(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

$$\rho_{T}\colon T^{\mathrm{an}}\to N_{\mathbb{R}}\colon x\mapsto(M\to\mathbb{R}\colon m\mapsto-\ln|m(x)|).$$

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].