(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

$$Y\to\mathbb{A}^{1}_{k}=\mathrm{Spec}\,k[t]$$

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

$$\mathbb{R}^{n}\times\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}\colon(u,v)\mapsto\iota\cdot v.$$

(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.