Jordan block of rank $n+1$, by [NX16a, 4.2.4(4)]. If $X$ is maximally degenerate and projective, then $\mathrm{Sk}(X)$ is a closed pseudomanifold (see [NX16a, 4.2.4(3)] – in the statement of that result, one should add the assumption that $X$ has a semistable snc-model, like in [NX16a, 4.1.7]). If we assume, moreover, that $X$ is geometrically simply connected and $h^{i,0}(X)=0$ for $0<i<n$, then it is expected that $\mathrm{Sk}(X)$ is homeomorphic to $S^{n}$. This has been proven in [KX16] when $n\leq 3$, and also when $n=4$ and $X$ has a minimal dlt-model that is also a semistable snc-model. One can prove in any dimension that $\mathrm{Sk}(X)$ has the $\mathbb{Q}$-rational homology of $S^{n}$, and that its fundamental group has trivial profinite completion; see [NX16a, 4.2.4(4)] and [HN17, 6.1.3(4)].

(3.1) Let $X$ be a maximally degenerate Calabi-Yau variety and let $\mathscr{X}$ be a good minimal dlt-model of $X$ with reduced special fiber. Then we will see in Corollary 4.6 that $\mathscr{X}$ satisfies assumption (2), so that it gives rise to a non-archimedean SYZ fibration $\rho_{\mathscr{X}}\colon X^{\mathrm{an}}\to\mathrm{Sk}(X)$ in the sense of Definition 2.5. The principal aim of this article is to study the fibers of $\rho_{\mathscr{X}}$. In the classical SYZ conjecture, the fibers of the SYZ fibration are expected to be special Lagrangian tori away from a codimension two subset of the base. We will now present the corresponding structure in non-archimedean geometry, which was introduced in [KS06, §4.1].

(3.2) Let $n$ be a positive integer, and let $T$ be a split algebraic $K$-torus of dimension $n$ with character module $M$ and cocharacter module $N=M^{\vee}$. We define the tropicalization map of $T$ by

This map is continuous, and its fibers are (not necessarily strictly) $K$-affinoid tori. The tropicalization map $\rho_{T}$ has a canonical continuous section $s\colon N_{\mathbb{R}}\to T^{\mathrm{an}}$ that maps each $n\in N_{\mathbb{R}}$ to the Gauss point of the affinoid torus $\rho^{-1}_{T}(n)$. The image of $s$ is called the canonical skeleton of $T$, and denoted by $\Delta(T)$. The map $s$ induces a homeomorphism $N_{\mathbb{R}}\to\Delta(T)$, which we will use to tacitly identify $\Delta(T)$ with $N_{\mathbb{R}}$.

(3.3) Let $Y$ be a $K$-analytic space, let $B$ be a topological space and let $f\colon Y\to B$ be a continuous map. Then we say that $f$ is an $n$-dimensional affinoid torus fibration if we can cover $B$ by open subsets $U$ such that there exist an open subset $V$ of $N_{\mathbb{R}}\cong\mathbb{R}^{n}$ and a commutative diagram

where the upper horizontal map is an isomorphism of $K$-analytic spaces and the lower horizontal map is a homeomorphism.

(3.4) If $f\colon Y\to B$ is an $n$-dimensional affinoid torus fibration, then $f$ induces an integral affine structure on the base $B$ [KS06, §4.1]. For every open $U$ in $B$ as in the definition, and every invertible analytic function $h$ on $f^{-1}(U)$, the absolute value of $h$ is constant on the fibers of $f$ by the maximum modulus principle. Thus $h$ induces a continuous function $|h|\colon U\to\mathbb{R}_{>0}$. The integral affine functions on $U$ are, by definition, the functions of the form $-\ln|h|$. If $U$ is connected, then it is proven in Theorem 1 of [KS06, §4.1] that under the homeomorphism $U\to V$, the ring of integral affine functions on $U$ is identified with the ring of polynomial functions of degree one with $\mathbb{Z}$-coefficients on $V\subset N_{\mathbb{R}}$, so that this construction indeed defines an integral affine structure on $B$ (to be precise, in [KS06] the authors consider affine functions with constant term in $\mathbb{R}$, rather than $\mathbb{Z}$, but since $K$ is discretely valued in our case, we get a slightly stronger result).

(3.6) As a first application, let us discuss the case of abelian varieties. Let $A$ be an abelian $K$-variety of dimension $n$, and denote by $\mathscr{A}$ its Néron model. Then Berkovich has constructed in [Be90, §6.5] a canonical skeleton $\Delta(A)$ in $A^{\mathrm{an}}$, together with a continuous retraction $\rho_{A}\colon A^{\mathrm{an}}\to\Delta(A)$, via the theory of non-archimedean uniformization. The dimension of $\Delta(A)$ is equal to the toric rank of $\mathscr{A}^{o}_{k}$ (the dimension of the maximal subtorus). Let us make this construction more precise in the maximally degenerate case. Assume that $A$ has purely toric reduction, that is, $\mathscr{A}^{o}_{k}$ is a torus. Let $e$ be the identity point on $A$. Then the universal pointed covering space of $(A,e)$ (with respect to the Berkovich topology) is isomorphic to the analytification of a split $n$-dimensional $K$-torus $T$. The kernel $L$ of the morphism $\pi\colon T^{an}\to A^{\mathrm{an}}$ is a lattice in $T(K)$ (called the period lattice), and the image $\rho_{T}(L)$ of $L$ in $N_{\mathbb{R}}$ is a lattice of rank $n$. By definition, the canonical skeleton $\Delta(A)$ is the image of $\Delta(T)$ under the map $\pi$. Moreover, we have a Cartesian diagram of topological spaces

such that $\rho_{A}$ sends $\Delta(A)$ homeomorphically onto $N_{\mathbb{R}}/\rho_{T}(L)$. In particular, $\Delta(A)$ is a real torus of dimension $n$, $\rho_{A}$ is an $n$-dimensional torus fibration, and the induced integral affine structure on $\Delta(A)$ coincides with the quotient structure on $N_{\mathbb{R}}/\rho_{T}(L)$.

(3.7) If $A$ has purely toric reduction, then we can interpret $\rho_{A}$ as a non-archimedean SYZ fibration by means of the theory of Mumford models [Mu72] and the refinements of Mumford’s construction given in [Kü98]. We say that a model $\mathscr{P}$ of $A$ is a Künnemann-Mumford model if it is a regular model that arises through the construction in the proof of [Kü98, 3.5].

(4.1) The aim of this section is to show that good minimal dlt-models with reduced special fibers of Calabi-Yau varieties over $K$ are snc along their one-dimensional strata (in fact, we will prove a more general result – see Theorem 4.5 and Corollary 4.6). A technical complication is that the full machinery of the MMP has only been written down for objects of finite type over a field. To circumvent this problem, we will first prove an approximation result (Proposition 4.3) that will allow us to reduce to that case.

(5.1) Let $\mathscr{X}$ be a regular flat $R$-scheme such that $\mathscr{X}_{k}$ is a strict normal crossings divisor. We write $\mathscr{X}_{k}=\sum_{i\in I}N_{i}E_{i}$, where $E_{i},\,i\in I$ are the prime divisors in $\mathscr{X}_{k}$ and the numbers $N_{i}$ are their multiplicities. By the definition of a strict normal crossings divisor, every stratum of $\mathscr{X}_{k}$ is a regular $k$-scheme. Let $C$ be a stratum of $\mathscr{X}_{k}$. We say that $\mathscr{X}$ is toric along $C$ if there exist a regular toric $R$-scheme $\mathscr{Y}$ and a stratum $D$ of $\mathscr{Y}_{k}$ such that $\mathscr{Y}_{k}$ is a strict normal crossings divisor and the formal $R$-schemes $\widehat{\mathscr{X}_{/C}}$ and $\widehat{\mathscr{Y}_{/D}}$ are isomorphic.

(5.2) Now let $C$ be a one-dimensional stratum of $\mathscr{X}_{k}$ that is proper over $k$. Let $E_{j},\,j\in J$ be the irreducible components of $\mathscr{X}_{k}$ that contain $C$. For every $j\in J$, we set

We write $C^{o}=C\setminus(\cup_{i\notin J}E_{i})$. We say that $\mathscr{X}$ is log Calabi-Yau along $C$ if $C\cong\mathbb{P}^{1}_{k}$ and $C\setminus C^{o}$ consists of precisely two points, which we denote by $c_{0}$ and $c_{\infty}$. Denote by $0$ and $\infty$ be the unique elements of $I\setminus J$ such that $\{c_{0}\}=C\cap E_{0}$ and $\{c_{\infty}\}=C\cap E_{\infty}$ (note that $0$ and $\infty$ are note necessarily distinct). Then the fact that $\sum_{i\in I}N_{i}E_{i}$ is a principal divisor on $\mathscr{X}$ implies that